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 require classifying the manifolds...
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| Format: | Article |
| Language: | English |
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American Physical Society
2018
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| Online Access: | http://hdl.handle.net/1721.1/116853 |
| _version_ | 1826193628401238016 |
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| author | Chung, SueYeon Lee, Daniel D. Sompolinsky, Haim |
| author2 | Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences |
| author_facet | Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences Chung, SueYeon Lee, Daniel D. Sompolinsky, Haim |
| author_sort | Chung, SueYeon |
| collection | MIT |
| description | 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 require 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. We show how special anchor points on the manifolds can be used to define novel geometrical measures of radius and dimension, which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including ℓ_{2} ellipsoids prototypical of strictly convex manifolds, ℓ_{1} balls representing polytopes with finite samples, and ring manifolds exhibiting nonconvex continuous structures that arise from modulating a continuous degree of freedom. The effects of label sparsity on the classification capacity of general manifolds are elucidated, displaying a universal scaling relation between label sparsity and the 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 perceptual neuronal responses as well as to artificial deep networks trained for object recognition tasks. |
| first_indexed | 2024-09-23T09:42:09Z |
| format | Article |
| id | mit-1721.1/116853 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:42:09Z |
| publishDate | 2018 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/1168532022-09-30T16:18:49Z Classification and Geometry of General Perceptual Manifolds Chung, SueYeon Lee, Daniel D. Sompolinsky, Haim Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences Chung, SueYeon 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 require 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. We show how special anchor points on the manifolds can be used to define novel geometrical measures of radius and dimension, which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including ℓ_{2} ellipsoids prototypical of strictly convex manifolds, ℓ_{1} balls representing polytopes with finite samples, and ring manifolds exhibiting nonconvex continuous structures that arise from modulating a continuous degree of freedom. The effects of label sparsity on the classification capacity of general manifolds are elucidated, displaying a universal scaling relation between label sparsity and the 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 perceptual neuronal responses as well as to artificial deep networks trained for object recognition tasks. 2018-07-09T14:34:16Z 2018-07-09T14:34:16Z 2018-07 2017-10 2018-07-06T18:00:18Z Article http://purl.org/eprint/type/JournalArticle 2160-3308 http://hdl.handle.net/1721.1/116853 Chung, SueYeon et al. "Classification and Geometry of General Perceptual Manifolds." Physical Review X 8, 3 (July 2018): 031003 en http://dx.doi.org/10.1103/PhysRevX.8.031003 Physical Review X Creative Commons Attribution http://creativecommons.org/licenses/by/3.0 application/pdf American Physical Society American Physical Society |
| spellingShingle | Chung, SueYeon Lee, Daniel D. Sompolinsky, Haim Classification and Geometry of General Perceptual Manifolds |
| title | Classification and Geometry of General Perceptual Manifolds |
| title_full | Classification and Geometry of General Perceptual Manifolds |
| title_fullStr | Classification and Geometry of General Perceptual Manifolds |
| title_full_unstemmed | Classification and Geometry of General Perceptual Manifolds |
| title_short | Classification and Geometry of General Perceptual Manifolds |
| title_sort | classification and geometry of general perceptual manifolds |
| url | http://hdl.handle.net/1721.1/116853 |
| work_keys_str_mv | AT chungsueyeon classificationandgeometryofgeneralperceptualmanifolds AT leedanield classificationandgeometryofgeneralperceptualmanifolds AT sompolinskyhaim classificationandgeometryofgeneralperceptualmanifolds |