Hyperspectral super-resolution (HSR) aims at fusing a pair of hyperspectral and multispectral images to recover a super-resolution image (SRI). This work revisits coupled tensor decomposition (CTD)-based HSR. The vast majority of the HSR approaches take a low-rank matrix recovery perspective. The challenge is that theoretical guarantees for recovering the SRI using low-rank matrix models are either elusive or derived under stringent conditions. A couple of recent CTD-based methods ensure recoverability for the SRI under relatively mild conditions, leveraging algebraic properties of the canonical polyadic decomposition (CPD) and the Tucker decomposition models, respectively. However, the latent factors of both the CPD and Tucker models have no physical interpretations in the context of spectral image analysis, which makes incorporating prior information challenging—but using priors is often essential for enhancing performance in noisy environments. This work employs an idea that models spectral images as tensors following the block-term decomposition model with multilinear rank-(Lr, Lr, 1) terms (i.e., the LL1 model) and formulates the HSR problem as a coupled LL1 tensor decomposition problem. Similar to the existing CTD approaches, recoverability of the SRI is shown under mild conditions. More importantly, the latent factors of the LL1 model can be interpreted as the key constituents of spectral images, i.e., the endmembers' spectral signatures and abundance maps. This connection allows us to incorporate prior information for performance enhancement. A flexible algorithmic framework that can work with a series of structural information is proposed to take advantages of the model interpretability. The effectiveness is showcased using simulated and real data.

Tensor analysis has received widespread attention in high-dimensional data learning. Unfortunately, the tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. How to recover the characteristics of the corrupted tensor data and make it compatible with the downstream clustering task remains a challenging problem. In this article, we study a generalized transformed tensor low-rank representation (TTLRR) model for simultaneously recovering and clustering the corrupted tensor data. The core idea is to find the latent low-rank tensor structure from the corrupted measurements using the transformed tensor singular value decomposition (SVD). Theoretically, we prove that TTLRR can recover the clean tensor data with a high probability guarantee under mild conditions. Furthermore, by using the transform adaptively learning from the data itself, the proposed TTLRR model can approximately represent and exploit the intrinsic subspace and seek out the cluster structure of the tensor data precisely. An effective algorithm is designed to solve the proposed model under the alternating direction method of multipliers (ADMMs) algorithm framework. The effectiveness and superiority of the proposed method against the compared methods are showcased over different tasks, including video/face data recovery and face/object/scene data clustering.

The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($ \rm{order} >3 $). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with ket augmentation, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.

The block-term tensor decomposition model with multilinear rank- $(L_{r},L_{r},1)$ terms (or, the “LL1 tensor decomposition” in short) offers a valuable alternative formulation for hyperspectral unmixing (HU), which ensures identifiability of the endmembers/abudnaces in cases where classic matrix factorization (MF) approaches cannot provide such guarantees. However, existing LL1 tensor decomposition-based HU algorithms use a three-factor parameterization of the tensor (i.e., the hyperspectral image cube), which causes difficulties in incorporating structural prior information arising in HU. Consequently, their algorithms often exhibit high per-iteration complexity and slow convergence. This work focuses on LL1 tensor decomposition under structural constraints and regularization terms in HU. Our algorithm uses a two-factor re-parameterization of the tensor model. Like in the MF-based approaches, the factors correspond to the endmembers and abundances in the context of HU. Thus, the proposed framework is natural to incorporate physics-motivated priors in HU. To tackle the formulated optimization problem, a two-block alternating gradient projection (GP)-based algorithm is proposed. Carefully designed projection solvers are proposed to implement the GP algorithm with a relatively low per-iteration complexity. An extrapolation-based acceleration strategy is proposed to expedite the GP algorithm. Such extrapolated multi-block algorithm only had asymptotic convergence assurances in the literature. Our analysis shows that the algorithm converges to the vicinity of a stationary point within finite iterations, under reasonable conditions. Empirical study shows that the proposed algorithm often attains orders-of-magnitude speedup and substantial HU performance gains compared to the existing LL1 decomposition-based HU algorithms.

Recently, low-rank tensor completion (LRTC) methods by exploiting the global low-rankness of the target tensor have shown great potential for traffic data imputation. However, in real-world transportation networks, traffic data usually suffer from more complicated structural missing patterns than random-missing patterns, e.g., tube-missing patterns due to disruptions in wireless connections or slice-missing mechanism caused by sensor maintenance. As the naturally low-rank structure of traffic data in several missing scenarios, the existing LRTC methods indeed refrain from desirable performance for imputing traffic data. To tackle the complicated missing scenarios, we propose a convolutional low-rank tensor representation (CLRTR). Especially, CLRTR represents each unfolding matrix of the tensor as a sum of convolutions between two-dimensional (2D) filters and the corresponding low-rank coefficients, which allows us to simultaneously reveal the local patterns and the low-rankness of traffic data. Based on the CLRTR, we introduce the corresponding low-rank metric CLRTR-rank. Based on the suggested low-rank metric, we propose a traffic data imputation model that is well-suited to the complicated missing data scenarios. To implement the resultant imputation model, we design the alternating direction method of multipliers (ADMM) based algorithm with a theoretical convergence guarantee. Extensive numerical experiments on several real-world traffic datasets for both traffic data imputation and downstream traffic data prediction highlight the superiority of our model over the existing state-of-the-art matrix/tensor models for extensive missing scenarios.

Multi-view subspace clustering (MVSC) separates the data with multiple views into multiple clusters, and each cluster corresponds to one certain subspace. Existing tensor-based MVSC methods construct self-representation subspace coefficient matrices of all views as a tensor, and introduce the tensor nuclear norm (TNN) to capture the complementary information hidden in different views. The key assumption is that the data samples of each subspace must be sufficient for subspace representation. This work proposes a nonconvex latent transformed low-rank tensor representation framework for MVSC. To deal with the insufficient sample problem, we study the latent low-rank representation in the multi-view case to supplement underlying observed samples. Moreover, we propose to use data-driven transformed TNN (TTNN), resulting from the intrinsic structure of multi-view samples, to preserve the consensus and complementary information in the transformed domain. Meanwhile, the proposed unified nonconvex low-rank tensor representation framework can better learn the high correlation among different views. To resolve the proposed nonconvex optimization model, we propose an effective algorithm under the framework of the alternating direction method of multipliers and theoretically prove that the iteration sequences converge to the critical point. Experiments on various datasets showcase outstanding performance.