Higher order tensor inversion is possible for even order. This due to the fact that a group endowed with contracted product isomorphic general linear of degree $n$. With these structures, we derive SVD which have shown be equivalent well-known canonical polyadic decomposition and multilinear provided some constraints are satisfied. Moreover, within this structure framework, systems derived solved problems high-dimensional PDEs large discrete quantum models. We also address do not fit...

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

We present a method for the numerical solution of Hamilton Jacobi Bellman PDE that arises in an infinite time optimal control problem. The can be higher order to reduce "the curse dimensionality". It proceeds two stages. First HJB is solved neighborhood origin using power series Al'brecht (1961). From boundary point this neighborhood, extremal trajectory computed backward Pontryagin maximum principle. Then ordinary differential equations are developed partial derivatives along extremal....

In this paper, we investigate and discuss in detail the structures of quaternion tensor SVD, rank decomposition, $\eta$-Hermitian decomposition with isomorphic group Einstein product. Then give expression Moore-Penrose inverse a by using SVD. Moreover, consider generalized Sylvester equation. We some necessary sufficient conditions for existence solution to equation terms inverses tensors. also present general when it is solvable. As applications equation, derive existences solutions...

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

Summary The goal of this paper is to find a low‐rank approximation for given n th tensor. Specifically, we give computable strategy on calculating the rank tensor, based approximating solution an NP‐hard problem. In paper, formulate sparse optimization problem via l 1 ‐regularization tensors. To solve problem, propose rescaling algorithm proximal alternating minimization and study theoretical convergence algorithm. Furthermore, discuss probabilistic consistency sparsity result suggest way...

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 solve the ARE. Since leads right-hand side rank equal number inputs, regardless weights, resulting Lyapunov equation can be more efficiently solved. A low-rank Cholesky-ADI algorithm is used at each step. The straightforward code. Performance illustrated with an example beam, different levels...

Abstract Dimension reduction is analytical methods for reconstructing high-order tensors that the intrinsic rank of these tensor data relatively much smaller than dimension ambient measurement space. Typically, this case most real world datasets in signals, images and machine learning. The CANDECOMP/PARAFAC (CP, aka Canonical Polyadic) completion a widely used approach to find low-rank approximation given tensor. In model (Sanogo Navasca 2018 52nd Asilomar conference on systems, computers,...

We present methods for locally solving the Dynamic Programming Equations (DPE) and Hamilton Jacobi Bellman (HJB) PDE that arise in infinite horizon optimal control problem. The method DPE is discrete time version of Al'brecht's procedure approximating solution HJB. also prove existence smooth solutions to has same Taylor series expansions as formal solutions. Our HJB numerically begins with local initial approximation uses some Lyapunov criteria piece together polynomial estimates....

We present a novel method for tensor dimensionality reduction. The rank reduction has many applications in signal and image processing including various blind techniques. In this paper, we generalize the trace class norm to higher-order tensors. Recently, matrix received much attention compressed sensing applications. It is known provide bounds minimum of matrix. new used formulate an optimization problem finding best low multilinear approximation. Our formulation leads set semidefinite...

Braman [B08] described a construction where third-order tensors are exactly the set of linear transformations acting on matrices with vectors as scalars. This extends familiar notion that form all over real-valued result is based upon circulant-based tensor multiplication due to Kilmer et al. [KMP08]. In this work, we generalize these observations further by viewing in its natural framework group rings.The products arise convolutions algebraic structures. Our generalization allows for any...

We study the symmetric outer product decomposition which decomposes a fully (partially) tensor into sum of rank-one tensors. present iterative algorithms for third-order partially and fourth-order tensor. The numerical examples indicate faster convergence rate new than standard method alternating least squares.

SUMMARY We study the least squares functional of canonical polyadic tensor decomposition for third order tensors by eliminating one factor matrix, which leads to a reduced functional. An analysis several equivalent optimization problem, such as Rayleigh quotient or projection. These formulations are basis new algorithms follows: Centroid Projection method efficient computation suboptimal solutions and fixed‐point iteration methods approximating best rank‐1 rank‐ R decompositions under...

In this paper, we discuss the acceleration of regularized alternating least-squares (RALS) algorithm for tensor approximations. We propose a fast iterative method using an Aitken-Stefensen-like update algorithm. Through numerical experiments, faster convergence rate accelerated version is demonstrated in comparison to both standard and algorithms. addition, analyze global based on Kurdyka-Łojasiewicz inequality, show that RALS has linear local rate.

Higher order tensor inversion is possible for even order. We have shown that a group endowed with the Einstein (contracted) product isomorphic to general linear of degree $n$. With structures, we derived new decompositions which be related well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this structure framework, systems are derived, specifically, solving high dimensional PDEs large discrete quantum models. also address do not fit framework in least-squares...

We develop a tensor based multilinear system framework. have shown that the TRIP gives conditions for sparse to unique solution. use this formulation leverage existing theory from compressed sensing efficiently infer unknown coefficients. Moreover, we also proven signals can be recovered through l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> minimization. In our future work, will provide some numerical methods on minimization...

This paper explores a new version of the Levenberg-Marquardt algorithm used for Tensor Canonical Polyadic (CP) decomposition with an emphasis on image compression and reconstruction. computation, especially CP decomposition, holds significant applications in data analysis. In this study, we formulate as nonlinear least squares optimization problem. Then, present iterative (LM) based computing decomposition. Ultimately, test various datasets, including randomly generated tensors RGB images....