AI News, What are kernels in machine learning and SVM and why do we need them?

What are kernels in machine learning and SVM and why do we need them?

Kernels are the idea of summing functions that imitate similarity (induce a positive-definite encoding of nearness) and support vector machines are the idea of solving a clever dual problem to maximize a quantity called margin.

Without the positive semidefinite property all of these optimization problems would be able to “run to negative infinity” or use negative terms (which are not possible from a kernel) to hide high error rates.

The limits of the kernel functions (not being able to turn distance penalties into bonuses) help ensure that the result of optimization is actually useful (and not just a flaw in our problem encoding).

The phi() transform can be arbitrarily magic except when transforming one vector it doesn’t know what the other vector is and phi() doesn’t even know which side of the inner product it is encoding.

For instance: if you had access to two functions q(.,.) and r(.,.) that claim to be kernels and you wanted the product kernel you would just, when asked to evaluate the kernel, just get the results for the two sub-kernels and multiply (so you never need to see the space phi(.) is implicitly working in).

To see the Gaussian is a kernel notice the following: And for all c ≥ 0 each of the three terms on the right is a kernel (the first because the Taylor series of exp() is absolutely convergent and non-negative and the second two are the f(u) f(v) form we listed as obvious kernels).

The great benefit of the support vector machine is that with access only to the data labels and the kernel function (in fact only the kernel function evaluated at pairs of training datums) the support vector machine can quickly solve for the optimal margin and data weights achieving this margin.

This kernel can be thought of as having a phi(.) function that takes a vector z and adds an extra coordinate of sqrt(c) and then projects the resulting vector onto the unit sphere.

So any knowledge of what sort of model we want (one class bounded or not) should greatly influence our choice of kernel functions (since the support vector machine can only pick weights for the kernel functions, not tune their shapes or bandwidths).

Both these claims are fallacious- you can’t fully use the higher order terms because they are entangled with other terms that are not orthogonal the outcome and the complexity of a kernel is not quite as simple as degrees of freedom (proofs about support vector machines are stated in terms of margin, not in terms of degrees of freedom or even in terms of VC dimension).

Figure 4: Squared magic kernel support vector model If you want higher order terms I feel you are much better off performing a primal transform so the terms are available in their most general form.

Support vector machine

In machine learning, support vector machines (SVMs, also support vector networks[1]) are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis.

Given a set of training examples, each marked as belonging to one or the other of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier (although methods such as Platt scaling exist to use SVM in a probabilistic classification setting).

An SVM model is a representation of the examples as points in space, mapped so that the examples of the separate categories are divided by a clear gap that is as wide as possible.

In addition to performing linear classification, SVMs can efficiently perform a non-linear classification using what is called the kernel trick, implicitly mapping their inputs into high-dimensional feature spaces.

When data is unlabelled, supervised learning is not possible, and an unsupervised learning approach is required, which attempts to find natural clustering of the data to groups, and then map new data to these formed groups.

algorithm created by Hava Siegelmann and Vladimir Vapnik, applies the statistics of support vectors, developed in the support vector machines algorithm, to categorize unlabeled data, and is one of the most widely used clustering algorithms in industrial applications.[citation needed]

If such a hyperplane exists, it is known as the maximum-margin hyperplane and the linear classifier it defines is known as a maximum margin classifier;

More formally, a support vector machine constructs a hyperplane or set of hyperplanes in a high- or infinite-dimensional space, which can be used for classification, regression, or other tasks like outliers detection[3].

Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training-data point of any class (so-called functional margin), since in general the larger the margin the lower the generalization error of the classifier[4].

Whereas the original problem may be stated in a finite dimensional space, it often happens that the sets to discriminate are not linearly separable in that space.

To keep the computational load reasonable, the mappings used by SVM schemes are designed to ensure that dot products of pairs input data vectors may be computed easily in terms of the variables in the original space, by defining them in terms of a kernel function

The hyperplanes in the higher-dimensional space are defined as the set of points whose dot product with a vector in that space is constant, where such a set of vector is an orthogonal (and thus minimal) set of vectors that defines a hyperplane.

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In this way, the sum of kernels above can be used to measure the relative nearness of each test point to the data points originating in one or the other of the sets to be discriminated.

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If the training data is linearly separable, we can select two parallel hyperplanes that separate the two classes of data, so that the distance between them is as large as possible.

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the second term in the loss function will become negligible, hence, it will behave similar to the hard-margin SVM, if the input data are linearly classifiable, but will still learn if a classification rule is viable or not.

Vapnik suggested a way to create nonlinear classifiers by applying the kernel trick (originally proposed by Aizerman et al.[14]) to maximum-margin hyperplanes.[12]

It is noteworthy that working in a higher-dimensional feature space increases the generalization error of support vector machines, although given enough samples the algorithm still performs well.[15]

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{\displaystyle k({\vec {x_{i}}},{\vec {x_{j}}})=\varphi ({\vec {x_{i}}})\cdot \varphi ({\vec {x_{j}}})}

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{\displaystyle \textstyle {\vec {w}}\cdot \varphi ({\vec {x}})=\sum _{i}\alpha _{i}y_{i}k({\vec {x}}_{i},{\vec {x}})}

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.) Suppose now that we would like to learn a nonlinear classification rule which corresponds to a linear classification rule for the transformed data points

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Both techniques have proven to offer significant advantages over the traditional approach when dealing with large, sparse datasets—sub-gradient methods are especially efficient when there are many training examples, and coordinate descent when the dimension of the feature space is high.

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As such, traditional gradient descent (or SGD) methods can be adapted, where instead of taking a step in the direction of the functions gradient, a step is taken in the direction of a vector selected from the function's sub-gradient.

The soft-margin support vector machine described above is an example of an empirical risk minimization (ERM) algorithm for the hinge loss.

Seen this way, support vector machines belong to a natural class of algorithms for statistical inference, and many of its unique features are due to the behavior of the hinge loss.

(for example, that they are generated by a finite Markov process), if the set of hypotheses being considered is small enough, the minimizer of the empirical risk will closely approximate the minimizer of the expected risk as

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In light of the above discussion, we see that the SVM technique is equivalent to empirical risk minimization with Tikhonov regularization, where in this case the loss function is the hinge loss

The difference between the three lies in the choice of loss function: regularized least-squares amounts to empirical risk minimization with the square-loss,

The difference between the hinge loss and these other loss functions is best stated in terms of target functions - the function that minimizes expected risk for a given pair of random variables

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Thus, in a sufficiently rich hypothesis space—or equivalently, for an appropriately chosen kernel—the SVM classifier will converge to the simplest function (in terms of

This extends the geometric interpretation of SVM—for linear classification, the empirical risk is minimized by any function whose margins lie between the support vectors, and the simplest of these is the max-margin classifier.[18]

The final model, which is used for testing and for classifying new data, is then trained on the whole training set using the selected parameters.[20]

Multiclass SVM aims to assign labels to instances by using support vector machines, where the labels are drawn from a finite set of several elements.

Crammer and Singer proposed a multiclass SVM method which casts the multiclass classification problem into a single optimization problem, rather than decomposing it into multiple binary classification problems.[25]

Transductive support vector machines extend SVMs in that they could also treat partially labeled data in semi-supervised learning by following the principles of transduction.

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The model produced by support vector classification (as described above) depends only on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin.

Analogously, the model produced by SVR depends only on a subset of the training data, because the cost function for building the model ignores any training data close to the model prediction.

This extended view allows for the application of Bayesian techniques to SVMs, such as flexible feature modeling, automatic hyperparameter tuning, and predictive uncertainty quantification.

There exist several specialized algorithms for quickly solving the QP problem that arises from SVMs, mostly relying on heuristics for breaking the problem down into smaller, more-manageable chunks.

To avoid solving a linear system involving the large kernel matrix, a low rank approximation to the matrix is often used in the kernel trick.

Another common method is Platt's sequential minimal optimization (SMO) algorithm, which breaks the problem down into 2-dimensional sub-problems that are solved analytically, eliminating the need for a numerical optimization algorithm and matrix storage.

The special case of linear support vector machines can be solved more efficiently by the same kind of algorithms used to optimize its close cousin, logistic regression;

Each convergence iteration takes time linear in the time taken to read the train data and the iterations also have a Q-Linear Convergence property, making the algorithm extremely fast.

Understanding Support Vector Machine algorithm from examples (along with code)

Note: This article was originally published on Oct 6th, 2015 and updated on Sept 13th, 2017 Mastering machine learning algorithms isn’t a myth at all.

Think of machine learning algorithms as an armory packed with axes, sword, blades, bow, dagger etc. You have various tools, but you ought to learn to use them at the right time.

In this article, I shall guide you through the basics to advanced knowledge of a crucial machine learning algorithm, support vector machines.

However,  it is mostly used in classification problems. In this algorithm, we plot each data item as a point in n-dimensional space (where n is number of features you have) with the value of each feature being the value of a particular coordinate.

In Python, scikit-learn is a widely used library for implementing machine learning algorithms, SVM is also available in scikit-learn library and follow the same structure (Import library, object creation, fitting model and prediction).

The creation of a support vector machine in R and Python follow similar approaches, let’s take a look now at the following code:

Let’s look at the example, where we’ve used linear kernel on two feature of iris data set to classify their class.

Example: Have linear kernel Example: Have rbf kernel Change the kernel type to rbf in below line and look at the impact.

would suggest you to go for linear kernel if you have large number of features (>1000) because it is more likely that the data is linearly separable in high dimensional space.

 I discussed its concept of working, process of implementation in python, the tricks to make the model efficient by tuning its parameters, Pros and Cons, and finally a problem to solve.

Short-term stock price forecasting using kernel principal component analysis and support vector machines: the case of Casablanca stock exchange

The simulation results show that, through KPCA attribute reduction, the structure of the investment decision system can be simplified significantly with improvement of the model performance.

The average performance of the integrated model that uses KPCA and SVR is significantly better than that of SVR model, which verifies the effectiveness and accuracy of the proposed method.

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