AI News, Kernels and Quantum Gravity Part 3: CoherentStates

Kernels and Quantum Gravity Part 3: CoherentStates

This would not be a pedagogic machine learning blog if I did not go into some overly abstract formalism&#8230;here we introduce the Kernel formalism using the languages of Coherent States: We define a space of labels (that is isomorphic to , or, more generally, just a locally compact space), and an abstract Hilbert space  We seek a map between the two:  such that

So rather than introduce an expression for , we introduce a operator acting on the Hilbert space that encapsulates the fact that our basis set is non-orthogonal (and perhaps overcomplete?)

The main difference is that in other fields, one (usually) tries to use their prior knowledge of the problem to actually find the solution and does not just guess random Kernels and crossvalidate (although there are important cases where it does seem like this, such as in Quantum Chemical Density Functional Theory).

would accept this recent paper on Reproducing Kernel Banach Spaces with the ℓ1 Norm In Physics, we may think of the labels as the Classical variables of phase space and the Hilbert space as   the space of Quantum Mechanical wavefunctions .

More importantly, for understanding machine learning, we will see the mathematical formulation of Frame Quantization and the attempts to capture the mathematics of coherent states under a single mathematical formalism (and how and when this is doable)  .

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