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Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differences
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Proposes stNCE, a binary-classification framework that learns energy-based models of stochastic-interpolant joint densities (x,t) via joint spatiotemporal differences, unifying temporal and spatial NCE/score-matching methods and yielding new objectives competitive on toy, MNIST/ImageNet64, and molecular data.
Problem
Learning energy-based models over data and corruption time from samples; existing temporal (NCE-style) methods fail under support mismatch and spatial (score-matching-style) methods fail under multimodality.
Approach
Learn the energy from joint spatiotemporal differences E(x',t')-E(x,t) via one logistic classification task whose perturbation kernel jointly perturbs space and time along a forward-reverse trajectory so pairs stay on the data manifold; many existing methods are special cases.
Key results
- On the 784-dim MNIST mixture, stNCE-s achieves MSE 121.6 and NormNLL -680.99 versus tNCE's 2e5 and -606.76, the only NCE-based method that learns the mixture well.
- On MNIST, stNCE-s+DSM reaches 1.00 bits/dim test NLL, competitive with FFJORD (0.99) and MintNet (0.98) and better than CTSM-v (1.03).
- On ImageNet64, stNCE-s+DSM achieves 2.94 bits/dim, beating Dual SM (3.36) and FM (3.31) and approaching TarFlow (2.99).
Surprising findings
- Spatial score matching variants with multi-level corruption still fail on the simple multimodal Gaussian mixture, contrary to a common belief.
- Standard NCE-based stNCE-s alone reaches 1.61 bpd on MNIST, whereas adding DSM is needed to reach state-of-the-art.
Limitations
- NCE-trained models can recover energies accurately but learn scores poorly, which hurts sampling.
- Mixture variant stNCE-m is slower due to naive summation of two losses.
- Molecular experiments show stNCE-s alone is behind FPE on JS and PMF, requiring DSM to be competitive.
Experimental setup
Datasets: MNIST ImageNet64 Alanine dipeptide MD data Chignolin MD data
Benchmarks: MNIST ImageNet64 Alanine dipeptide Chignolin MNIST Gaussian mixture
Metrics: MSE Ratio NormMSE NormNLL NLL in bits per dimension JS divergence PMF
Baselines: tNCE tCNCE DSM CTSM RNE Dual SM FPE TSM Flow Matching Glow TarFlow
Builds on: Noise-Contrastive Estimation Conditional NCE Dual Score Matching Telescoping density ratio estimation Time score matching
Closest papers by meaning
- Iterated Denoising Energy Matching for Sampling from Boltzmann Densities (ICML 2024 · similarity 0.88)
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