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Latent Optimal Paths by Gumbel Propagation for Variational Bayesian Dynamic Programming
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Proposes Bayesian dynamic programming (BDP), which softens optimal path problems on DAGs into Gibbs distributions over paths, uses Gumbel propagation for linear-time sampling, likelihood and KL, and applies it as BDP-VAE for end-to-end TTS and SVS.
Problem
Optimal paths in DAGs are non-differentiable (max operator) and sparse, so integrating unobserved structured alignments into neural generative models usually requires external DP aligners or multi-stage training.
Approach
Treat paths in a DAG as random variables following a Gibbs distribution with path score as energy; use Gumbel max and shift properties to show the normalizer and path marginals follow message-passing recursions, giving closed-form linear-time sampling, likelihood and KL for variational inference over latent sparse optimal paths.
Key results
- On RyanSpeech, BDPVAE-TTS reaches MCD 8.49 vs FastSpeech2 9.96, Tacotron2 11.39, Glow-TTS 8.58, and VAENAR-TTS 8.18 (VAENAR-TTS better).
- On TIMIT duration MAE, BDP-VAE achieves 3.93 at inference vs 5.69 for a random uniform-weight DTW baseline.
- Temperature alpha=1 to 1.5 gives the best inference MAE (3.89-3.93); alpha=3 degrades to 4.53.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace FastSpeech2's external-alignment setup with end-to-end BDPVAE-TTS and discrete alignment | TTS · RyanSpeech test | MCD: 9.96 ± 1.01 -> 8.49 ± 0.96 | Helped |
| replace FastSpeech2's external-alignment setup with end-to-end BDPVAE-TTS and discrete alignment | TTS · RyanSpeech test | RTF: 3.87 × 10^{-4} -> 3.00 × 10^{-4} | Hurt |
| use BDPVAE-TTS instead of VAENAR-TTS | TTS · RyanSpeech test | MCD: 8.18 ± 0.87 -> 8.49 ± 0.96 | Hurt |
| use BDPVAE-TTS instead of VAENAR-TTS | TTS · RyanSpeech test | RTF: 1.10 × 10^{-4} -> 3.00 × 10^{-4} | Hurt |
| use BDPVAE-TTS instead of Glow-TTS | TTS · RyanSpeech test | MCD: 8.58 ± 0.89 -> 8.49 ± 0.96 | Helped |
| use BDPVAE-TTS instead of Glow-TTS | TTS · RyanSpeech test | RTF: 2.87 × 10^{-4} -> 3.00 × 10^{-4} | Hurt |
| learn edge weights with BDP-VAE instead of using uniform random DTW weights | BDP-VAE · TIMIT train | phoneme duration MAE: 5.67 -> 2.92 | Helped |
| learn edge weights with BDP-VAE instead of using uniform random DTW weights | BDP-VAE · TIMIT inference | phoneme duration MAE: 5.69 ± 0.31 -> 3.93 ± 0.37 | Helped |
| set α to 0.5 rather than α=1 (ablation) | BDP-VAE · TIMIT inference | phoneme duration MAE: 3.93 ± 0.37 -> 4.40 ± 0.46 | Hurt |
| set α to 1.5 rather than α=1 (ablation) | BDP-VAE · TIMIT inference | phoneme duration MAE: 3.93 ± 0.37 -> 3.89 ± 0.35 | Helped |
| set α to 2 rather than α=1 (ablation) | BDP-VAE · TIMIT inference | phoneme duration MAE: 3.93 ± 0.37 -> 4.08 ± 0.27 | Hurt |
| set α to 3 rather than α=1 (ablation) | BDP-VAE · TIMIT inference | phoneme duration MAE: 3.93 ± 0.37 -> 4.53 ± 0.10 | Hurt |
Surprising findings
- BDPVAE-TTS has a higher MCD than the VAENAR-TTS baseline despite being end-to-end with discrete alignment, which the authors attribute to the trade-off in interpretability.
Limitations
- REINFORCE estimator for the posterior gradient may have high variance and slow convergence.
- Inference speed is slower than VAENAR-TTS due to the DP computation.
- Experiments are on small-scale datasets with known DAG structure; learning DAG structure is not evaluated.
Experimental setup
Datasets: RyanSpeech popcs TIMIT
Benchmarks: RyanSpeech popcs TIMIT
Metrics: Mel cepstral distortion real-time factor mean absolute error of phoneme duration
Baselines: FastSpeech2 Tacotron2 VAENAR-TTS Glow-TTS
Builds on: FastSpeech2 Glow-TTS VAENAR-TTS variational autoencoder soft dynamic time warping
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