Related papers: Symmetric Tensor Decompositions over Finite Fields
Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging…
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of…
It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear…
In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$\underset{ rank(X)\leq k}{\min} \sum^m_{i=1}\left \Vert A_i - B_i XB_i^T \right \Vert^2_F,$$ where…
For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count the number of semilinear endomorphisms of a g-dimensional k-vector space which have rank r and stable rank s. Such endomorphisms show up naturally in the…
This paper studies the problem of decomposing a low-rank positive-semidefinite matrix into symmetric factors with binary entries, either $\{\pm 1\}$ or $\{0,1\}$. This research answers fundamental questions about the existence and…
We show that the linear map defined by multiplication with a general bi-homogeneous form between two bi-graduated pieces of the first cohomology of a nonsingular quadric in the projective space is of maximal rank. This is the first non…
Finding the symmetric and orthogonal decomposition (SOD) of a tensor is a recurring problem in signal processing, machine learning and statistics. In this paper, we review, establish and compare the perturbation bounds for two natural types…
We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models we consider an infinite family of linear maximum rank distance codes obtained by puncturing…
Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address…
The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce the first-order and second-order T-derivatives for the multi-vector real-valued function with the tensor T-product; and…
We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. These…
We study central configurations when the set of positions is symmetric. We use a theorem from representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies…
This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the…
We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end we introduce Igusa-type rational…
We study tensor powers of representations of finite monoids, focusing on the growth behavior of their composition length and the number of indecomposable summands. Special attention is given to diagram monoids such as the Temperley-Lieb,…
Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing…
Low-rank tensor estimation offers a powerful approach to addressing high-dimensional data challenges and can substantially improve solutions to ill-posed inverse problems, such as image reconstruction under noisy or undersampled conditions.…
We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of…
In this paper, with the help of trinomial coefficients we study some arithmetic properties of certain determiants involving reciprocals of binary quadratic forms over finite fields.