Low-rank Tensor Recovery

Low-rank tensor recovery is an interesting subject from both the theoretical and application point of view. On one side, it is a natural extension of the sparse vector and low-rank matrix recovery problem. On the other side, estimating a low-rank tensor has applications in many different areas such as machine learning, video compression, and seismic data interpolation. In this thesis, two approaches are introduced. The first approach is a convex optimization approach and could be considered as a tractable extension of $ell_1$-minimization for sparse vector and nuclear norm minimization for matrix recovery to tensor scenario. It is based on theta bodies – a recently introduced tool from real algebraic geometry. In particular, theta bodies of appropriately defined polynomial ideal correspond to the unit-theta norm balls. These unit-theta norm balls are relaxations of the unit-tensor-nuclear norm ball. Thus, in this case, we consider a canonical tensor format. The method requires computing the reduced Groebner basis (with respect to the graded reverse lexicographic ordering) of the appropriately defined polynomial ideal. Numerical results for third-order tensor recovery via $theta_1$-norm are provided. The second approach is a generalization of iterative hard thresholding algorithm for sparse vector and low-rank matrix recovery to tensor scenario (tensor IHT or TIHT algorithm). Here, we consider the Tucker format, the tensor train decomposition, and the hierarchical Tucker decomposition. The analysis of the algorithm is based on a version of the restricted isometry property (tensor RIP or TRIP) adapted to the tensor decomposition at hand. We show that subgaussian measurement ensembles satisfy TRIP with high probability under an almost optimal condition on the number of measurements. Additionally, we show that partial Fourier maps combined with random sign flips of the tensor entries satisfy TRIP with high probability. Under the assumption that the linear operator satisfies TRIP and under an additional assumption on the thresholding operator, we provide a linear convergence result for the TIHT algorithm. Finally, we present numerical results on low-Tucker-rank third-order tensors via partial Fourier maps combined with random sign flips of tensor entries, tensor completion, and Gaussian measurement ensembles

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

Tensor completion in hierarchical tensor representations

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the reconstruction of tensors of low multi-linear rank in recently introduced hierarchical tensor formats from a small number of measurements. Hierarchical tensors are a flexible generalization of the well-known Tucker representation, which have the advantage that the number of degrees of freedom of a low rank tensor does not scale exponentially with the order of the tensor. While corresponding tensor decompositions can be computed efficiently via successive applications of (matrix) singular value decompositions, some important properties of the singular value decomposition do not extend from the matrix to the tensor case. This results in major computational and theoretical difficulties in designing and analyzing algorithms for low rank tensor recovery. For instance, a canonical analogue of the tensor nuclear norm is NP-hard to compute in general, which is in stark contrast to the matrix case. In this book chapter we consider versions of iterative hard thresholding schemes adapted to hierarchical tensor formats. A variant builds on methods from Riemannian optimization and uses a retraction mapping from the tangent space of the manifold of low rank tensors back to this manifold. We provide first partial convergence results based on a tensor version of the restricted isometry property (TRIP) of the measurement map. Moreover, an estimate of the number of measurements is provided that ensures the TRIP of a given tensor rank with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral

Low rank tensor recovery via iterative hard thresholding

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.Comment: 34 page