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DiffATS: Diffusion in Aligned Tensor Space
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Proposes DiffATS, which trains diffusion models on Orthogonal-Procrustes-aligned Tucker factors (matrix and tensor Grassmannian primitives) for autoencoder-free generation of images, videos and PDE trajectories.
Problem
Diffusion over high-resolution spatiotemporal tensors is computationally expensive, and Tucker-based parametrizations are degenerate (rotational gauge ambiguity), which hurts diffusion training; pretrained compression autoencoders are often unavailable for scientific data.
Approach
Resolve Tucker's orthogonal gauge ambiguity by aligning each factor matrix to a medoid anchor via orthogonal Procrustes, yielding canonical matrix/tensor Grassmannian primitives that are compact and decodable by multilinear reconstruction; prove the primitive map is a homeomorphism under an anchor-overlap condition and diffuse in that space.
Key results
- On CelebA-HQ, DiffATS reaches FID 23.57 (Inception-V3) and 381.58 (DINOv2), beating AvgPool (32.89 / 559.67) and DCTDiff (52.22 / 787.11).
- On Moving MNIST, DiffATS attains FVD 415.42 versus 487.82 for AvgPool among autoencoder-free baselines.
- On 1-d reaction-diffusion, relative l2 error is 0.0085 versus 0.0713 for the best baseline (DCTDiff), roughly 9x lower.
Surprising findings
- Factor-space augmentation severely degrades generation (FID 339.01 vs 23.57 on CelebA-HQ), and unaligned data-dependent bases give FID 76.94.
Limitations
- Theoretical guarantees rely on anchor-overlap conditions; authors note future work on multi-anchor alignment is needed to relax assumptions.
- Does not aim to match state-of-the-art image/video models that use pretrained autoencoders; image and video results are on low-resolution-adjacent benchmarks only.
Experimental setup
Datasets: CelebA-HQ Moving MNIST 1-d Burgers 1-d reaction-diffusion 2-d Burgers 2-d Karman vortex street
Benchmarks: CelebA-HQ Moving MNIST 1-d Burgers 1-d reaction-diffusion 2-d Burgers 2-d Karman vortex street
Metrics: FID (Inception-V3) FID (DINOv2) FVD relative l2 error
Baselines: DCTDiff SDIFT FNO AvgPool
Builds on: DDPM DDIM DiT Tucker decomposition (HOOI) DCTDiff
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