Tensor factorization has been steadily used to represent high-dimensional data. In particular, the canonical polyadic decomposition (CPD) is very appreciated for its remarkable uniqueness properties. However, computing the high-order CPD is challenging: numerical issues and high needs for storage and processing can make algorithms diverge. Furthermore, the recovery of the CP factors is an ill-posed problem. One way to circumvent this limitation is to exploit the equivalence between the CPD and the Tensor Train Decomposition (TTD). This paper formulates the CPD as a dimension reduction using a TTD followed by a global marginally convex optimization problem . This global optimization scheme estimates the CP factors with minimal error. The resulting approach, Dimensionality Reduction, joint Estimation of the Ambiguity Matrices and the CP FACtors (DREAMFAC), relies on a block-coordinate descent that reaches a first-order stationary point when estimating the CP factors. DREAMFAC is also shown to be an optimal estimator that reaches the corresponding constrained Cramér–Rao bound. It therefore appears as a state-of-the-art solution to estimate the best rank- K CPD of a tensor (when it exists). Its performance is illustrated on the problem of parameter estimation in a dual-polarized MIMO system . Numerical experiments show the excellent practical performance of DREAMFAC, even with very low SNR.

We present a new method for solving simultaneously hyperspectral super-resolution and spectral unmixing of the unknown super-resolution image. Our method relies on three key elements: (1) the nonnegative decomposition in rank-(Lr, Lr, 1) block-terms, (2) joint tensor factorization with multiplicative updates, and (3) the formulation of a family of optimization problems with β-divergences objective functions. We come up with a family of simple, robust and efficient algorithms, adaptable to various noise statistics. Experiments show that our approach competes favorably with state-of-the-art methods for solving both problems at hand for various noise statistics.

Hyperspectral super-resolution based on coupled Tucker decomposition has been recently considered in the remote sensing community. The state-of-the-art approaches did not fully exploit the coupling of information contained in hyperspectral and multispectral images of the same scene. This paper proposes a new algorithm that overcomes this limitation. It accounts for both the high-resolution and the low-resolution information in the model by solving a set of least-squares problems. In addition, we provide exact recovery conditions for the super-resolution image in the noiseless case. Our simulations show that the proposed algorithm achieves very good reconstruction quality with a very low computational complexity.

In this paper, we propose to jointly solve the hyperspectral super-resolution problem and the unmixing problem of the underlying super-resolution image using a coupled LL1 block-tensor decomposition. We consider a spectral variability phenomenon occurring between the observed low-resolution images. Exact recovery conditions for the image and mixing factors are provided. We propose two algorithms, an unconstrained one and another one subject to nonnegativity constraints, to solve the problems at hand. We showcase performance of the proposed approach on synthetic and real images.

Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurements. Indeed, most of factors impacting the asymptotic estimation performance of the parameters of interest can be taken into account via equality constraints. In this communication, we introduce a new constrained Cramér-Rao- like bound for observations where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. In this setting, it is now possible to consider random equality constraints, i.e., equality constraints on the unknown deterministic parameters depending on the random parameters, which can not be addressed with the usual constrained Cramér-Rao bound. The usefulness of the proposed bound is illustrated by way of a coupled canonical polyadic model with linear constraints applied to the hyperspectral super-resolution problem.

We propose a novel approach for hyperspectral super-resolution that is based on low-rank tensor approximation for a coupled low-rank multilinear (Tucker) model. We show that the correct recovery holds for a wide range of multilinear ranks. For coupled tensor approximation, we propose an SVD-based algorithm that is simple and fast, but with a performance comparable to that of the state-of-the-art methods.