In this paper, we propose three new tensor decompositions for even-order tensors corresponding respectively to the rank-one decompositions of some unfolded matrices. Consequently such new decompositions lead to three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank in this paper. We discuss the bounds between these new te...

An n×n matrix X is called completely positive semidefinite (cpsd) if there exist d×d Hermitian positive semidefinite matrices {Pi}i=1 (for some d ≥ 1) such that Xij = Tr(PiPj), for all i, j ∈ {1, . . . , n}. The cpsd-rank of a cpsd matrix is the smallest d ≥ 1 for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the c...

The first 2 PARI-GP files below give the sets of vertices and edges of the Voronoi graph first in dimensions 2 to 6, then in dimension 7. The numerical data are extracted from Jaquet’s thesis [Ja]. The third file, based on Chapters 9 and 14 of [Mar], is devoted to minimal classes in dimensions 2 to 4. We present below a short account of Voronoi’s theory and minimal classes. 1. The perfection ra...

In this paper we study the structure of standard Einstein solvmanifolds of arbitrary rank. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.

Canonical Polyadic Decomposition (CPD) of a higher-order tensor is an important tool in mathematical engineering. In many applications at least one of the matrix factors is constrained to be column-wise orthonormal. We first derive a relaxed condition that guarantees uniqueness of the CPD under this constraint. Second, we give a simple proof of the existence of the optimal low-rank approximatio...

Imagine we have an n1×n2 matrix from which we only get to see a small number of the entries. Is it possible from the available entries to guess the many entries that are missing? In general it is an impossible task because the unknown entries could be anything. However, if one knows that the matrix is low rank and makes a few reasonable assumptions, then the matrix can indeed be reconstructed a...

We proposed two randomized tensor algorithms for reducing multilinear ranks in the Tucker format. The basis of these randomized algorithms is from the randomized SVD of Halko, Martinsson and Tropp [9]. Here we provide randomized versions of the higher order SVD and higher order orthogonal iteration. Moreover, we provide a sharper probabilistic error bounds for the matrix low rank approximation....

We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert–Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the...

The task of matrix completion involves estimating the entries of a matrix, M ∈ Rm×n, when a subset, Ω ⊂ {(i, j) : 1 ≤ i ≤ m, 1 ≤ j ≤ n} of the entries are observed. A particular set of low rank models for this task approximate the matrix as a product of two low rank matrices, M̂ = UV T , where U ∈ Rm×k and V ∈ Rn×k and k min{m,n}. A popular algorithm of choice in practice for recovering M from t...