Fast Trainable Multilinear Bases for Image Compression

2026-07-26

AI Summary

Traditional image compression relies on fixed orthogonal bases such as the discrete cosine transform (DCT), which are ill-suited to the underlying data distribution and thus limit compression efficiency. This work proposes a trainable isometric multilinear basis, introducing for the first time tensor networks from quantum many-body theory into image compression. By employing block-wise multilinear transforms optimized via Riemannian optimization over the Stiefel manifold of unitary matrices, the method achieves data-adaptive basis learning while preserving near-linear computational complexity, exact invertibility, and an extremely low parameter count. Experiments demonstrate consistent superiority over DCT on both natural images and the Quick Draw dataset; compared to JPEG’s 8×8 DCT, the proposed approach reduces storage requirements by approximately 20% at equivalent reconstruction quality.

Abstract

The Discrete Fourier Transform, the Discrete Cosine Transform, and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: they run in near-linear time (linear up to a polylogarithmic factor), they are exactly invertible, and they carry few to no parameters. In this work, we generalize these bases to isometric multilinear bases, allowing a small number of extra parameters, polylogarithmic in the image size, while preserving all three properties. Given an image dataset, we develop a systematic framework that searches this family for the basis compressing the dataset most effectively: the basis is parameterized as an isometric tensor network, inspired by quantum many-body theory, and trained with Riemannian optimization on the manifold of unitary matrices. Across natural photographs and line drawings, the trained bases consistently improve on their fixed, non-parametric counterparts. On Quick Draw line-drawing compression, they store images in roughly $20\%$ fewer bytes than JPEG's $8 \times 8$ block cosine transform at the same reconstruction quality.

Problem

• image compression
• multilinear bases
• isometric transforms
• learned representations
• tensor networks

Innovation

• isometric multilinear bases
• tensor network
• Riemannian optimization
• image compression
• learnable transform