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Solving Multilinear Systems via Tensor Inversion

Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree

Approximation of low rank solutions for linear quadratic control of partial differential equations

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Iterative Solution of Algebraic Riccati Equations for Damped Systems

. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.

Video detection anomaly via low-rank and sparse decompositions

Algebraic Riccati equations (ARE) of large dimension arise when using approximations to design controllers for systems modeled by partial differential equations. We use a modified Newton method to solve the ARE that takes advantage of several special features of these problems. The modified Newton method leads to a right-hand side of rank equal to the number of inputs regardless of the weights. Thus, the resulting Lyapunov equation can be more efficiently solved. The Cholesky-ADI algorithm is used to solve the Lyapunov equation resulting at each step. The algorithm is straightforward to code. Performance is illustrated with a number of standard examples. An example on controlling the deflection of the Euler-Bernoulli beam indicates that for weakly damped problems a low rank solution to the ARE may not exist. Further analysis supports this point.

Adaptive Low Rank Approximation for Tensors

Algebraic Riccati equations (ARE) of large dimension arise when using approximations to design controllers for systems modelled by partial differential equations. We use a modified Newton method to solve the ARE. Since the modified Newton method leads to a right-hand side of rank equal to the number of inputs, regardless of the weights, the resulting Lyapunov equation can be more efficiently solved. A low-rank Cholesky-ADI algorithm is used to solve the Lyapunov equation resulting at each step. The algorithm is straightforward to code. Performance is illustrated with an example of a beam, with different levels of damping. Results indicate that for weakly damped problems a low rank solution to the ARE may not exist. Further analysis supports this point

Low‐rank approximation of tensors via sparse optimization

In this paper, we purpose a method for anomaly detection in surveillance video in a tensor framework. We treat a video as a tensor and utilize a stable PCA to decompose it into two tensors, the first tensor is a low rank tensor that consists of background pixels and the second tensor is a sparse tensor that consists of the foreground pixels. The sparse tensor is then analyzed to detect anomaly. The proposed method is a one-shot framework to determine frames that are anomalous in a video.

Sparseness constraints on nonnegative tensor decomposition

In this paper, we propose a novel framework for finding low rank approximation of a given tensor. This framework is based on the adaptive lasso with coefficient weights for sparse computation in tensor rank detection. We also provide an algorithm for solving the adaptive lasso model problem for tensor approximation. In a special case, the convergence of the algorithm and the probabilistic consistency of the sparsity have been addressed [15] when each weight equals to one. The method is applied to background extraction and video compression problems.

Recovering tensor data from incomplete measurement via compressive sampling

The goal of this paper is to find a low-rank approximation for a given tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP-hard problem. In this paper, we formulate a sparse optimization problem via an

Random Projections for Low Multilinear Rank Tensors

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Tensor restricted isometry property for multilinear sparse system of genomic interactions

-regularization to find a low-rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of the proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiments, the performance of our algorithm supports that our method provides an efficient estimate on the number of rank-one tensor components in a given tensor. Moreover, this algorithm is also applied to surveillance videos for low-rank approximation.