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.
Low-rank tensor decomposition (TD) is usually effective on clean, fully observed data, but it often degrades under severe missingness or noise. Low-rankness is itself a useful but limited structural prior, and additional handcrafted priors (e.g., sparsity or smoothness) still fall short of capturing the rich statistics of real-world data. To compensate for this weak inductive bias under heavy corruption, one would like to inject a learned, data-driven prior; however, the state-of-the-art diffusion models are not readily compatible with current TD and tractable posterior inference. To address these challenges, we introduce DiffBCP, a hybrid-prior Bayesian CP decomposition framework that couples a cumulative shrinkage process prior over the CP factors for automatic rank selection with an off-the-shelf pre-trained diffusion model as an implicit data prior on the reconstructed tensor. To make posterior inference tractable despite the coupling among the likelihood, low-rank constraint, and diffusion prior, we develop a split Gibbs sampler: CP factors admit conjugate updates, while the diffusion block is sampled via low-rank-guided denoising. A noise-adaptive coupling schedule further reduces sensitivity to hand-tuned annealing. Experiments on image inpainting and denoising, including high-resolution out-of-distribution images, show consistent gains over Bayesian, nonlinear, and plug-and-play TD baselines.
Tensor singular value decomposition (T-SVD), which is built upon the tensor-tensor product (t-product), has emerged as a powerful tool for processing high-dimensional visual data such as color images and videos. However, the standard t-product imposes strict dimensional compatibility constraints. Although extensions based on the semi-tensor product (STP) relax this restriction, their single-term formulations still suffer from limited approximation accuracy. Moreover, these deterministic methods incur high computational costs when processing large-scale tensor data. To address these issues, this paper introduces a novel semi-tensor product for third-order tensors under the t-product framework induced by arbitrary invertible linear transforms. The resulting tensor semi-tensor product breaks the rigid dimension matching requirement of the standard t-product, while retaining the closed-form property of T-SVD. Based on this construction, we develop a multi-term semi-tensor product singular value decomposition (MSTP-SVD), which integrates multiple orthogonal decomposition terms to significantly improve low-rank approximation accuracy compared with single-term schemes. To reduce the computational cost of multi-term modeling, we incorporate randomized projection and power iteration techniques into the MSTP-SVD framework, yielding an accelerated multi-term randomized semi-tensor product SVD (MRSTP-SVD) algorithm that achieves a balance between reconstruction accuracy and computational efficiency. Experiments on image and video compression and completion tasks demonstrate the effectiveness of the proposed method.
Xingchen Xiao (School of Mathematics and Statistics, Southwest University, Chongqing +13
Large language models have driven recent progress in language and multimodal AI, yet pre-training them at scale is prohibitively expensive. Low-rank pre-training, which factorizes each weight matrix into a rank-r product to reduce both parameters and FLOPs, is a promising response but typically lags full-rank training in quality. We propose Duplicated Latent Residual (DLR), a training-only, parameter-free, foldable plug-in for low-rank pre-training. DLR augments the standard low-rank output Bz with a fixed structured residual alpha/sqrt(K) * Expand_K(z) that replicates each latent coordinate K = ceil(d_out/r) times across the output. With alpha fixed, DLR adds zero learnable parameters per layer; after training, it is absorbed into the up-projection in closed form, B* = B + alpha/sqrt(K) R^T, so deployment parameter count, FLOPs and memory match the underlying low-rank backbone exactly. Across LLaMA models from 60M to 7B parameters, DLR strengthens low-rank pre-training on C4 validation perplexity in most settings, with the clearest gains at 130M and above; folded checkpoints transfer cleanly to supervised fine-tuning on standard benchmarks.