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Distributional Autoencoders Know the Score
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Gives exact theory for Distributional Principal Autoencoders (DPA): optimal encoder level sets satisfy a closed-form score-geometry identity, and extraneous latents beyond intrinsic dimension are conditionally independent of the data; validated on Gaussian toy data and Müller–Brown MFEP recovery.
Problem
Theoretical understanding of why DPA (energy-score-based distributional autoencoder) disentangles factors of variation and learns intrinsic dimension, and whether it can recover the data distribution's score and manifold dimension simultaneously.
Approach
Derives, via first-variation of the energy-score-based DPA objective, a pointwise balance equation (for beta=2) linking the normal components of optimal encoder level sets to the data score, and proves that for a manifold approximable by the encoder, extraneous latent coordinates are conditionally independent of X given the informative ones.
Key results
- Score-alignment cosine similarity between level-set normal vectors and data score is 1.00 (mean abs, std ~0) for standard Normal and Gaussian mixture.
- On Müller–Brown, DPA achieves best MFEP agreement, e.g., Chamfer 0.262 ± 0.053 vs 0.387 for AE and 0.515 for VAE.
- Extraneous latents on toy manifolds show R^2 ≈ 0.999 predicting from Z and near-zero ID-drop; conditional randomization test on Gaussian line gives KS p=0.822.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DPA instead of AE | Müller–Brown MFEP parameterization · Müller–Brown MFEP | Chamfer: 0.387 ± 0.113 -> 0.262 ± 0.053 | Helped |
| Use DPA instead of VAE | Müller–Brown MFEP parameterization · Müller–Brown MFEP | Chamfer: 0.515 ± 0.469 -> 0.262 ± 0.053 | Helped |
| Use DPA instead of β-VAE | Müller–Brown MFEP parameterization · Müller–Brown MFEP | Chamfer: 0.450 ± 0.288 -> 0.262 ± 0.053 | Helped |
| Use DPA instead of β-TCVAE | Müller–Brown MFEP parameterization · Müller–Brown MFEP | Chamfer: 0.377 ± 0.077 -> 0.262 ± 0.053 | Helped |
| Use DPA instead of β-TCVAE | Müller–Brown MFEP parameterization · Müller–Brown MFEP | Path-to-MFEP distance: 0.164 ± 0.101 -> 0.236 ± 0.047 | Hurt |
| Fit DPA to samples from the known distribution | DPA encoder level sets · Standard Normal and Gaussian Mixture | Mean absolute cosine similarity: 1.00 | Helped |
| Use DPA’s first component as the MFEP parameterization | DPA · Müller–Brown MFEP | Parameterization distance: 0.00 ± 0.00 | Helped |
| Train DPA on manifold datasets with extraneous latents | DPA extraneous latents · Gaussian line, Parabola, Exponential, Helix slice, Grid sum, S-curve, and Swiss-roll (3D) | R² of regressing U on Z: 0.9997; 0.9997; 0.9996; 0.9995; 0.9986; 0.9996; 0.9872 | Helped |
| Test conditional independence of the extraneous latent given the informative latent | DPA on Gaussian line · Gaussian line | Conditional randomization test p-value: p=0.822 | No clear effect |
Surprising findings
- Extraneous-latent determinism holds empirically even with beta=2, outside the theoretical beta in (0,2) requirement.
- Level-set score alignment is essentially perfect (cosine ~1.00) in the reported examples.
Limitations
- Results hold at the population optimum; finite-sample and optimization errors can cause deviations.
- Requires full-rank encoder Jacobian on level sets and sufficient encoder expressiveness; silent otherwise.
- Theory uses beta in (0,2) for decoder uniqueness, while the explicit score identity uses beta=2 where decoder uniqueness is not guaranteed.
Experimental setup
Datasets: Standard Normal (synthetic) Gaussian mixture (synthetic) Müller–Brown potential Boltzmann samples Gaussian line Parabola Exponential Helix slice Grid sum S-curve Swiss-roll (3D)
Benchmarks: Müller–Brown MFEP parameterization Gaussian examples score alignment toy manifold determinism diagnostics
Metrics: absolute cosine similarity Chamfer distance Hausdorff distance R^2 intrinsic dimension drop conditional entropy KS test p-value
Baselines: AE VAE beta-VAE beta-TCVAE
Builds on: Distributional Principal Autoencoder Engression energy score denoising autoencoders beta-VAE
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