Abstract
Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object recognition and discrimination requires classifying the manifolds in a manner that is insensitive to variability within a manifold. How neuronal systems give rise to invariant object classification and recognition is a fundamental problem in brain theory as well as in machine learning. Here we study the ability of a readout network to classify objects from their perceptual manifold representations. We develop a statistical mechanical theory for the linear classification of manifolds with arbitrary geometry revealing a remarkable relation to the mathematics of conic decomposition. Novel geometrical measures of manifold radius and manifold dimension are introduced which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including L2 ellipsoids prototypical of strictly convex manifolds, L1 balls representing polytopes consisting of finite sample points, and orientation manifolds which arise from neurons tuned to respond to a continuous angle variable, such as object orientation. The effects of label sparsity on the classification capacity of manifolds are elucidated, revealing a scaling relation between label sparsity and manifold radius. Theoretical predictions are corroborated by numerical simulations using recently developed algorithms to compute maximum margin solutions for manifold dichotomies. Our theory and its extensions provide a powerful and rich framework for applying statistical mechanics of linear classification to data arising from neuronal responses to object stimuli, as well as to artificial deep networks trained for object recognition tasks.
Highlights
One fundamental cognitive task performed by animals and humans is the invariant perception of objects, requiring the nervous system to discriminate between different objects despite substantial variability in each object’s physical features
The anchor point is given by the following boundary point on the ellipse, given by Eq (B4) with v⃗ 1⁄4 ⃗t
This regime holds for t0, obeying the inequality t0 − κ ≥ ttouchð⃗tÞ, with Eq (22), yielding ttouchð⃗tÞ 1⁄4 k⃗t ∘ R⃗ k: ðB6Þ
Summary
One fundamental cognitive task performed by animals and humans is the invariant perception of objects, requiring the nervous system to discriminate between different objects despite substantial variability in each object’s physical features. Studies of high-level sensory systems, e.g., the inferotemporal cortex (IT) in vision [1], auditory cortex in audition [2], and piriform cortex in olfaction [3], reveal that even the late sensory stages exhibit significant sensitivity of neuronal responses to physical variables This suggests that sensory hierarchies generate representations of objects that, not entirely invariant to changes in physical features, are still readily decoded in an invariant manner by a downstream system. This formalism allows us to generate generic bounds on the manifold separability capacity from the limits of small manifold sizes (classification of isolated points) to that of large sizes (classification of entire affine subspaces) These bounds highlight the fact that for large ambient dimension N, the maximal number P of separable finite-dimensional manifolds is proportional to N, even though each consists of an infinite number of points, setting the stage for a statistical mechanical evaluation of the maximal α 1⁄4 ðP=NÞ. The theory makes an important contribution to the development of statistical mechanical theories of neural information processing in realistic conditions
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