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Continuity-Preserving Convolutional Autoencoders for Learning Continuous Latent Dynamical Models from Images
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Proposes continuity-preserving convolutional autoencoders (CpAEs) that regularize convolution filters to be continuous so that image-encoded latent states evolve continuously and yield more accurate latent dynamical models.
Problem
Learning continuous latent dynamical models from discrete pixel image observations of continuous trajectories; standard CNN autoencoders produce latent states that are discontinuous in time.
Approach
Formalizes a δ-continuity notion for pixel observations, proves that CNN encoders map to Lipschitz-continuous latent trajectories when their large early-layer filters are Lipschitz, and enforces this with a nonlocal (Gaussian-kernel) filter-smoothness regularizer plus VPNet-based dynamics regularization.
Key results
- On the damped pendulum, CpAE achieves VPT 99.2 and VPF 99.2 versus 50.7 and 23.3 for AE+Neural ODE.
- On the double pendulum, CpAE achieves VPT 69.1 and VPF 40.0 versus 24.3 and 0.0 for AE+Neural ODE and 11.0 for AE+HNN.
- On the swing stick, CpAE achieves VPT 57.4 and VPF 11.1 versus 13.7 for the hybrid scheme and 24.1 for AE+HNN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CpAE instead of the Hybrid scheme | Latent dynamics prediction · Damped pendulum | VPT: 95.4$\pm$15.0 -> 99.2$\pm$8.5 | Helped |
| Use CpAE instead of the Hybrid scheme | Latent dynamics prediction · Elastic pendulum | VPT: 49.5$\pm$24.2 -> 72.1$\pm$27.2 | Helped |
| Use CpAE instead of the Hybrid scheme | Latent dynamics prediction · Double pendulum | VPT: 46.8$\pm$21.4 -> 69.1$\pm$31.5 | Helped |
| Use CpAE instead of AE + HNN | Latent dynamics prediction · Swing stick | VPT: 24.1$\pm$14.5 -> 57.4$\pm$20.4 | Helped |
| Add the continuity regularizer to the standard autoencoder (ablation) | Single-hidden-layer 48 × 48 CNN autoencoder followed by Neural ODE · Circular motion | latent-state continuity | Helped |
| Add conventional L2 regularization instead of the continuity regularizer (ablation) | Single-hidden-layer 48 × 48 CNN autoencoder followed by Neural ODE · Circular motion | latent-state continuity | Hurt |
Surprising findings
- Standard CNN encoders with random filters produce discontinuous latent states with probability one in the uniform-motion analysis.
- Adding a conventional L2 regularizer on filters does not yield continuous latent states, while the nonlocal continuity regularizer does.
Limitations
- Continuity theory is restricted to rigid body motion in a 2D plane; non-rigid and 3D-projected motion not covered.
- Performance declines on visually complex data such as swing stick with complex backgrounds and noise.
- Uses the simplest filter-regularization approach; more effective alternatives such as hypernetworks remain unexplored.
Experimental setup
Datasets: damped pendulum (simulated) elastic pendulum (simulated) double pendulum swing stick
Benchmarks: damped pendulum elastic pendulum double pendulum swing stick
Metrics: Valid Prediction Time (VPT) Valid Prediction Frequency (VPF) Pixel MSE
Baselines: AE+Neural ODE AE+HNN AE+SympNet Hybrid scheme with neural state variables
Builds on: Neural ODE VPNets Hamiltonian Neural Networks SympNets neural state variables
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