# Classification and Geometry of General Perceptual Manifolds

arXiv: https://arxiv.org/abs/1710.06487 · HTML: https://mlsanity.com/paper/1710.06487/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2017.

## Built on by (most cited first, 30 of 62)

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- [Learning Efficient Coding of Natural Images with Maximum Manifold Capacity Representations](https://mlsanity.com/paper/2303.03307/) (2023)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
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- [The Geometry of Prompting: Unveiling Distinct Mechanisms of Task Adaptation in Language Models](https://mlsanity.com/paper/2502.08009/) (2025)
- [Investigating Power laws in Deep Representation Learning](https://mlsanity.com/paper/2202.05808/) (2022)
- [The Intrinsic Dimension of Prompts in Internal Representations of Large Language Models](https://mlsanity.com/paper/2501.10573/) (2025)
- [Geometric Signatures of Compositionality Across a Language Model's Lifetime](https://mlsanity.com/paper/2410.01444/) (2024)
- [Syntactic Perturbations Reveal Representational Correlates of Hierarchical Phrase Structure in Pretrained Language Models](https://mlsanity.com/paper/2104.07578/) (2021)
- [Neural Population Geometry Reveals the Role of Stochasticity in Robust Perception](https://mlsanity.com/paper/2111.06979/) (2021)
- [Interpretable statistical representations of neural population dynamics and geometry](https://mlsanity.com/paper/2304.03376/) (2023)
- [Stable Anisotropic Regularization](https://mlsanity.com/paper/2305.19358/) (2023)
- [High-dimensional learning of narrow neural networks](https://mlsanity.com/paper/2409.13904/) (2024)
- [Data Representations' Study of Latent Image Manifolds](https://mlsanity.com/paper/2305.19730/) (2023)
- [On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theory](https://mlsanity.com/paper/2412.11521/) (2024)
- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [Linear Classification of Neural Manifolds with Correlated Variability](https://mlsanity.com/paper/2211.14961/) (2022)
- [Neural population geometry and optimal coding of tasks with shared latent structure](https://mlsanity.com/paper/2402.16770/) (2024)
- [Feature Learning beyond the Lazy-Rich Dichotomy: Insights from Representational Geometry](https://mlsanity.com/paper/2503.18114/) (2025)
- [Statistical mechanics of extensive-width Bayesian neural networks near interpolation](https://mlsanity.com/paper/2505.24849/) (2025)
- [Towards an Improved Understanding and Utilization of Maximum Manifold Capacity Representations](https://mlsanity.com/paper/2406.09366/) (2024)
- [Analyzing Data-Centric Properties for Graph Contrastive Learning](https://mlsanity.com/paper/2208.02810/) (2022)
- [Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalisation](https://mlsanity.com/paper/2303.05161/) (2023)
- [Random features and polynomial rules](https://mlsanity.com/paper/2402.10164/) (2024)
- [Unveiling and Mitigating Generalized Biases of DNNs through the Intrinsic Dimensions of Perceptual Manifolds](https://mlsanity.com/paper/2404.13859/) (2024)
- [Restoring balance: principled under/oversampling of data for optimal classification](https://mlsanity.com/paper/2405.09535/) (2024)
- [What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold](https://mlsanity.com/paper/2505.11128/) (2025)

## Abstract (arXiv)

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.