Bayesian Tensor Decomposition with Diffusion Model Prior

Direct diffusion modeling of high-resolution spatiotemporal fields is computationally challenging. Parameter-efficient primitives address this by representing high-dimensional data with a compact set of parameters. In this paper, we construct data-dependent tensor primitives without pretrained compression autoencoders. Our construction starts from Tucker decomposition, which captures low-rank multilinear structure through a core tensor and mode-wise factors. However, Tucker factors are non-unique: the same tensor can be represented by different rotated factors, which complicates generative modeling. We address this issue with orthogonal Procrustes (OP) alignment. Specifically, we select medoid anchor matrices from the data and align the factor matrices to resolve the gauge ambiguity. This yields matrix Grassmannian primitives and tensor Grassmannian primitives that are compact, data-adaptive, and directly decodable by explicit multilinear reconstruction. Theoretically, we prove that the proposed primitive maps are homeomorphisms between low-rank tensors and their corresponding primitive spaces, certifying that the representations are non-degenerate and topologically faithful. Building on these primitives, we propose Diffusion in Aligned Tensor Space (DiffATS), a generative framework that trains diffusion models directly on aligned tensor primitives. Across images, videos, and PDE solutions, DiffATS achieves strong unconditional and conditional generation performance while compressing original data by $3.9\times$ to $210\times$, without relying on any pretrained deep compression autoencoders.

Recently, tensor decompositions are prevalent for multi-dimensional image representation, which learn the instance-specific structure of each image from scratch. However, tensor decompositions neglect the common structure across different images, leading to limited semantic modeling capability, high computational cost, and a large number of learnable parameters. To address this challenge, we suggest the first pre-trained low-rank tensor decomposition (PLTD) framework, which organically integrates the pre-trained large vision model into the classical tensor decomposition framework. Beyond the shallow and untrained deep tensor decomposition, the suggested PLTD achieves an unprecedented balance among higher recovery fidelity, fewer learnable parameters, and smaller carbon footprint. Specifically, PLTD factorizes the target tensor into a latent tensor and a learnable transform that maps the latent tensor back to the original data domain. The latent tensor consists of two indispensable and complementary terms, i.e., a fixed pre-trained latent tensor and a learnable low-rank latent tensor. The fixed pre-trained latent tensor is distilled from a pre-trained large vision model (i.e., DINOv3) to capture the common structure of the target tensor, while the learnable low-rank latent tensor characterizes the instance-specific structure of the target tensor. To examine the potential of PLTD, we develop the corresponding multi-dimensional image recovery model and theoretically justify the advantages of this framework. Additionally, we discuss the connections between PLTD and classical tensor decomposition frameworks. Extensive experiments on multi-dimensional image recovery demonstrate that PLTD consistently achieves superior performance compared with state-of-the-art methods.

Obtaining a reliable estimate of the joint probability mass function (PMF) of a set of random variables from observed data is a significant objective in statistical signal processing and machine learning. Modelling the joint PMF as a tensor that admits a low-rank canonical polyadic decomposition (CPD) has enabled the development of efficient PMF estimation algorithms. However, these algorithms require the rank (model order) of the tensor to be specified beforehand. In real-world applications, the true rank is unknown. Therefore, an appropriate rank is usually selected from a candidate set either by observing validation errors or by computing various likelihood-based information criteria, a procedure that could be costly in terms of computational time or hardware resources, or could result in mismatched models which affect the model accuracy. This paper presents a novel Bayesian framework for estimating the low-rank components of a joint PMF tensor and simultaneously inferring its rank from the observed data. We specify a Bayesian PMF estimation model and employ appropriate prior distributions for the model parameters, allowing the rank to be inferred without cross-validation.We then derive a deterministic solution based on variational inference (VI) to approximate the posterior distributions of various model parameters. Numerical experiments involving both synthetic data and real classification and item recommendation data illustrate the advantages of our VI-based method in terms of estimation accuracy, automatic rank detection, and computational efficiency.