# AI News, Marginally Interesting by Mikio L. Braun

- On Thursday, June 28, 2018
- By Read More

## Marginally Interesting by Mikio L. Braun

Some people even questioned my choice to study computer science at all, because apparently programming computers was supposed to become so easy no specialists are required anymore. Actually,

I went through the first two years of training, made sure I picked up enough math for whatever would lie ahead, and finally arrived in my first AI lectured held by Joachim Buhmann, back then professor at the University of Bonn (where Sebastian Thrun was just about to leave for the US). I

As I now know, people stopped working on symbolic AI and instead stuck to more statistical approaches to learning, where learning essentially was reduced to the problem of picking the right function based on a finite amount of observations. The

computer vision lecture was even less about learning and relied more on explicit physical modelling to derive the right estimators, for example, to reconstruct motion from a video.

This was back in the days when NIPS had full tracks devoted to studying the physiology and working mechanisms of neurons, and there were people who believed there is something fundamentally different happening in our brains, maybe on a quantum level, that gives rise to the human mind, and that this difference is a blocker for having truly intelligent machines. While

graduated in October 2000 with a pretty farfetched master thesis trying to make a connection between learning and hard optimization problems, then started on my PhD thesis and stuck around in this area of research till 2015. While

One odd thing about machine learning as a whole is that there exist only a handful of fundamentally different problems like classification, regression, clustering, and so on, but a whole zoo of approaches.

It is not like in physics (I assume) or mathematics where some generally agreed upon unsolved hard problems exist whose solution would advance the state of the art.

Large scale learning always existed, for example for genomic data in bioinformatics, but one usually tried to solve problems by finding more efficient algorithms and approximations, not by parallelizing brute force. Companies

Many come from physics, mathematics or other sciences, where their rigorous mathematical training was an excellent fit for the algorithm and modeling heavy approach central to machine learning. Hadoop

Python is definitely one if the languages of choice for doing data analysis, and technologies like Spark try to tap into that by providing Python bindings, whether it makes sense from a performance point of view or not. The

Some people like Yann LeCun have always stuck to this approach, but maybe ten years ago, there where a few works which showed how to use layerwise pretraining and other tricks to train “deep” networks, that is larger networks than one previously thought possible. The

first there were a few papers sparking the interest in neural networks, then people started using them again, and successively achieved excellent results for a number of application areas.

Google published that ingenious deep dream paper where they let a learned network generate some data, and we humans with our immediate readiness to read structure and attribute intelligence picked up quickly on this. I

academics I have talked to are unhappy about the dominance of deep learning right now, because it is an approach which works well, maybe even too well, but doesn’t bring us much closer to really understand how the human mind works. I

Some people even questioned my choice to study computer science at all, because apparently programming computers was supposed to become so easy no specialists are required anymore.

I went through the first two years of training, made sure I picked up enough math for whatever would lie ahead, and finally arrived in my first AI lectured held by Joachim Buhmann, back then professor at the University of Bonn (where Sebastian Thrun was just about to leave for the US).

As I now know, people stopped working on symbolic AI and instead stuck to more statistical approaches to learning, where learning essentially was reduced to the problem of picking the right function based on a finite amount of observations.

The computer vision lecture was even less about learning and relied more on explicit physical modelling to derive the right estimators, for example, to reconstruct motion from a video.

This was back in the days when NIPS had full tracks devoted to studying the physiology and working mechanisms of neurons, and there were people who believed there is something fundamentally different happening in our brains, maybe on a quantum level, that gives rise to the human mind, and that this difference is a blocker for having truly intelligent machines.

graduated in October 2000 with a pretty farfetched master thesis trying to make a connection between learning and hard optimization problems, then started on my PhD thesis and stuck around in this area of research till 2015.

One odd thing about machine learning as a whole is that there exist only a handful of fundamentally different problems like classification, regression, clustering, and so on, but a whole zoo of approaches.

It is not like in physics (I assume) or mathematics where some generally agreed upon unsolved hard problems exist whose solution would advance the state of the art.

Large scale learning always existed, for example for genomic data in bioinformatics, but one usually tried to solve problems by finding more efficient algorithms and approximations, not by parallelizing brute force.

Many come from physics, mathematics or other sciences, where their rigorous mathematical training was an excellent fit for the algorithm and modeling heavy approach central to machine learning.

Still, there was no way around it, so they rebranded everything they did as Big Data, or began to stress, that Big Data only provides the infrastructure for large scale computations, but you need someone who “knows what he is doing” to make sense of the data.

Python is definitely one if the languages of choice for doing data analysis, and technologies like Spark try to tap into that by providing Python bindings, whether it makes sense from a performance point of view or not.

Some people like Yann LeCun have always stuck to this approach, but maybe ten years ago, there where a few works which showed how to use layerwise pretraining and other tricks to train “deep” networks, that is larger networks than one previously thought possible.

Then Google published that ingenious deep dream paper where they let a learned network generate some data, and we humans with our immediate readiness to read structure and attribute intelligence picked up quickly on this.

Many academics I have talked to are unhappy about the dominance of deep learning right now, because it is an approach which works well, maybe even too well, but doesn’t bring us much closer to really understand how the human mind works.

- On Thursday, June 28, 2018
- By Read More

## Artificial neural network

Artificial neural networks (ANNs) or connectionist systems are computing systems vaguely inspired by the biological neural networks that constitute animal brains.[1] Such systems 'learn' to perform tasks by considering examples, generally without being programmed with any task-specific rules.

For example, in image recognition, they might learn to identify images that contain cats by analyzing example images that have been manually labeled as 'cat' or 'no cat' and using the results to identify cats in other images.

An ANN is based on a collection of connected units or nodes called artificial neurons which loosely model the neurons in a biological brain.

In common ANN implementations, the signal at a connection between artificial neurons is a real number, and the output of each artificial neuron is computed by some non-linear function of the sum of its inputs.

Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.

ANNs have been used on a variety of tasks, including computer vision, speech recognition, machine translation, social network filtering, playing board and video games and medical diagnosis.

With mathematical notation, Rosenblatt described circuitry not in the basic perceptron, such as the exclusive-or circuit that could not be processed by neural networks at the time.[8] In 1959, a biological model proposed by Nobel laureates Hubel and Wiesel was based on their discovery of two types of cells in the primary visual cortex: simple cells and complex cells.[9] The first functional networks with many layers were published by Ivakhnenko and Lapa in 1965, becoming the Group Method of Data Handling.[10][11][12] Neural network research stagnated after machine learning research by Minsky and Papert (1969),[13] who discovered two key issues with the computational machines that processed neural networks.

Much of artificial intelligence had focused on high-level (symbolic) models that are processed by using algorithms, characterized for example by expert systems with knowledge embodied in if-then rules, until in the late 1980s research expanded to low-level (sub-symbolic) machine learning, characterized by knowledge embodied in the parameters of a cognitive model.[citation needed] A

key trigger for renewed interest in neural networks and learning was Werbos's (1975) backpropagation algorithm that effectively solved the exclusive-or problem and more generally accelerated the training of multi-layer networks.

Backpropagation distributed the error term back up through the layers, by modifying the weights at each node.[8] In the mid-1980s, parallel distributed processing became popular under the name connectionism.

Rumelhart and McClelland (1986) described the use of connectionism to simulate neural processes.[14] Support vector machines and other, much simpler methods such as linear classifiers gradually overtook neural networks in machine learning popularity.

However, using neural networks transformed some domains, such as the prediction of protein structures.[15][16] In 1992, max-pooling was introduced to help with least shift invariance and tolerance to deformation to aid in 3D object recognition.[17][18][19] In 2010, Backpropagation training through max-pooling was accelerated by GPUs and shown to perform better than other pooling variants.[20] The vanishing gradient problem affects many-layered feedforward networks that used backpropagation and also recurrent neural networks (RNNs).[21][22] As errors propagate from layer to layer, they shrink exponentially with the number of layers, impeding the tuning of neuron weights that is based on those errors, particularly affecting deep networks.

To overcome this problem, Schmidhuber adopted a multi-level hierarchy of networks (1992) pre-trained one level at a time by unsupervised learning and fine-tuned by backpropagation.[23] Behnke (2003) relied only on the sign of the gradient (Rprop)[24] on problems such as image reconstruction and face localization.

(2006) proposed learning a high-level representation using successive layers of binary or real-valued latent variables with a restricted Boltzmann machine[25] to model each layer.

Once sufficiently many layers have been learned, the deep architecture may be used as a generative model by reproducing the data when sampling down the model (an 'ancestral pass') from the top level feature activations.[26][27] In 2012, Ng and Dean created a network that learned to recognize higher-level concepts, such as cats, only from watching unlabeled images taken from YouTube videos.[28] Earlier challenges in training deep neural networks were successfully addressed with methods such as unsupervised pre-training, while available computing power increased through the use of GPUs and distributed computing.

Nanodevices[29] for very large scale principal components analyses and convolution may create a new class of neural computing because they are fundamentally analog rather than digital (even though the first implementations may use digital devices).[30] Ciresan and colleagues (2010)[31] in Schmidhuber's group showed that despite the vanishing gradient problem, GPUs makes back-propagation feasible for many-layered feedforward neural networks.

Between 2009 and 2012, recurrent neural networks and deep feedforward neural networks developed in Schmidhuber's research group won eight international competitions in pattern recognition and machine learning.[32][33] For example, the bi-directional and multi-dimensional long short-term memory (LSTM)[34][35][36][37] of Graves et al.

won three competitions in connected handwriting recognition at the 2009 International Conference on Document Analysis and Recognition (ICDAR), without any prior knowledge about the three languages to be learned.[36][35] Ciresan and colleagues won pattern recognition contests, including the IJCNN 2011 Traffic Sign Recognition Competition,[38] the ISBI 2012 Segmentation of Neuronal Structures in Electron Microscopy Stacks challenge[39] and others.

Their neural networks were the first pattern recognizers to achieve human-competitive or even superhuman performance[40] on benchmarks such as traffic sign recognition (IJCNN 2012), or the MNIST handwritten digits problem.

Researchers demonstrated (2010) that deep neural networks interfaced to a hidden Markov model with context-dependent states that define the neural network output layer can drastically reduce errors in large-vocabulary speech recognition tasks such as voice search.

GPU-based implementations[41] of this approach won many pattern recognition contests, including the IJCNN 2011 Traffic Sign Recognition Competition,[38] the ISBI 2012 Segmentation of neuronal structures in EM stacks challenge,[39] the ImageNet Competition[42] and others.

Deep, highly nonlinear neural architectures similar to the neocognitron[43] and the 'standard architecture of vision',[44] inspired by simple and complex cells, were pre-trained by unsupervised methods by Hinton.[45][26] A team from his lab won a 2012 contest sponsored by Merck to design software to help find molecules that might identify new drugs.[46] As of 2011, the state of the art in deep learning feedforward networks alternated convolutional layers and max-pooling layers,[41][47] topped by several fully or sparsely connected layers followed by a final classification layer.

Such supervised deep learning methods were the first to achieve human-competitive performance on certain tasks.[40] ANNs were able to guarantee shift invariance to deal with small and large natural objects in large cluttered scenes, only when invariance extended beyond shift, to all ANN-learned concepts, such as location, type (object class label), scale, lighting and others.

This was realized in Developmental Networks (DNs)[48] whose embodiments are Where-What Networks, WWN-1 (2008)[49] through WWN-7 (2013).[50] An artificial neural network is a network of simple elements called artificial neurons, which receive input, change their internal state (activation) according to that input, and produce output depending on the input and activation.

of predecessor neurons and typically has the form[51] The learning rule is a rule or an algorithm which modifies the parameters of the neural network, in order for a given input to the network to produce a favored output.

This learning process typically amounts to modifying the weights and thresholds of the variables within the network.[51] Neural network models can be viewed as simple mathematical models defining a function

A common use of the phrase 'ANN model' is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons or their connectivity).

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(commonly referred to as the activation function[52]) is some predefined function, such as the hyperbolic tangent or sigmoid function or softmax function or rectifier function.

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For applications where the solution is data dependent, the cost must necessarily be a function of the observations, otherwise the model would not relate to the data.

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While it is possible to define an ad hoc cost function, frequently a particular cost (function) is used, either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (e.g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost).

The basics of continuous backpropagation[10][53][54][55] were derived in the context of control theory by Kelley[56] in 1960 and by Bryson in 1961,[57] using principles of dynamic programming.

In 1962, Dreyfus published a simpler derivation based only on the chain rule.[58] Bryson and Ho described it as a multi-stage dynamic system optimization method in 1969.[59][60] In 1970, Linnainmaa finally published the general method for automatic differentiation (AD) of discrete connected networks of nested differentiable functions.[61][62] This corresponds to the modern version of backpropagation which is efficient even when the networks are sparse.[10][53][63][64] In 1973, Dreyfus used backpropagation to adapt parameters of controllers in proportion to error gradients.[65] In 1974, Werbos mentioned the possibility of applying this principle to ANNs,[66] and in 1982, he applied Linnainmaa's AD method to neural networks in the way that is widely used today.[53][67] In 1986, Rumelhart, Hinton and Williams noted that this method can generate useful internal representations of incoming data in hidden layers of neural networks.[68] In 1993, Wan was the first[10] to win an international pattern recognition contest through backpropagation.[69] The weight updates of backpropagation can be done via stochastic gradient descent using the following equation: where,

The choice of the cost function depends on factors such as the learning type (supervised, unsupervised, reinforcement, etc.) and the activation function.

For example, when performing supervised learning on a multiclass classification problem, common choices for the activation function and cost function are the softmax function and cross entropy function, respectively.

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The network is trained to minimize L2 error for predicting the mask ranging over the entire training set containing bounding boxes represented as masks.

commonly used cost is the mean-squared error, which tries to minimize the average squared error between the network's output,

Minimizing this cost using gradient descent for the class of neural networks called multilayer perceptrons (MLP), produces the backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation).

The cost function is dependent on the task (the model domain) and any a priori assumptions (the implicit properties of the model, its parameters and the observed variables).

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whereas in statistical modeling, it could be related to the posterior probability of the model given the data (note that in both of those examples those quantities would be maximized rather than minimized).

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The aim is to discover a policy for selecting actions that minimizes some measure of a long-term cost, e.g., the expected cumulative cost.

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ANNs are frequently used in reinforcement learning as part of the overall algorithm.[77][78] Dynamic programming was coupled with ANNs (giving neurodynamic programming) by Bertsekas and Tsitsiklis[79] and applied to multi-dimensional nonlinear problems such as those involved in vehicle routing,[80] natural resources management[81][82] or medicine[83] because of the ability of ANNs to mitigate losses of accuracy even when reducing the discretization grid density for numerically approximating the solution of the original control problems.

In 2004 a recursive least squares algorithm was introduced to train CMAC neural network online.[84] This algorithm can converge in one step and update all weights in one step with any new input data.

Based on QR decomposition, this recursive learning algorithm was simplified to be O(N).[85] Training a neural network model essentially means selecting one model from the set of allowed models (or, in a Bayesian framework, determining a distribution over the set of allowed models) that minimizes the cost.

Backpropagation training algorithms fall into three categories: Evolutionary methods,[87] gene expression programming,[88] simulated annealing,[89] expectation-maximization, non-parametric methods and particle swarm optimization[90] are other methods for training neural networks.

It used a deep feedforward multilayer perceptron with eight layers.[92] It is a supervised learning network that grows layer by layer, where each layer is trained by regression analysis.

convolutional neural network (CNN) is a class of deep, feed-forward networks, composed of one or more convolutional layers with fully connected layers (matching those in typical ANNs) on top.

In particular, max-pooling[18] is often structured via Fukushima's convolutional architecture.[94] This architecture allows CNNs to take advantage of the 2D structure of input data.

CNNs are easier to train than other regular, deep, feed-forward neural networks and have many fewer parameters to estimate.[97] Examples of applications in computer vision include DeepDream[98] and robot navigation.[99] Long short-term memory (LSTM) networks are RNNs that avoid the vanishing gradient problem.[100] LSTM is normally augmented by recurrent gates called forget gates.[101] LSTM networks prevent backpropagated errors from vanishing or exploding.[21] Instead errors can flow backwards through unlimited numbers of virtual layers in space-unfolded LSTM.

That is, LSTM can learn 'very deep learning' tasks[10] that require memories of events that happened thousands or even millions of discrete time steps ago.

Stacks of LSTM RNNs[103] trained by Connectionist Temporal Classification (CTC)[104] can find an RNN weight matrix that maximizes the probability of the label sequences in a training set, given the corresponding input sequences.

In 2003, LSTM started to become competitive with traditional speech recognizers.[105] In 2007, the combination with CTC achieved first good results on speech data.[106] In 2009, a CTC-trained LSTM was the first RNN to win pattern recognition contests, when it won several competitions in connected handwriting recognition.[10][36] In 2014, Baidu used CTC-trained RNNs to break the Switchboard Hub5'00 speech recognition benchmark, without traditional speech processing methods.[107] LSTM also improved large-vocabulary speech recognition,[108][109] text-to-speech synthesis,[110] for Google Android,[53][111] and photo-real talking heads.[112] In 2015, Google's speech recognition experienced a 49% improvement through CTC-trained LSTM.[113] LSTM became popular in Natural Language Processing.

Unlike previous models based on HMMs and similar concepts, LSTM can learn to recognise context-sensitive languages.[114] LSTM improved machine translation,[115][116] language modeling[117] and multilingual language processing.[118] LSTM combined with CNNs improved automatic image captioning.[119] Deep Reservoir Computing and Deep Echo State Networks (deepESNs)[120][121] provide a framework for efficiently trained models for hierarchical processing of temporal data, while enabling the investigation of the inherent role of RNN layered composition.[clarification needed] A

This allows for both improved modeling and faster convergence of the fine-tuning phase.[123] Large memory storage and retrieval neural networks (LAMSTAR)[124][125] are fast deep learning neural networks of many layers that can use many filters simultaneously.

Its speed is provided by Hebbian link-weights[126] that integrate the various and usually different filters (preprocessing functions) into its many layers and to dynamically rank the significance of the various layers and functions relative to a given learning task.

This grossly imitates biological learning which integrates various preprocessors (cochlea, retina, etc.) and cortexes (auditory, visual, etc.) and their various regions.

Its deep learning capability is further enhanced by using inhibition, correlation and its ability to cope with incomplete data, or 'lost' neurons or layers even amidst a task.

The link-weights allow dynamic determination of innovation and redundancy, and facilitate the ranking of layers, of filters or of individual neurons relative to a task.

LAMSTAR has been applied to many domains, including medical[127][128][129] and financial predictions,[130] adaptive filtering of noisy speech in unknown noise,[131] still-image recognition,[132] video image recognition,[133] software security[134] and adaptive control of non-linear systems.[135] LAMSTAR had a much faster learning speed and somewhat lower error rate than a CNN based on ReLU-function filters and max pooling, in 20 comparative studies.[136] These applications demonstrate delving into aspects of the data that are hidden from shallow learning networks and the human senses, such as in the cases of predicting onset of sleep apnea events,[128] of an electrocardiogram of a fetus as recorded from skin-surface electrodes placed on the mother's abdomen early in pregnancy,[129] of financial prediction[124] or in blind filtering of noisy speech.[131] LAMSTAR was proposed in 1996 (A U.S. Patent 5,920,852 A) and was further developed Graupe and Kordylewski from 1997–2002.[137][138][139] A modified version, known as LAMSTAR 2, was developed by Schneider and Graupe in 2008.[140][141] The auto encoder idea is motivated by the concept of a good representation.

The whole process of auto encoding is to compare this reconstructed input to the original and try to minimize the error to make the reconstructed value as close as possible to the original.

This idea was introduced in 2010 by Vincent et al.[142] with a specific approach to good representation, a good representation is one that can be obtained robustly from a corrupted input and that will be useful for recovering the corresponding clean input.

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might be either the cross-entropy loss with an affine-sigmoid decoder, or the squared error loss with an affine decoder.[142] In order to make a deep architecture, auto encoders stack.[143] Once the encoding function

of the first denoising auto encoder is learned and used to uncorrupt the input (corrupted input), the second level can be trained.[142] Once the stacked auto encoder is trained, its output can be used as the input to a supervised learning algorithm such as support vector machine classifier or a multi-class logistic regression.[142] A

deep stacking network (DSN)[144] (deep convex network) is based on a hierarchy of blocks of simplified neural network modules.

It was introduced in 2011 by Deng and Dong.[145] It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization.[146] Each DSN block is a simple module that is easy to train by itself in a supervised fashion without backpropagation for the entire blocks.[147] Each block consists of a simplified multi-layer perceptron (MLP) with a single hidden layer.

Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block.

It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer.[148] TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor.

While parallelization and scalability are not considered seriously in conventional DNNs,[149][150][151] all learning for DSNs and TDSNs is done in batch mode, to allow parallelization.[145][144] Parallelization allows scaling the design to larger (deeper) architectures and data sets.

The need for deep learning with real-valued inputs, as in Gaussian restricted Boltzmann machines, led to the spike-and-slab RBM (ssRBM), which models continuous-valued inputs with strictly binary latent variables.[152] Similar to basic RBMs and its variants, a spike-and-slab RBM is a bipartite graph, while like GRBMs, the visible units (input) are real-valued.

A spike is a discrete probability mass at zero, while a slab is a density over continuous domain;[153] their mixture forms a prior.[154] An extension of ssRBM called µ-ssRBM provides extra modeling capacity using additional terms in the energy function.

Features can be learned using deep architectures such as DBNs,[26] DBMs,[155] deep auto encoders,[156] convolutional variants,[157][158] ssRBMs,[153] deep coding networks,[159] DBNs with sparse feature learning,[160] RNNs,[161] conditional DBNs,[162] de-noising auto encoders.[163] This provides a better representation, allowing faster learning and more accurate classification with high-dimensional data.

However, these architectures are poor at learning novel classes with few examples, because all network units are involved in representing the input (a distributed representation) and must be adjusted together (high degree of freedom).

It is a full generative model, generalized from abstract concepts flowing through the layers of the model, which is able to synthesize new examples in novel classes that look 'reasonably' natural.

All the levels are learned jointly by maximizing a joint log-probability score.[169] In a DBM with three hidden layers, the probability of a visible input ν is: where

deep predictive coding network (DPCN) is a predictive coding scheme that uses top-down information to empirically adjust the priors needed for a bottom-up inference procedure by means of a deep, locally connected, generative model.

DPCNs predict the representation of the layer, by using a top-down approach using the information in upper layer and temporal dependencies from previous states.[170] DPCNs can be extended to form a convolutional network.[170] Integrating external memory with ANNs dates to early research in distributed representations[171] and Kohonen's self-organizing maps.

For example, in sparse distributed memory or hierarchical temporal memory, the patterns encoded by neural networks are used as addresses for content-addressable memory, with 'neurons' essentially serving as address encoders and decoders.

Preliminary results demonstrate that neural Turing machines can infer simple algorithms such as copying, sorting and associative recall from input and output examples.

They out-performed Neural turing machines, long short-term memory systems and memory networks on sequence-processing tasks.[180][181][182][183][184] Approaches that represent previous experiences directly and use a similar experience to form a local model are often called nearest neighbour or k-nearest neighbors methods.[185] Deep learning is useful in semantic hashing[186] where a deep graphical model the word-count vectors[187] obtained from a large set of documents.[clarification needed] Documents are mapped to memory addresses in such a way that semantically similar documents are located at nearby addresses.

Unlike sparse distributed memory that operates on 1000-bit addresses, semantic hashing works on 32 or 64-bit addresses found in a conventional computer architecture.

These models have been applied in the context of question answering (QA) where the long-term memory effectively acts as a (dynamic) knowledge base and the output is a textual response.[190] Deep neural networks can be potentially improved by deepening and parameter reduction, while maintaining trainability.

While training extremely deep (e.g., 1 million layers) neural networks might not be practical, CPU-like architectures such as pointer networks[191] and neural random-access machines[192] overcome this limitation by using external random-access memory and other components that typically belong to a computer architecture such as registers, ALU and pointers.

The key characteristic of these models is that their depth, the size of their short-term memory, and the number of parameters can be altered independently – unlike models like LSTM, whose number of parameters grows quadratically with memory size.

In that work, an LSTM RNN or CNN was used as an encoder to summarize a source sentence, and the summary was decoded using a conditional RNN language model to produce the translation.[196] These systems share building blocks: gated RNNs and CNNs and trained attention mechanisms.

For the sake of dimensionality reduction of the updated representation in each layer, a supervised strategy selects the best informative features among features extracted by KPCA.

more straightforward way to use kernel machines for deep learning was developed for spoken language understanding.[199] The main idea is to use a kernel machine to approximate a shallow neural net with an infinite number of hidden units, then use stacking to splice the output of the kernel machine and the raw input in building the next, higher level of the kernel machine.

The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network.[200] Using ANNs requires an understanding of their characteristics.

ANN capabilities fall within the following broad categories:[citation needed] Because of their ability to reproduce and model nonlinear processes, ANNs have found many applications in a wide range of disciplines.

Application areas include system identification and control (vehicle control, trajectory prediction,[201] process control, natural resource management), quantum chemistry,[202] game-playing and decision making (backgammon, chess, poker), pattern recognition (radar systems, face identification, signal classification,[203] object recognition and more), sequence recognition (gesture, speech, handwritten and printed text recognition), medical diagnosis, finance[204] (e.g.

automated trading systems), data mining, visualization, machine translation, social network filtering[205] and e-mail spam filtering.

ANNs have been used to diagnose cancers, including lung cancer,[206] prostate cancer, colorectal cancer[207] and to distinguish highly invasive cancer cell lines from less invasive lines using only cell shape information.[208][209] ANNs have been used to accelerate reliability analysis of infrastructures subject to natural disasters.[210][211] ANNs have also been used for building black-box models in geoscience: hydrology,[212][213] ocean modelling and coastal engineering,[214][215] and geomorphology,[216] are just few examples of this kind.

They range from models of the short-term behavior of individual neurons,[217] models of how the dynamics of neural circuitry arise from interactions between individual neurons and finally to models of how behavior can arise from abstract neural modules that represent complete subsystems.

These include models of the long-term, and short-term plasticity, of neural systems and their relations to learning and memory from the individual neuron to the system level.

specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a universal Turing machine,[218] using a finite number of neurons and standard linear connections.

Further, the use of irrational values for weights results in a machine with super-Turing power.[219] Models' 'capacity' property roughly corresponds to their ability to model any given function.

It is related to the amount of information that can be stored in the network and to the notion of complexity.[citation needed] Models may not consistently converge on a single solution, firstly because many local minima may exist, depending on the cost function and the model.

However, for CMAC neural network, a recursive least squares algorithm was introduced to train it, and this algorithm can be guaranteed to converge in one step.[84] Applications whose goal is to create a system that generalizes well to unseen examples, face the possibility of over-training.

but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting.

Supervised neural networks that use a mean squared error (MSE) cost function can use formal statistical methods to determine the confidence of the trained model.

A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in a component-based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities.

common criticism of neural networks, particularly in robotics, is that they require too much training for real-world operation.[citation needed] Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example and by grouping examples in so-called mini-batches.

For example, by introducing a recursive least squares algorithm for CMAC neural network, the training process only takes one step to converge.[84] No neural network has solved computationally difficult problems such as the n-Queens problem, the travelling salesman problem, or the problem of factoring large integers.

Back propagation is a critical part of most artificial neural networks, although no such mechanism exists in biological neural networks.[220] How information is coded by real neurons is not known.

Sensor neurons fire action potentials more frequently with sensor activation and muscle cells pull more strongly when their associated motor neurons receive action potentials more frequently.[221] Other than the case of relaying information from a sensor neuron to a motor neuron, almost nothing of the principles of how information is handled by biological neural networks is known.

Alexander Dewdney commented that, as a result, artificial neural networks have a 'something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are.

Weng[224] argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies.

Large and effective neural networks require considerable computing resources.[225] While the brain has hardware tailored to the task of processing signals through a graph of neurons, simulating even a simplified neuron on von Neumann architecture may compel a neural network designer to fill many millions of database rows for its connections – which can consume vast amounts of memory and storage.

Schmidhuber notes that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by GPGPUs (on GPUs), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before.[226] The use of parallel GPUs can reduce training times from months to days.[225] Neuromorphic engineering addresses the hardware difficulty directly, by constructing non-von-Neumann chips to directly implement neural networks in circuitry.

Another chip optimized for neural network processing is called a Tensor Processing Unit, or TPU.[227] Arguments against Dewdney's position are that neural networks have been successfully used to solve many complex and diverse tasks, ranging from autonomously flying aircraft[228] to detecting credit card fraud to mastering the game of Go.

Technology writer Roger Bridgman commented: Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be 'an opaque, unreadable table...valueless as a scientific resource'.

In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers.

An unreadable table that a useful machine could read would still be well worth having.[229] Although it is true that analyzing what has been learned by an artificial neural network is difficult, it is much easier to do so than to analyze what has been learned by a biological neural network.

Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful.

For example, local vs non-local learning and shallow vs deep architecture.[230] Advocates of hybrid models (combining neural networks and symbolic approaches), claim that such a mixture can better capture the mechanisms of the human mind.[231][232] Artificial neural networks have many variations.

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The various types of neural networks are explained and demonstrated, applications of neural networks like ANNs in medicine are described, and a detailed historical background is provided.

It is composed of a large number of highly interconnected processing elements (neurones) working in unison to solve specific problems.

Minsky and Papert, published a book (in 1969) in which they summed up a general feeling of frustration (against neural networks) among researchers, and was thus accepted by most without further analysis.

For a more detailed description of the history click here The first artificial neuron was produced in 1943 by the neurophysiologist Warren McCulloch and the logician Walter Pits.

Neural networks, with their remarkable ability to derive meaning from complicated or imprecise data, can be used to extract patterns and detect trends that are too complex to be noticed by either humans or other computer techniques.

The network is composed of a large number of highly interconnected processing elements(neurones) working in parallel to solve a specific problem.

Even more, a large number of tasks, require systems that use a combination of the two approaches (normally a conventional computer is used to supervise the neural network) in order to perform at maximum efficiency.

In the human brain, a typical neuron collects signals from others through a host of fine structures called dendrites.

The neuron sends out spikes of electrical activity through a long, thin stand known as an axon, which splits into thousands of branches.

At the end of each branch, a structure called a synapse converts the activity from the axon into electrical effects that inhibit or excite activity from the axon into electrical effects that inhibit or excite activity in the connected neurones.

When a neuron receives excitatory input that is sufficiently large compared with its inhibitory input, it sends a spike of electrical activity down its axon.

However because our knowledge of neurones is incomplete and our computing power is limited, our models are necessarily gross idealisations of real networks of neurones.

In the using mode, when a taught input pattern is detected at the input, its associated output becomes the current output.

If the input pattern does not belong in the taught list of input patterns, the firing rule is used to determine whether to fire or not.

The rule goes as follows: Take a collection of training patterns for a node, some of which cause it to fire (the 1-taught set of patterns) and others which prevent it from doing so (the 0-taught set).

Then the patterns not in the collection cause the node to fire if, on comparison , they have more input elements in common with the 'nearest' pattern in the 1-taught set than with the 'nearest' pattern in the 0-taught set.

For example, a 3-input neuron is taught to output 1 when the input (X1,X2 and X3) is 111 or 101 and to output 0 when the input is 000 or 001.

It differs from 000 in 1 element, from 001 in 2 elements, from 101 in 3 elements and from 111 in 2 elements.

Therefore the firing rule gives the neuron a sense of similarity and enables it to respond 'sensibly' to patterns not seen during training.

If we represent black squares with 0 and white squares with 1 then the truth tables for the 3 neurones after generalisation are;

Top neuron Middle neuron Bottom neuron  From the tables it can be seen the following associasions can be extracted:

Feedback architectures are also referred to as interactive or recurrent, although the latter term is often used to denote feedback connections in single-layer organisations.

The commonest type of artificial neural network consists of three groups, or layers, of units: a layer of "input"

The activity of each hidden unit is determined by the activities of the input units and the weights on the connections between the input and the hidden units.

The weights between the input and hidden units determine when each hidden unit is active, and so by modifying these weights, a hidden unit can choose what it represents.

The single-layer organisation, in which all units are connected to one another, constitutes the most general case and is of more potential computational power than hierarchically structured multi-layer organisations.

The perceptron (figure 4.4) turns out to be an MCP model ( neuron with weighted inputs ) with some additional, fixed, pre--processing.

Units labelled A1, A2, Aj , Ap are called association units and their task is to extract specific, localised featured from the input images.

The book was very well written and showed mathematically that single layer perceptrons could not do some basic pattern recognition operations like determining the parity of a shape or determining whether a shape is connected or not.

associative mapping in which the network learns to produce a particular pattern on the set of input units whenever another particular pattern is applied on the set of input units.

This is used to provide pattern completition, ie to produce a pattern whenever a portion of it or a distorted pattern is presented.

nearest-neighbour recall, where the output pattern produced corresponds to the input pattern stored, which is closest to the pattern presented, and interpolative recall, where the output pattern is a similarity dependent interpolation of the patterns stored corresponding to the pattern presented.

Yet another paradigm, which is a variant associative mapping is classification, ie when there is a fixed set of categories into which the input patterns are to be classified.

Whereas in asssociative mapping the network stores the relationships among patterns, in regularity detection the response of each unit has a particular 'meaning'.

Supervised learning which incorporates an external teacher, so that each output unit is told what its desired response to input signals ought to be.

important issue conserning supervised learning is the problem of error convergence, ie the minimisation of error between the desired and computed unit values.

For threshold units, the output is set at one of two levels, depending on whether the total input is greater than or less than some threshold value.

Sigmoid units bear a greater resemblance to real neurones than do linear or threshold units, but all three must be considered rough approximations.

To make a neural network that performs some specific task, we must choose how the units are connected to one another (see figure 4.1), and we must set the weights on the connections appropriately.

We can teach a three-layer network to perform a particular task by using the following procedure: Assume that we want a network to recognise hand-written digits.

The network would therefore need 256 input units (one for each sensor), 10 output units (one for each kind of digit) and a number of hidden units.

For each kind of digit recorded by the sensors, the network should produce high activity in the appropriate output unit and low activity in the other output units.

To train the network, we present an image of a digit and compare the actual activity of the 10 output units with the desired activity.

Next we change the weight of each connection so as to reduce the error.We repeat this training process for many different images of each different images of each kind of digit until the network classifies every image correctly.

To implement this procedure we need to calculate the error derivative for the weight (EW) in order to change the weight by an amount that is proportional to the rate at which the error changes as the weight is changed.

It was developed independently by two teams, one (Fogelman-Soulie, Gallinari and Le Cun) in France, the other (Rumelhart, Hinton and Williams) in U.S. In order to train a neural network to perform some task, we must adjust the weights of each unit in such a way that the error between the desired output and the actual output is reduced.

To compute the EA for a hidden unit in the layer just before the output layer, we first identify all the weights between that hidden unit and the output units to which it is connected.

After calculating all the EAs in the hidden layer just before the output layer, we can compute in like fashion the EAs for other layers, moving from layer to layer in a direction opposite to the way activities propagate through the network.

Since neural networks are best at identifying patterns or trends in data, they are well suited for prediction or forecasting needs including:

Neural networks are ideal in recognising diseases using scans since there is no need to provide a specific algorithm on how to identify the disease.

Diagnosis can be achieved by building a model of the cardiovascular system of an individual and comparing it with the real time physiological measurements taken from the patient.

If this routine is carried out regularly, potential harmful medical conditions can be detected at an early stage and thus make the process of combating the disease much easier.

model of an individual's cardiovascular system must mimic the relationship among physiological variables (i.e., heart rate, systolic and diastolic blood pressures, and breathing rate) at different physical activity levels.

Sensor fusion enables the ANNs to learn complex relationships among the individual sensor values, which would otherwise be lost if the values were individually analysed.

In medical modelling and diagnosis, this implies that even though each sensor in a set may be sensitive only to a specific physiological variable, ANNs are capable of detecting complex medical conditions by fusing the data from the individual biomedical sensors.

trained an autoassociative memory neural network to store a large number of medical records, each of which includes information on symptoms, diagnosis, and treatment for a particular case.

There is also a strong potential for using neural networks for database mining, that is, searching for patterns implicit within the explicitly stored information in databases.

A feedforward neural network is integrated with the AMT and was trained using back-propagation to assist the marketing control of airline seat allocations.

Stephens, 1987] While it is significant that neural networks have been applied to this problem, it is also important to see that this intelligent technology can be integrated with expert systems and other approaches to make a functional system.

Finally, I would like to state that even though neural networks have a huge potential we will only get the best of them when they are intergrated with computing, AI, fuzzy logic and related subjects.

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## Machine learning

Machine learning is a subset of artificial intelligence in the field of computer science that often uses statistical techniques to give computers the ability to 'learn' (i.e., progressively improve performance on a specific task) with data, without being explicitly programmed.[1] The name machine learning was coined in 1959 by Arthur Samuel.[2] Evolved from the study of pattern recognition and computational learning theory in artificial intelligence,[3] machine learning explores the study and construction of algorithms that can learn from and make predictions on data[4] – such algorithms overcome following strictly static program instructions by making data-driven predictions or decisions,[5]:2 through building a model from sample inputs.

Mitchell provided a widely quoted, more formal definition of the algorithms studied in the machine learning field: 'A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E.'[13] This definition of the tasks in which machine learning is concerned offers a fundamentally operational definition rather than defining the field in cognitive terms.

Machine learning tasks are typically classified into two broad categories, depending on whether there is a learning 'signal' or 'feedback' available to a learning system: Another categorization of machine learning tasks arises when one considers the desired output of a machine-learned system:[5]:3 Among other categories of machine learning problems, learning to learn learns its own inductive bias based on previous experience.

Developmental learning, elaborated for robot learning, generates its own sequences (also called curriculum) of learning situations to cumulatively acquire repertoires of novel skills through autonomous self-exploration and social interaction with human teachers and using guidance mechanisms such as active learning, maturation, motor synergies, and imitation.

Probabilistic systems were plagued by theoretical and practical problems of data acquisition and representation.[17]:488 By 1980, expert systems had come to dominate AI, and statistics was out of favor.[18] Work on symbolic/knowledge-based learning did continue within AI, leading to inductive logic programming, but the more statistical line of research was now outside the field of AI proper, in pattern recognition and information retrieval.[17]:708–710;

Machine learning and data mining often employ the same methods and overlap significantly, but while machine learning focuses on prediction, based on known properties learned from the training data, data mining focuses on the discovery of (previously) unknown properties in the data (this is the analysis step of knowledge discovery in databases).

Much of the confusion between these two research communities (which do often have separate conferences and separate journals, ECML PKDD being a major exception) comes from the basic assumptions they work with: in machine learning, performance is usually evaluated with respect to the ability to reproduce known knowledge, while in knowledge discovery and data mining (KDD) the key task is the discovery of previously unknown knowledge.

Jordan, the ideas of machine learning, from methodological principles to theoretical tools, have had a long pre-history in statistics.[20] He also suggested the term data science as a placeholder to call the overall field.[20] Leo Breiman distinguished two statistical modelling paradigms: data model and algorithmic model,[21] wherein 'algorithmic model' means more or less the machine learning algorithms like Random forest.

Multilinear subspace learning algorithms aim to learn low-dimensional representations directly from tensor representations for multidimensional data, without reshaping them into (high-dimensional) vectors.[26] Deep learning algorithms discover multiple levels of representation, or a hierarchy of features, with higher-level, more abstract features defined in terms of (or generating) lower-level features.

In machine learning, genetic algorithms found some uses in the 1980s and 1990s.[30][31] Conversely, machine learning techniques have been used to improve the performance of genetic and evolutionary algorithms.[32] Rule-based machine learning is a general term for any machine learning method that identifies, learns, or evolves 'rules' to store, manipulate or apply, knowledge.

They seek to identify a set of context-dependent rules that collectively store and apply knowledge in a piecewise manner in order to make predictions.[34] Applications for machine learning include: In 2006, the online movie company Netflix held the first 'Netflix Prize' competition to find a program to better predict user preferences and improve the accuracy on its existing Cinematch movie recommendation algorithm by at least 10%.

A joint team made up of researchers from AT&T Labs-Research in collaboration with the teams Big Chaos and Pragmatic Theory built an ensemble model to win the Grand Prize in 2009 for $1 million.[40] Shortly after the prize was awarded, Netflix realized that viewers' ratings were not the best indicators of their viewing patterns ('everything is a recommendation') and they changed their recommendation engine accordingly.[41] In 2010 The Wall Street Journal wrote about the firm Rebellion Research and their use of Machine Learning to predict the financial crisis.

[42] In 2012, co-founder of Sun Microsystems Vinod Khosla predicted that 80% of medical doctors jobs would be lost in the next two decades to automated machine learning medical diagnostic software.[43] In 2014, it has been reported that a machine learning algorithm has been applied in Art History to study fine art paintings, and that it may have revealed previously unrecognized influences between artists.[44] Although machine learning has been very transformative in some fields, effective machine learning is difficult because finding patterns is hard and often not enough training data are available;

as a result, machine-learning programs often fail to deliver.[45][46] Classification machine learning models can be validated by accuracy estimation techniques like the Holdout method, which splits the data into a training and test sets (conventionally 2/3 training set and 1/3 test set designation) and evaluates the performance of the training model on the test set.

Systems which are trained on datasets collected with biases may exhibit these biases upon use (algorithmic bias), thus digitizing cultural prejudices.[49] For example, using job hiring data from a firm with racist hiring policies may lead to a machine learning system duplicating the bias by scoring job applicants against similarity to previous successful applicants.[50][51] Responsible collection of data and documentation of algorithmic rules used by a system thus is a critical part of machine learning.

There is huge potential for machine learning in health care to provide professionals a great tool to diagnose, medicate, and even plan recovery paths for patients, but this will not happen until the personal biases mentioned previously, and these 'greed' biases are addressed.[53] Software suites containing a variety of machine learning algorithms include the following :

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## Using neural nets to recognize handwritten digits

Simple intuitions about how we recognize shapes - 'a 9 has a loop at the top, and a vertical stroke in the bottom right' - turn out to be not so simple to express algorithmically.

As a prototype it hits a sweet spot: it's challenging - it's no small feat to recognize handwritten digits - but it's not so difficult as to require an extremely complicated solution, or tremendous computational power.

But along the way we'll develop many key ideas about neural networks, including two important types of artificial neuron (the perceptron and the sigmoid neuron), and the standard learning algorithm for neural networks, known as stochastic gradient descent.

Today, it's more common to use other models of artificial neurons - in this book, and in much modern work on neural networks, the main neuron model used is one called the sigmoid neuron.

A perceptron takes several binary inputs, $x_1, x_2, \ldots$, and produces a single binary output: In the example shown the perceptron has three inputs, $x_1, x_2, x_3$.

The neuron's output, $0$ or $1$, is determined by whether the weighted sum $\sum_j w_j x_j$ is less than or greater than some threshold value.

To put it in more precise algebraic terms: \begin{eqnarray} \mbox{output} & = & \left\{ \begin{array}{ll} 0 & \mbox{if } \sum_j w_j x_j \leq \mbox{ threshold} \\ 1 & \mbox{if } \sum_j w_j x_j > \mbox{ threshold} \end{array} \right.

And it should seem plausible that a complex network of perceptrons could make quite subtle decisions: In this network, the first column of perceptrons - what we'll call the first layer of perceptrons - is making three very simple decisions, by weighing the input evidence.

The first change is to write $\sum_j w_j x_j$ as a dot product, $w \cdot x \equiv \sum_j w_j x_j$, where $w$ and $x$ are vectors whose components are the weights and inputs, respectively.

Using the bias instead of the threshold, the perceptron rule can be rewritten: \begin{eqnarray} \mbox{output} = \left\{ \begin{array}{ll} 0 & \mbox{if } w\cdot x + b \leq 0 \\ 1 & \mbox{if } w\cdot x + b > 0 \end{array} \right.

This requires computing the bitwise sum, $x_1 \oplus x_2$, as well as a carry bit which is set to $1$ when both $x_1$ and $x_2$ are $1$, i.e., the carry bit is just the bitwise product $x_1 x_2$: To get an equivalent network of perceptrons we replace all the NAND gates by perceptrons with two inputs, each with weight $-2$, and an overall bias of $3$.

Note that I've moved the perceptron corresponding to the bottom right NAND gate a little, just to make it easier to draw the arrows on the diagram: One notable aspect of this network of perceptrons is that the output from the leftmost perceptron is used twice as input to the bottommost perceptron.

(If you don't find this obvious, you should stop and prove to yourself that this is equivalent.) With that change, the network looks as follows, with all unmarked weights equal to -2, all biases equal to 3, and a single weight of -4, as marked: Up to now I've been drawing inputs like $x_1$ and $x_2$ as variables floating to the left of the network of perceptrons.

In fact, it's conventional to draw an extra layer of perceptrons - the input layer - to encode the inputs: This notation for input perceptrons, in which we have an output, but no inputs, is a shorthand.

Then the weighted sum $\sum_j w_j x_j$ would always be zero, and so the perceptron would output $1$ if $b > 0$, and $0$ if $b \leq 0$.

Instead of explicitly laying out a circuit of NAND and other gates, our neural networks can simply learn to solve problems, sometimes problems where it would be extremely difficult to directly design a conventional circuit.

If it were true that a small change in a weight (or bias) causes only a small change in output, then we could use this fact to modify the weights and biases to get our network to behave more in the manner we want.

In fact, a small change in the weights or bias of any single perceptron in the network can sometimes cause the output of that perceptron to completely flip, say from $0$ to $1$.

We'll depict sigmoid neurons in the same way we depicted perceptrons: Just like a perceptron, the sigmoid neuron has inputs, $x_1, x_2, \ldots$.

Instead, it's $\sigma(w \cdot x+b)$, where $\sigma$ is called the sigmoid function* *Incidentally, $\sigma$ is sometimes called the logistic function, and this new class of neurons called logistic neurons.

\tag{3}\end{eqnarray} To put it all a little more explicitly, the output of a sigmoid neuron with inputs $x_1,x_2,\ldots$, weights $w_1,w_2,\ldots$, and bias $b$ is \begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)}.

In fact, there are many similarities between perceptrons and sigmoid neurons, and the algebraic form of the sigmoid function turns out to be more of a technical detail than a true barrier to understanding.

var data = d3.range(sample).map(function(d){ return { x: x1(d), y: s(x1(d))};

var y = d3.scale.linear() .domain([0, 1]) .range([height, 0]);

}) var graph = d3.select('#sigmoid_graph') .append('svg') .attr('width', width + m[1] + m[3]) .attr('height', height + m[0] + m[2]) .append('g') .attr('transform', 'translate(' + m[3] + ',' + m[0] + ')');

var xAxis = d3.svg.axis() .scale(x) .tickValues(d3.range(-4, 5, 1)) .orient('bottom') graph.append('g') .attr('class', 'x axis') .attr('transform', 'translate(0, ' + height + ')') .call(xAxis);

var yAxis = d3.svg.axis() .scale(y) .tickValues(d3.range(0, 1.01, 0.2)) .orient('left') .ticks(5) graph.append('g') .attr('class', 'y axis') .call(yAxis);

graph.append('text') .attr('class', 'x label') .attr('text-anchor', 'end') .attr('x', width/2) .attr('y', height+35) .text('z');

graph.append('text') .attr('x', (width / 2)) .attr('y', -10) .attr('text-anchor', 'middle') .style('font-size', '16px') .text('sigmoid function');

var data = d3.range(sample).map(function(d){ return { x: x1(d), y: s(x1(d))};

var y = d3.scale.linear() .domain([0,1]) .range([height, 0]);

}) var graph = d3.select('#step_graph') .append('svg') .attr('width', width + m[1] + m[3]) .attr('height', height + m[0] + m[2]) .append('g') .attr('transform', 'translate(' + m[3] + ',' + m[0] + ')');

var xAxis = d3.svg.axis() .scale(x) .tickValues(d3.range(-4, 5, 1)) .orient('bottom') graph.append('g') .attr('class', 'x axis') .attr('transform', 'translate(0, ' + height + ')') .call(xAxis);

var yAxis = d3.svg.axis() .scale(y) .tickValues(d3.range(0, 1.01, 0.2)) .orient('left') .ticks(5) graph.append('g') .attr('class', 'y axis') .call(yAxis);

graph.append('text') .attr('class', 'x label') .attr('text-anchor', 'end') .attr('x', width/2) .attr('y', height+35) .text('z');

graph.append('text') .attr('x', (width / 2)) .attr('y', -10) .attr('text-anchor', 'middle') .style('font-size', '16px') .text('step function');

If $\sigma$ had in fact been a step function, then the sigmoid neuron would be a perceptron, since the output would be $1$ or $0$ depending on whether $w\cdot x+b$ was positive or negative* *Actually, when $w \cdot x +b = 0$ the perceptron outputs $0$, while the step function outputs $1$.

The smoothness of $\sigma$ means that small changes $\Delta w_j$ in the weights and $\Delta b$ in the bias will produce a small change $\Delta \mbox{output}$ in the output from the neuron.

In fact, calculus tells us that $\Delta \mbox{output}$ is well approximated by \begin{eqnarray} \Delta \mbox{output} \approx \sum_j \frac{\partial \, \mbox{output}}{\partial w_j} \Delta w_j + \frac{\partial \, \mbox{output}}{\partial b} \Delta b, \tag{5}\end{eqnarray} where the sum is over all the weights, $w_j$, and $\partial \, \mbox{output} / \partial w_j$ and $\partial \, \mbox{output} /\partial b$ denote partial derivatives of the $\mbox{output}$ with respect to $w_j$ and $b$, respectively.

While the expression above looks complicated, with all the partial derivatives, it's actually saying something very simple (and which is very good news): $\Delta \mbox{output}$ is a linear function of the changes $\Delta w_j$ and $\Delta b$ in the weights and bias.

If it's the shape of $\sigma$ which really matters, and not its exact form, then why use the particular form used for $\sigma$ in Equation (3)\begin{eqnarray} \sigma(z) \equiv \frac{1}{1+e^{-z}} \nonumber\end{eqnarray}$('#margin_778862672352_reveal').click(function() {$('#margin_778862672352').toggle('slow', function() {});});?

In fact, later in the book we will occasionally consider neurons where the output is $f(w \cdot x + b)$ for some other activation function $f(\cdot)$.

The main thing that changes when we use a different activation function is that the particular values for the partial derivatives in Equation (5)\begin{eqnarray} \Delta \mbox{output} \approx \sum_j \frac{\partial \, \mbox{output}}{\partial w_j} \Delta w_j + \frac{\partial \, \mbox{output}}{\partial b} \Delta b \nonumber\end{eqnarray}$('#margin_726336021933_reveal').click(function() {$('#margin_726336021933').toggle('slow', function() {});});

It turns out that when we compute those partial derivatives later, using $\sigma$ will simplify the algebra, simply because exponentials have lovely properties when differentiated.

But in practice we can set up a convention to deal with this, for example, by deciding to interpret any output of at least $0.5$ as indicating a '9', and any output less than $0.5$ as indicating 'not a 9'.

Exercises Sigmoid neurons simulating perceptrons, part I $\mbox{}$ Suppose we take all the weights and biases in a network of perceptrons, and multiply them by a positive constant, $c > 0$.

Show that the behaviour of the network doesn't change.Sigmoid neurons simulating perceptrons, part II $\mbox{}$ Suppose we have the same setup as the last problem - a network of perceptrons.

Suppose the weights and biases are such that $w \cdot x + b \neq 0$ for the input $x$ to any particular perceptron in the network.

Now replace all the perceptrons in the network by sigmoid neurons, and multiply the weights and biases by a positive constant $c > 0$.

Suppose we have the network: As mentioned earlier, the leftmost layer in this network is called the input layer, and the neurons within the layer are called input neurons.

The term 'hidden' perhaps sounds a little mysterious - the first time I heard the term I thought it must have some deep philosophical or mathematical significance - but it really means nothing more than 'not an input or an output'.

For example, the following four-layer network has two hidden layers: Somewhat confusingly, and for historical reasons, such multiple layer networks are sometimes called multilayer perceptrons or MLPs, despite being made up of sigmoid neurons, not perceptrons.

If the image is a $64$ by $64$ greyscale image, then we'd have $4,096 = 64 \times 64$ input neurons, with the intensities scaled appropriately between $0$ and $1$.

The output layer will contain just a single neuron, with output values of less than $0.5$ indicating 'input image is not a 9', and values greater than $0.5$ indicating 'input image is a 9 '.

A trial segmentation gets a high score if the individual digit classifier is confident of its classification in all segments, and a low score if the classifier is having a lot of trouble in one or more segments.

So instead of worrying about segmentation we'll concentrate on developing a neural network which can solve the more interesting and difficult problem, namely, recognizing individual handwritten digits.

As discussed in the next section, our training data for the network will consist of many $28$ by $28$ pixel images of scanned handwritten digits, and so the input layer contains $784 = 28 \times 28$ neurons.

The input pixels are greyscale, with a value of $0.0$ representing white, a value of $1.0$ representing black, and in between values representing gradually darkening shades of grey.

A seemingly natural way of doing that is to use just $4$ output neurons, treating each neuron as taking on a binary value, depending on whether the neuron's output is closer to $0$ or to $1$.

The ultimate justification is empirical: we can try out both network designs, and it turns out that, for this particular problem, the network with $10$ output neurons learns to recognize digits better than the network with $4$ output neurons.

In a similar way, let's suppose for the sake of argument that the second, third, and fourth neurons in the hidden layer detect whether or not the following images are present:

Of course, that's not the only sort of evidence we can use to conclude that the image was a $0$ - we could legitimately get a $0$ in many other ways (say, through translations of the above images, or slight distortions).

Assume that the first $3$ layers of neurons are such that the correct output in the third layer (i.e., the old output layer) has activation at least $0.99$, and incorrect outputs have activation less than $0.01$.

We'll use the MNIST data set, which contains tens of thousands of scanned images of handwritten digits, together with their correct classifications.

To make this a good test of performance, the test data was taken from a different set of 250 people than the original training data (albeit still a group split between Census Bureau employees and high school students).

For example, if a particular training image, $x$, depicts a $6$, then $y(x) = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0)^T$ is the desired output from the network.

We use the term cost function throughout this book, but you should note the other terminology, since it's often used in research papers and other discussions of neural networks.

\tag{6}\end{eqnarray} Here, $w$ denotes the collection of all weights in the network, $b$ all the biases, $n$ is the total number of training inputs, $a$ is the vector of outputs from the network when $x$ is input, and the sum is over all training inputs, $x$.

If we instead use a smooth cost function like the quadratic cost it turns out to be easy to figure out how to make small changes in the weights and biases so as to get an improvement in the cost.

Even given that we want to use a smooth cost function, you may still wonder why we choose the quadratic function used in Equation (6)\begin{eqnarray} C(w,b) \equiv \frac{1}{2n} \sum_x \|

This is a well-posed problem, but it's got a lot of distracting structure as currently posed - the interpretation of $w$ and $b$ as weights and biases, the $\sigma$ function lurking in the background, the choice of network architecture, MNIST, and so on.

And for neural networks we'll often want far more variables - the biggest neural networks have cost functions which depend on billions of weights and biases in an extremely complicated way.

We could do this simulation simply by computing derivatives (and perhaps some second derivatives) of $C$ - those derivatives would tell us everything we need to know about the local 'shape' of the valley, and therefore how our ball should roll.

So rather than get into all the messy details of physics, let's simply ask ourselves: if we were declared God for a day, and could make up our own laws of physics, dictating to the ball how it should roll, what law or laws of motion could we pick that would make it so the ball always rolled to the bottom of the valley?

To make this question more precise, let's think about what happens when we move the ball a small amount $\Delta v_1$ in the $v_1$ direction, and a small amount $\Delta v_2$ in the $v_2$ direction.

Calculus tells us that $C$ changes as follows: \begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial v_1} \Delta v_1 + \frac{\partial C}{\partial v_2} \Delta v_2.

To figure out how to make such a choice it helps to define $\Delta v$ to be the vector of changes in $v$, $\Delta v \equiv (\Delta v_1, \Delta v_2)^T$, where $T$ is again the transpose operation, turning row vectors into column vectors.

We denote the gradient vector by $\nabla C$, i.e.: \begin{eqnarray} \nabla C \equiv \left( \frac{\partial C}{\partial v_1}, \frac{\partial C}{\partial v_2} \right)^T.

In fact, it's perfectly fine to think of $\nabla C$ as a single mathematical object - the vector defined above - which happens to be written using two symbols.

With these definitions, the expression (7)\begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial v_1} \Delta v_1 + \frac{\partial C}{\partial v_2} \Delta v_2 \nonumber\end{eqnarray}$('#margin_832985330775_reveal').click(function() {$('#margin_832985330775').toggle('slow', function() {});});

\tag{9}\end{eqnarray} This equation helps explain why $\nabla C$ is called the gradient vector: $\nabla C$ relates changes in $v$ to changes in $C$, just as we'd expect something called a gradient to do.

In particular, suppose we choose \begin{eqnarray} \Delta v = -\eta \nabla C, \tag{10}\end{eqnarray} where $\eta$ is a small, positive parameter (known as the learning rate).

\nabla C \|^2 \geq 0$, this guarantees that $\Delta C \leq 0$, i.e., $C$ will always decrease, never increase, if we change $v$ according to the prescription in (10)\begin{eqnarray} \Delta v = -\eta \nabla C \nonumber\end{eqnarray}$('#margin_39079991636_reveal').click(function() {$('#margin_39079991636').toggle('slow', function() {});});.

to compute a value for $\Delta v$, then move the ball's position $v$ by that amount: \begin{eqnarray} v \rightarrow v' = v -\eta \nabla C.

To make gradient descent work correctly, we need to choose the learning rate $\eta$ to be small enough that Equation (9)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_663076476028_reveal').click(function() {$('#margin_663076476028').toggle('slow', function() {});});

In practical implementations, $\eta$ is often varied so that Equation (9)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_362932327599_reveal').click(function() {$('#margin_362932327599').toggle('slow', function() {});});

Then the change $\Delta C$ in $C$ produced by a small change $\Delta v = (\Delta v_1, \ldots, \Delta v_m)^T$ is \begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v, \tag{12}\end{eqnarray} where the gradient $\nabla C$ is the vector \begin{eqnarray} \nabla C \equiv \left(\frac{\partial C}{\partial v_1}, \ldots, \frac{\partial C}{\partial v_m}\right)^T.

\tag{13}\end{eqnarray} Just as for the two variable case, we can choose \begin{eqnarray} \Delta v = -\eta \nabla C, \tag{14}\end{eqnarray} and we're guaranteed that our (approximate) expression (12)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_398945612724_reveal').click(function() {$('#margin_398945612724').toggle('slow', function() {});});

This gives us a way of following the gradient to a minimum, even when $C$ is a function of many variables, by repeatedly applying the update rule \begin{eqnarray} v \rightarrow v' = v-\eta \nabla C.

The rule doesn't always work - several things can go wrong and prevent gradient descent from finding the global minimum of $C$, a point we'll return to explore in later chapters.

But, in practice gradient descent often works extremely well, and in neural networks we'll find that it's a powerful way of minimizing the cost function, and so helping the net learn.

It can be proved that the choice of $\Delta v$ which minimizes $\nabla C \cdot \Delta v$ is $\Delta v = - \eta \nabla C$, where $\eta = \epsilon / \|\nabla C\|$ is determined by the size constraint $\|\Delta v\|

Hint: If you're not already familiar with the Cauchy-Schwarz inequality, you may find it helpful to familiarize yourself with it.

If there are a million such $v_j$ variables then we'd need to compute something like a trillion (i.e., a million squared) second partial derivatives* *Actually, more like half a trillion, since $\partial^2 C/ \partial v_j \partial v_k = \partial^2 C/ \partial v_k \partial v_j$.

The idea is to use gradient descent to find the weights $w_k$ and biases $b_l$ which minimize the cost in Equation (6)\begin{eqnarray} C(w,b) \equiv \frac{1}{2n} \sum_x \|

In other words, our 'position' now has components $w_k$ and $b_l$, and the gradient vector $\nabla C$ has corresponding components $\partial C / \partial w_k$ and $\partial C / \partial b_l$.

Writing out the gradient descent update rule in terms of components, we have \begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\eta \frac{\partial C}{\partial w_k} \tag{16}\\ b_l & \rightarrow & b_l' = b_l-\eta \frac{\partial C}{\partial b_l}.

In practice, to compute the gradient $\nabla C$ we need to compute the gradients $\nabla C_x$ separately for each training input, $x$, and then average them, $\nabla C = \frac{1}{n} \sum_x \nabla C_x$.

To make these ideas more precise, stochastic gradient descent works by randomly picking out a small number $m$ of randomly chosen training inputs.

Provided the sample size $m$ is large enough we expect that the average value of the $\nabla C_{X_j}$ will be roughly equal to the average over all $\nabla C_x$, that is, \begin{eqnarray} \frac{\sum_{j=1}^m \nabla C_{X_{j}}}{m} \approx \frac{\sum_x \nabla C_x}{n} = \nabla C, \tag{18}\end{eqnarray} where the second sum is over the entire set of training data.

Swapping sides we get \begin{eqnarray} \nabla C \approx \frac{1}{m} \sum_{j=1}^m \nabla C_{X_{j}}, \tag{19}\end{eqnarray} confirming that we can estimate the overall gradient by computing gradients just for the randomly chosen mini-batch.

Then stochastic gradient descent works by picking out a randomly chosen mini-batch of training inputs, and training with those, \begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial w_k} \tag{20}\\ b_l & \rightarrow & b_l' = b_l-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial b_l}, \tag{21}\end{eqnarray} where the sums are over all the training examples $X_j$ in the current mini-batch.

And, in a similar way, the mini-batch update rules (20)\begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial w_k} \nonumber\end{eqnarray}$('#margin_255037324417_reveal').click(function() {$('#margin_255037324417').toggle('slow', function() {});});

and (21)\begin{eqnarray} b_l & \rightarrow & b_l' = b_l-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial b_l} \nonumber\end{eqnarray}$('#margin_141169455106_reveal').click(function() {$('#margin_141169455106').toggle('slow', function() {});});

We can think of stochastic gradient descent as being like political polling: it's much easier to sample a small mini-batch than it is to apply gradient descent to the full batch, just as carrying out a poll is easier than running a full election.

For example, if we have a training set of size $n = 60,000$, as in MNIST, and choose a mini-batch size of (say) $m = 10$, this means we'll get a factor of $6,000$ speedup in estimating the gradient!

Of course, the estimate won't be perfect - there will be statistical fluctuations - but it doesn't need to be perfect: all we really care about is moving in a general direction that will help decrease $C$, and that means we don't need an exact computation of the gradient.

In practice, stochastic gradient descent is a commonly used and powerful technique for learning in neural networks, and it's the basis for most of the learning techniques we'll develop in this book.

That is, given a training input, $x$, we update our weights and biases according to the rules $w_k \rightarrow w_k' = w_k - \eta \partial C_x / \partial w_k$ and $b_l \rightarrow b_l' = b_l - \eta \partial C_x / \partial b_l$.

Name one advantage and one disadvantage of online learning, compared to stochastic gradient descent with a mini-batch size of, say, $20$.

In neural networks the cost $C$ is, of course, a function of many variables - all the weights and biases - and so in some sense defines a surface in a very high-dimensional space.

I won't go into more detail here, but if you're interested then you may enjoy reading this discussion of some of the techniques professional mathematicians use to think in high dimensions.

We'll leave the test images as is, but split the 60,000-image MNIST training set into two parts: a set of 50,000 images, which we'll use to train our neural network, and a separate 10,000 image validation set.

We won't use the validation data in this chapter, but later in the book we'll find it useful in figuring out how to set certain hyper-parameters of the neural network - things like the learning rate, and so on, which aren't directly selected by our learning algorithm.

When I refer to the 'MNIST training data' from now on, I'll be referring to our 50,000 image data set, not the original 60,000 image data set* *As noted earlier, the MNIST data set is based on two data sets collected by NIST, the United States' National Institute of Standards and Technology.

for x, y in zip(sizes[:-1], sizes[1:])]

So, for example, if we want to create a Network object with 2 neurons in the first layer, 3 neurons in the second layer, and 1 neuron in the final layer, we'd do this with the code: net = Network([2, 3, 1])

The biases and weights in the Network object are all initialized randomly, using the Numpy np.random.randn function to generate Gaussian distributions with mean $0$ and standard deviation $1$.

Note that the Network initialization code assumes that the first layer of neurons is an input layer, and omits to set any biases for those neurons, since biases are only ever used in computing the outputs from later layers.

The big advantage of using this ordering is that it means that the vector of activations of the third layer of neurons is: \begin{eqnarray} a' = \sigma(w a + b).

(This is called vectorizing the function $\sigma$.) It's easy to verify that Equation (22)\begin{eqnarray} a' = \sigma(w a + b) \nonumber\end{eqnarray}$('#margin_552886241220_reveal').click(function() {$('#margin_552886241220').toggle('slow', function() {});});

gives the same result as our earlier rule, Equation (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_7421600236_reveal').click(function() {$('#margin_7421600236').toggle('slow', function() {});});, for computing the output of a sigmoid neuron.

in component form, and verify that it gives the same result as the rule (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_347257101140_reveal').click(function() {$('#margin_347257101140').toggle('slow', function() {});});

We then add a feedforward method to the Network class, which, given an input a for the network, returns the corresponding output* *It is assumed that the input a is an (n, 1) Numpy ndarray, not a (n,) vector.

Although using an (n,) vector appears the more natural choice, using an (n, 1) ndarray makes it particularly easy to modify the code to feedforward multiple inputs at once, and that is sometimes convenient.

All the method does is applies Equation (22)\begin{eqnarray} a' = \sigma(w a + b) \nonumber\end{eqnarray}$('#margin_335258165235_reveal').click(function() {$('#margin_335258165235').toggle('slow', function() {});});

training_data[k:k+mini_batch_size]

for k in xrange(0, n, mini_batch_size)]

self.update_mini_batch(mini_batch, eta)

print "Epoch {0}: {1} / {2}".format(

j, self.evaluate(test_data), n_test)

print "Epoch {0} complete".format(j)

This is done by the code self.update_mini_batch(mini_batch, eta), which updates the network weights and biases according to a single iteration of gradient descent, using just the training data in mini_batch.

delta_nabla_b, delta_nabla_w = self.backprop(x, y)

nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]

nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

for w, nw in zip(self.weights, nabla_w)]

for b, nb in zip(self.biases, nabla_b)]

Most of the work is done by the line delta_nabla_b, delta_nabla_w = self.backprop(x, y)

The self.backprop method makes use of a few extra functions to help in computing the gradient, namely sigmoid_prime, which computes the derivative of the $\sigma$ function, and self.cost_derivative, which I won't describe here.

for x, y in zip(sizes[:-1], sizes[1:])]

training_data[k:k+mini_batch_size]

for k in xrange(0, n, mini_batch_size)]

self.update_mini_batch(mini_batch, eta)

print "Epoch {0}: {1} / {2}".format(

j, self.evaluate(test_data), n_test)

print "Epoch {0} complete".format(j)

delta_nabla_b, delta_nabla_w = self.backprop(x, y)

nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]

nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

for w, nw in zip(self.weights, nabla_w)]

for b, nb in zip(self.biases, nabla_b)]

activations = [x] # list to store all the activations, layer by layer

zs = [] # list to store all the z vectors, layer by layer

# l = 1 means the last layer of neurons, l = 2 is the

delta = np.dot(self.weights[-l+1].transpose(), delta) * sp

for (x, y) in test_data]

Finally, we'll use stochastic gradient descent to learn from the MNIST training_data over 30 epochs, with a mini-batch size of 10, and a learning rate of $\eta = 3.0$, >>>

As was the case earlier, if you're running the code as you read along, you should be warned that it takes quite a while to execute (on my machine this experiment takes tens of seconds for each training epoch), so it's wise to continue reading in parallel while the code executes.

At least in this case, using more hidden neurons helps us get better results* *Reader feedback indicates quite some variation in results for this experiment, and some training runs give results quite a bit worse.

Using the techniques introduced in chapter 3 will greatly reduce the variation in performance across different training runs for our networks..

(If making a change improves things, try doing more!) If we do that several times over, we'll end up with a learning rate of something like $\eta = 1.0$ (and perhaps fine tune to $3.0$), which is close to our earlier experiments.

Exercise Try creating a network with just two layers - an input and an output layer, no hidden layer - with 784 and 10 neurons, respectively.

The data structures used to store the MNIST data are described in the documentation strings - it's straightforward stuff, tuples and lists of Numpy ndarray objects (think of them as vectors if you're not familiar with ndarrays): """mnist_loader~~~~~~~~~~~~A library to load the MNIST image data.

In some sense, the moral of both our results and those in more sophisticated papers, is that for some problems: sophisticated algorithm $\leq$ simple learning algorithm + good training data.

We could attack this problem the same way we attacked handwriting recognition - by using the pixels in the image as input to a neural network, with the output from the network a single neuron indicating either 'Yes, it's a face' or 'No, it's not a face'.

The end result is a network which breaks down a very complicated question - does this image show a face or not - into very simple questions answerable at the level of single pixels.

It does this through a series of many layers, with early layers answering very simple and specific questions about the input image, and later layers building up a hierarchy of ever more complex and abstract concepts.

Comparing a deep network to a shallow network is a bit like comparing a programming language with the ability to make function calls to a stripped down language with no ability to make such calls.

- On Thursday, June 28, 2018
- By Read More

## Handbook of Neural Computing Applications

Other applications involve signal processing, spatio-temporal pattern recognition, medical diagnoses, fault diagnoses, robotics, business, data communications, data compression, and adaptive man-machine systems.

One paper describes data compression and dimensionality reduction methods that have characteristics, such as high compression ratios to facilitate data storage, strong discrimination of novel data from baseline, rapid operation for software and hardware, as well as the ability to recognized loss of data during compression or reconstruction.

- On Tuesday, February 18, 2020

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