Mathematics of the Neural Response

Abstract

We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure between functions which we call the derived kernel. Based on a hierarchical architecture, we give a recursive definition of the neural response and associated derived kernel. The derived kernel can be used in a variety of application domains such as classification of images, strings of text and genomics data.

Hodge Theory on Metric Spaces

“…Spaces of images are important for developing a mathematics of vision (see e.g. Smale, Rosasco, Bouvrie, Caponnetto, and Poggio [35]), but these spaces are far from possessing manifold structures. Other settings include spaces occurring in quantum field theory, such as manifolds with singularities and/or non-uniform measures.…”

The computational magic of the ventral stream

“…Appendix: Mathematics of the Invariant Neural Response (with L. Rosasco). We provide a mathematical description of the neural response architecture [29] of Figure 1, which is designed to be robust to transformations encoded implicitly in sets of templates. Robustness is achieved by mean of suitable pooling operations across the responses to such templates.…”

“…Robustness is achieved by mean of suitable pooling operations across the responses to such templates. The setting we describe is a modification of the one introduced in [29]. 7.6.1.…”

Does Cognitive Science Need Kernels?

“…Such kernels might be useful for modelling categorization of interestingly structured stimuli, like sentences or visual objects. Of particular interest in this context are recursively defined kernels (Haussler, 1999;Smale, Rosasco, Bouvire, Caponnetto, & Poggio, 2008).…”