Classification and Geometry of General Perceptual Manifolds

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.Comment: 24 pages, 12 figures, Supplementary Material

A Spectral Theory of Neural Prediction and Alignment

The representations of neural networks are often compared to those of biological systems by performing regression between the neural network responses and those measured from biological systems. Many different state-of-the-art deep neural networks yield similar neural predictions, but it remains unclear how to differentiate among models that perform equally well at predicting neural responses. To gain insight into this, we use a recent theoretical framework that relates the generalization error from regression to the spectral bias of the model activations and the alignment of the neural responses onto the learnable subspace of the model. We extend this theory to the case of regression between model activations and neural responses, and define geometrical properties describing the error embedding geometry. We test a large number of deep neural networks that predict visual cortical activity and show that there are multiple types of geometries that result in low neural prediction error as measured via regression. The work demonstrates that carefully decomposing representational metrics can provide interpretability of how models are capturing neural activity and points the way towards improved models of neural activity.Comment: First two authors contributed equally. To appear at NeurIPS 202

Linear Classification of Neural Manifolds with Correlated Variability

Understanding how the statistical and geometric properties of neural activations relate to network performance is a key problem in theoretical neuroscience and deep learning. In this letter, we calculate how correlations between object representations affect the capacity, a measure of linear separability. We show that for spherical object manifolds, introducing correlations between centroids effectively pushes the spheres closer together, while introducing correlations between the spheres' axes effectively shrinks their radii, revealing a duality between neural correlations and geometry. We then show that our results can be used to accurately estimate the capacity with real neural data.Comment: 6 pages and 5 figures in main text. 11 pages and 1 figure in supplementary materia

Learning Efficient Coding of Natural Images with Maximum Manifold Capacity Representations

The efficient coding hypothesis proposes that the response properties of sensory systems are adapted to the statistics of their inputs such that they capture maximal information about the environment, subject to biological constraints. While elegant, information theoretic properties are notoriously difficult to measure in practical settings or to employ as objective functions in optimization. This difficulty has necessitated that computational models designed to test the hypothesis employ several different information metrics ranging from approximations and lower bounds to proxy measures like reconstruction error. Recent theoretical advances have characterized a novel and ecologically relevant efficiency metric, the manifold capacity, which is the number of object categories that may be represented in a linearly separable fashion. However, calculating manifold capacity is a computationally intensive iterative procedure that until now has precluded its use as an objective. Here we outline the simplifying assumptions that allow manifold capacity to be optimized directly, yielding Maximum Manifold Capacity Representations (MMCR). The resulting method is closely related to and inspired by advances in the field of self supervised learning (SSL), and we demonstrate that MMCRs are competitive with state of the art results on standard SSL benchmarks. Empirical analyses reveal differences between MMCRs and representations learned by other SSL frameworks, and suggest a mechanism by which manifold compression gives rise to class separability. Finally we evaluate a set of SSL methods on a suite of neural predictivity benchmarks, and find MMCRs are higly competitive as models of the ventral stream.Comment: Accepted at NeurIPS 202

Unsupervised learning on spontaneous retinal activity leads to efficient neural representation geometry

Prior to the onset of vision, neurons in the developing mammalian retina spontaneously fire in correlated activity patterns known as retinal waves. Experimental evidence suggests that retinal waves strongly influence the emergence of sensory representations before visual experience. We aim to model this early stage of functional development by using movies of neurally active developing retinas as pre-training data for neural networks. Specifically, we pre-train a ResNet-18 with an unsupervised contrastive learning objective (SimCLR) on both simulated and experimentally-obtained movies of retinal waves, then evaluate its performance on image classification tasks. We find that pre-training on retinal waves significantly improves performance on tasks that test object invariance to spatial translation, while slightly improving performance on more complex tasks like image classification. Notably, these performance boosts are realized on held-out natural images even though the pre-training procedure does not include any natural image data. We then propose a geometrical explanation for the increase in network performance, namely that the spatiotemporal characteristics of retinal waves facilitate the formation of separable feature representations. In particular, we demonstrate that networks pre-trained on retinal waves are more effective at separating image manifolds than randomly initialized networks, especially for manifolds defined by sets of spatial translations. These findings indicate that the broad spatiotemporal properties of retinal waves prepare networks for higher order feature extraction