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Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders

arXiv 2408.16138 ↗First posted 28 Aug 2024cs.LG math.DG stat.MLcited by 0

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Abstract (arXiv)

Conformal Autoencoders are a neural network architecture that imposes orthogonality conditions between the gradients of latent variables to obtain disentangled representations of data. In this work we show that orthogonality relations within the latent layer of the network can be leveraged to infer the intrinsic dimensionality of nonlinear manifold data sets (locally characterized by the dimension of their tangent space), while simultaneously computing encoding and decoding (embedding) maps. We outline the relevant theory relying on differential geometry, and describe the corresponding gradient-descent optimization algorithm. The method is applied to several data sets and we highlight its applicability, advantages, and shortcomings. In addition, we demonstrate that the same computational technology can be used to build coordinate invariance to local group actions when defined only on a (reduced) submanifold of the embedding space.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 15 of its 51 references that are in the map. First posted on arXiv 28 Aug 2024.

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References · 15

  1. 2019Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networkscited by 290
  2. 2017Sobolev Training for Neural Networkscited by 136
  3. 2022Equivariance with Learned Canonicalization Functionscited by 106
  4. 2020Training robust neural networks using Lipschitz boundscited by 98
  5. 2018Variational Autoencoders Pursue PCA Directions (by Accident)cited by 84
  6. 2020On Implicit Regularization in β-VAEscited by 34
  7. 2019Chart Auto-Encoders for Manifold Structured Datacited by 32
  8. 2021Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equationscited by 28
  9. 2019Improved Surrogates in Inertial Confinement Fusion with Manifold and Cycle Consistenciescited by 14
  10. 2021On the Parameter Combinations That Matter and on Those That do Notcited by 12
  11. 2022Group invariant machine learning by fundamental domain projectionscited by 11
  12. 2023Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML eracited by 7
3 more
  1. 2022Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reductioncited by 5
  2. 2024Gamma-VAE: Curvature regularized variational autoencoders for uncovering emergent low dimensional geometric structure in high dimensional datacited by 2
  3. 2023Gentlest ascent dynamics on manifolds defined by adaptively sampled point-cloudscited by 1

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