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Mathematical Physics Seminar

Geometry of Concept Manifolds

Haim Sompolinsky, Hebrew University and Harvard University

Location:  zoom
Date & time: Wednesday, 05 May 2021 at 10:45AM - 11:45AM

Abstract: A remarkable capability of animals and humans is invariant perception - the ability to extract abstract features, such as the identity of objects, from high-dimensional sensory streams, which are sensitive to variation in many nuisance physical variables (e.g., in the case of visual objects-orientation, pose, scale, location, and intensity). How neuronal systems give rise to invariant object classification and recognition is a fundamental problem in brain theory, as well as in machine learning. A related outstanding cognitive skill is the ability to learn new concepts or categories based on only a few examples (a property known as few-shot learning).
I will address these challenges by introducing the notion of a concept manifold, i.e., the set of all population response vectors induced by different physical instantiations of a single concept. I will discuss a recent statistical mechanics theory of concept manifolds that we developed. The theory derived new geometric measures, manifold radius and dimension, and established the relationship between them and invariant perception, focusing on two classes of tasks:
first, the capacity to categorize a large number of familiar concepts, and second, the ability to perform few-shot learning of new concepts.
We have applied our theory to illuminate the principles underlying changes in object representations across layers of artificial Deep Convolutional Neural Networks. Electrical recordings from neurons in several stages of visual cortex, responding to object and face stimuli, have been similarly analyzed, shedding light on the correspondence between artificial deep networks and the visual hierarchy in the primate brain.
Overall, this work extends the reach of statistical mechanics to important problems in information processing of high dimensional data with rich, naturalistic statistical structures.