Statistical Mechanics of Neural Processing of Object Manifolds
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Invariant object recognition is one of the most fundamental cognitive tasks performed by the brain. In the neural state space, different objects with stimulus variabilities are represented as different manifolds. In this geometrical perspective, object recognition becomes the problem of linearly separating different object manifolds. In feedforward visual hierarchy, it has been suggested that the object manifold representations are reformatted across the layers, to become more linearly separable. Thus, a complete theory of perception requires characterizing the ability of linear readout networks to classify object manifolds from variable neural responses. A theory of the perceptron of isolated points was pioneered by E. Gardner who formulated it as a statistical mechanics problem and analyzed it using replica theory. In this thesis, we generalize Gardner's analysis and establish a theory of linear classification of manifolds synthesizing statistical and geometric properties of high dimensional signals. [..] Next, we generalize our theory further to linear classification of general perceptual manifolds, such as point clouds. We identify that the capacity of a manifold is determined that effective radius, R_M, and effective dimension, D_M. Finally, we show extensions relevant for applications to real data, incorporating correlated manifolds, heterogenous manifold geometries, sparse labels and nonlinear classifications. Then, we demonstrate how object-based manifolds transform in standard deep networks. This thesis lays the groundwork for a computational theory of neuronal processing of objects, providing quantitative measures for linear separability of object manifolds. We hope this theory will provide new insights into the computational principles underlying processing of sensory representations in biological and artificial neural networks.
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