Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStructured low rank decomposition of multivariate Hankel matrices
341
0
0.0
(
0
)
Authors
Jouhayna Harmouch
Ask ChatGPT about the research
No Arabic abstract
We study the decomposition of a multivariate Hankel matrix H_$sigma$ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol $sigma$ as a sum of polynomial-exponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the properties of the associated Artinian Gorenstein quotient algebra A_$sigma$. A basis of A_$sigma$ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix H_$sigma$. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of H $sigma$. Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Prony-type decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments.
rate research
Read More
Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ over $F$ have rank $leq r$ ? This question is classical, and the answer ($q^{2r}$ when $rleqminleft{ p,qright} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},ldots,x_{k-1}$ for some $kleq rleqminleft{ p,qright} $, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},ldots,x_{p+q}$ such that the resulting Hankel matrix $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ has rank $leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.
We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n times n$ diagonal matrix and $U, V inmathbb{C}^{n times k}$. The proposed algorithm for the real case exploits a two--stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted sub-diagonals. It is shown that the novel method requires $O(n^2k)$ arithmetic operations and it is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of $Re(D)$ induces a structured reduction on $A$ in a block staircase CMV--type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and we show how to generalize the sub-diagonal elimination to this shape, while still being able to provide a condensed representation for the reduced matrix. In this way the complexity still remains linear in $k$ and, moreover, the resulting algorithm can be adapted to deal efficiently with block companion matrices.
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal neighbor and nearest singular neighbor of a given matrix, both of which play central roles in matrix decompositions, and then against this visual backdrop examine the polar and singular value decompositions and some of their applications.
The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
