Related papers: Efficient Removal Lemmas for Matrices
We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the…
We study sigma-ideals and regularity properties related to the "filter-Laver" and "dual-filter-Laver" forcing partial orders. An important innovation which enables this study is a dichotomy theorem proved recently by Miller [1]. [1] Arnold…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
Szemer\'edi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that…
In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing…
Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of…
Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…
In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces $L^{2}(\mu)$, with $\mu$ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix to the…
The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping…
In our previous works, we proved that the inverse of the stiffness matrix of an $h$-version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block…
We present a matrix-factorization algorithm that scales to input matrices with both huge number of rows and columns. Learned factors may be sparse or dense and/or non-negative, which makes our algorithm suitable for dictionary learning,…
Originally developed for imputing missing entries in low rank, or approximately low rank matrices, matrix completion has proven widely effective in many problems where there is no reason to assume low-dimensional linear structure in the…
In 1986 Lovasz, Spencer, and Vesztergombi proved a lower bound for the hereditary a discrepancy of a set system F in terms of determinants of square submatrices of the incidence matrix of F. As shown by an example of Hoffman, this bound can…
We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are $F$-pure when the size of matrices is equal to 3.…
This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We…
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…
We equip the complex polynomial algebra C[t] with the involution which is the identity on C and sends t to -t. Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this…
The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating…