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Continuity-Preserving Convolutional Autoencoders for Learning Continuous Latent Dynamical Models from Images

Aiqing Zhu, Yuting Pan, Qianxiao Li

TL;DR

The paper tackles learning continuous latent dynamical models from sequences of image observations, where pixel data are discrete yet the underlying dynamics are continuous. It introduces continuity-preserving autoencoders (CpAEs) that enforce δ-continuity by constraining early CNN filters to be Lipschitz and by employing a nonlocal regularizer to promote smooth filter changes, ensuring that latent states $Z$ evolve coherently with the true dynamics. A theoretical result connects δ-continuity of the encoder to the continuity of the latent trajectories, and the model is trained in two steps: (i) learn a CpAE with a continuity-aware loss, and (ii) fit a continuous latent dynamic model (e.g., Neural ODE) for $Z$. Empirical evaluation on synthetic and real-world motion datasets shows CpAEs yield continuous latent representations and improved predictive performance over standard autoencoders and several baselines, with potential for future improvements via hypernetworks and transformer-based architectures.

Abstract

Continuous dynamical systems are cornerstones of many scientific and engineering disciplines. While machine learning offers powerful tools to model these systems from trajectory data, challenges arise when these trajectories are captured as images, resulting in pixel-level observations that are discrete in nature. Consequently, a naive application of a convolutional autoencoder can result in latent coordinates that are discontinuous in time. To resolve this, we propose continuity-preserving convolutional autoencoders (CpAEs) to learn continuous latent states and their corresponding continuous latent dynamical models from discrete image frames. We present a mathematical formulation for learning dynamics from image frames, which illustrates issues with previous approaches and motivates our methodology based on promoting the continuity of convolution filters, thereby preserving the continuity of the latent states. This approach enables CpAEs to produce latent states that evolve continuously with the underlying dynamics, leading to more accurate latent dynamical models. Extensive experiments across various scenarios demonstrate the effectiveness of CpAEs.

Continuity-Preserving Convolutional Autoencoders for Learning Continuous Latent Dynamical Models from Images

TL;DR

The paper tackles learning continuous latent dynamical models from sequences of image observations, where pixel data are discrete yet the underlying dynamics are continuous. It introduces continuity-preserving autoencoders (CpAEs) that enforce δ-continuity by constraining early CNN filters to be Lipschitz and by employing a nonlocal regularizer to promote smooth filter changes, ensuring that latent states evolve coherently with the true dynamics. A theoretical result connects δ-continuity of the encoder to the continuity of the latent trajectories, and the model is trained in two steps: (i) learn a CpAE with a continuity-aware loss, and (ii) fit a continuous latent dynamic model (e.g., Neural ODE) for . Empirical evaluation on synthetic and real-world motion datasets shows CpAEs yield continuous latent representations and improved predictive performance over standard autoencoders and several baselines, with potential for future improvements via hypernetworks and transformer-based architectures.

Abstract

Continuous dynamical systems are cornerstones of many scientific and engineering disciplines. While machine learning offers powerful tools to model these systems from trajectory data, challenges arise when these trajectories are captured as images, resulting in pixel-level observations that are discrete in nature. Consequently, a naive application of a convolutional autoencoder can result in latent coordinates that are discontinuous in time. To resolve this, we propose continuity-preserving convolutional autoencoders (CpAEs) to learn continuous latent states and their corresponding continuous latent dynamical models from discrete image frames. We present a mathematical formulation for learning dynamics from image frames, which illustrates issues with previous approaches and motivates our methodology based on promoting the continuity of convolution filters, thereby preserving the continuity of the latent states. This approach enables CpAEs to produce latent states that evolve continuously with the underlying dynamics, leading to more accurate latent dynamical models. Extensive experiments across various scenarios demonstrate the effectiveness of CpAEs.

Paper Structure

This paper contains 29 sections, 8 theorems, 62 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Learning latent dynamical models using autoencoders
  4. Continuity-Preserving Autoencoder
  5. Mathematical formulation
  6. Modeling of motion
  7. Why standard CNN autoencoders fail
  8. Quantifying continuity of CNN autoencoders
  9. Method to preserve continuity of CNN autoencoders
  10. Experiments
  11. Continuity of latent states
  12. Simulation data
  13. Real-world data
  14. Summary
  15. Appendix
  16. ...and 14 more sections

Key Result

Theorem 3.1

Assume that the underlying dynamical system is a rigid body motion (eq:ode2) on a two-dimensional plane. If Assumption ass:eva1 hold, let $c_{\mathcal{W}}$ be constants satisfying and if $s_l=2$ for $l=1, \cdots,L^*-1$, then for any $z_1=(z_1^t, z_1^r), z_2=(z_2^t, z_2^r) \in \mathcal{Z}$, we have Here $C$ is a constant independent of $\delta$ and $z$.

Figures (14)

  • Figure 1: Illustration of pixel observations of continuous motion. A single pixel white square, initially located at the leftmost position, moves uniformly to the right against a black background (plotted in gray for clarity). Its position is recorded at each pixel step. This translational motion results in pixel coordinates that first increase by one, then decrease by one, and finally remain constant. As an illustrative example, we show the evolution of the value at the third pixel position.
  • Figure 2: Illustration of the mathematical formulation. The continuous dynamics of the system are captured in pixel form. We learn latent dynamical models by encoding this recorded pixel data.
  • Figure 3: Illustration of discrete nature of pixel observations for continuous motion. Considering a motion similar to that depicted in Figure \ref{['fig:discrete_env']}, we assume that the object occupies a very small volume and its motion is recorded in continuous time periods. The left side illustrates the underlying motion of the object, while the middle side shows the evolution of pixel values during the recording process. The right side depicts the observed motion derived from these pixel values, reflecting the discrete nature of pixel observations.
  • Figure 4: (A) Illustration of convolution operation. The red boxes represent the filter of size $\mathcal{O}(1)$ or $\mathcal{O}(1/\delta)$. The blue box represents the object. The solid line indicates its initial position, while the dashed line represents its position after motion. (B) The variation of latent states divided by $\Delta t$ for the two-body system, where the encoder is a one-layer CNN with parameters uniformly sampled from $[-1,1]$. The shaded region represents one standard deviation. (C) Examples of motion where the positions of the objects after variation only partially overlap with their positions before variation.
  • Figure 5: The latent states and the corresponding learned dynamics derived from them
  • ...and 9 more figures

Theorems & Definitions (16)

  • Definition 3.1
  • Theorem 3.1
  • Theorem A.1
  • Definition A.1
  • Lemma A.1
  • proof
  • Corollary A.1
  • Corollary A.2
  • proof
  • Lemma A.2
  • ...and 6 more