Related papers: Support-based lower bounds for the positive semide…
Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the…
Let $1 \leq k \leq n$ be a positive integer. A {\em nonnegative signed $k$-subdominating function} is a function $f:V(G) \rightarrow \{-1,1\}$ satisfying $\sum_{u\in N_G[v]}f(u) \geq 0$ for at least $k$ vertices $v$ of $G$. The value…
In this work, we focus on the high-dimensional trace regression model with a low-rank coefficient matrix. We establish a nearly optimal in-sample prediction risk bound for the rank-constrained least-squares estimator under no assumptions on…
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In…
Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in $l^p$ when $p \geq 1$ and its dual version, the upper bounds when $0<p…
We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad--Schreyer--Shafiei lower bounds for the Waring ranks and cactus ranks…
This paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix. Maximizing this smallest eigenvalue has applications to selecting input…
We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of…
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct…
Recently, the \textit{Tensor Nuclear Norm~(TNN)} regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to…
Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in…
We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is…
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum…
Classical latent-score ranking models often fail to distinguish objects' intrinsic scores from contextual effects, which are typically nonlinear and can dominate the observed outcomes. To address this, we introduce a semiparametric ranking…
We characterize the infimum of a matrix norm of a square matrix A induced by an absolute norm, over the fields of real and complex numbers. Usually this infimum is greater than the spectral radius of A. If A is sign equivalent to a…
The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew symmetric matrices, whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. In this paper we…
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…
This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear…
We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages…