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Dequantified Diffusion-Schrödinger Bridge for Density Ratio Estimation
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Proposes D^3RE, a density ratio estimation framework built on diffusion-bridge interpolants with Gaussian dequantization and optimal-transport rearrangement (Sinkhorn) to handle density and support chasms and stabilize time scores.
Problem
Density ratio estimation fails when the two distributions differ strongly (density-chasm) or have little support overlap (support-chasm), and prior interpolant-based time scores diverge near boundaries t->1, causing instability.
Approach
Replace deterministic interpolants with a Brownian-bridge-based diffusion bridge interpolant (DBI), add Gaussian dequantization (GD) of boundary densities to bound the time score (DDBI), and pair samples via entropic OT/Sinkhorn rearrangement (OTR) to obtain a Schrödinger-bridge-like interpolant (DSBI).
Key results
- On MNIST with Gaussian noise, D^3RE (DSBI) reaches 1.293 BPD versus 1.33 for DRE-infinity and 1.96 for NCE.
- On MNIST with Copula noise, DSBI achieves 1.170 BPD versus 1.21 for DRE-infinity.
- The paper reports that DSBI converges faster and stays closer to ground-truth MI than DDBI and DRE-infinity in high dimension (d=120), with OTR reducing NFE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace DRE-$\infty$ with $\text{D}^3\text{RE}$ (DSBI) | MNIST density estimation · MNIST, Gaussian noise | BPD: 1.33 -> 1.293 | Helped |
| replace DRE-$\infty$ with $\text{D}^3\text{RE}$ (DSBI) | MNIST density estimation · MNIST, Copula noise | BPD: 1.21 -> 1.170 | Helped |
| replace DRE-$\infty$ with $\text{D}^3\text{RE}$ (DSBI) | MNIST density estimation · MNIST, RQ-NSF noise | BPD: 1.09 -> 1.066 | Helped |
| add OTR to DDBI to form DSBI (ablation) | $\text{D}^3\text{RE}$ · MNIST, Gaussian noise | BPD: 1.297 -> 1.293 | Helped |
| add OTR to DDBI to form DSBI (ablation) | $\text{D}^3\text{RE}$ · MNIST, Copula noise | BPD: 1.193 -> 1.170 | Helped |
| add OTR to DDBI to form DSBI (ablation) | $\text{D}^3\text{RE}$ · MNIST, RQ-NSF noise | BPD: 1.072 -> 1.066 | Helped |
| vary bridge noise strength $\gamma^2$ (ablation) | $\text{D}^3\text{RE}$ density estimation · 2-D synthetic datasets | density estimation quality | Mixed |
| use smaller $\gamma^2$ values ($\leq 0.01$) (ablation) | $\text{D}^3\text{RE}$ mutual information estimation · MI estimation, dimensions $d = \{40, 80, 120\}$ | convergence speed | Mixed |
| use moderate $\gamma^2 = 0.1$ instead of smaller or larger values (ablation) | $\text{D}^3\text{RE}$ mutual information estimation · MI estimation, dimensions $d = \{40, 80, 120\}$ | stability | Helped |
| add Gaussian dequantization (ablation) | DDBI and DSBI · Eight 2-D synthetic datasets | density estimation quality | Helped |
| add OTR (ablation) | DRE interpolation · Density estimation on checkerboard | intermediate distribution alignment | Helped |
| use DSBI with OTR rather than DDBI without OTR (ablation) | $\text{D}^3\text{RE}$ · MI estimation, dimensions $d=80, 120$ | convergence speed | Helped |
Surprising findings
- On MNIST with RQ-NSF noise, DDBI matches DRE-infinity+OTR (1.072 BPD) and DSBI gives only a marginal gain (1.066), with NCE, TRE and DRE-infinity all at 1.09.
Limitations
- Theoretical bounds rely on strong assumptions (C^2 densities, Lipschitz conditional scores, inf q0 > 0), and the paper's own proofs contain loose steps.
- Gains over DRE-infinity on MNIST are small (e.g., 1.066 vs 1.072 for RQ-NSF), and experiments are limited to 2-D toy data, MNIST EBMs, and Gaussian MI.
- Dequantization and noise factor gamma require tuning; overly small or large gamma degrades results according to the ablation.
Experimental setup
Datasets: swissroll circles rings moons 8gaussians pinwheel 2spirals checkerboard MNIST
Benchmarks: MNIST energy-based density estimation (BPD) 2-D synthetic density estimation mutual information estimation on correlated Gaussians (d=40,80,120)
Metrics: bits per dimension mutual information estimate convergence NFE
Baselines: DRE-infinity TRE NCE DRE-infinity+OTR DI+OTR
Builds on: DRE-infinity telescoping ratio estimation denoising diffusion bridge models Schrödinger bridge Sinkhorn algorithm
Closest papers by meaning
- Diffusion Bridge Implicit Models (ICLR 2025 · similarity 0.87)
- Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr\"{o}dinger Bridge Estimation (NeurIPS 2025 · similarity 0.87)
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.87)
- Exponential Convergence Guarantees for Iterative Markovian Fitting (NeurIPS 2025 · similarity 0.87)
- Variational Schrödinger Diffusion Models (ICML 2024 · similarity 0.86)
- Improved sampling via learned diffusions (ICLR 2024 · similarity 0.86)
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