Related papers: A Semidefinite Hierarchy for Containment of Spectr…
In previous work [Adv. Math. 298, pp. 325-368, 2016], the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix were described. Key ingredients in the proof included a novel stratification of the cone…
Semidefinite Optimization has become a standard technique in the landscape of Mathematical Programming that has many applications in finite dimensional Quantum Information Theory. This paper presents a way for finite-dimensional relaxations…
Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $\pi\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of…
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative…
Phase retrieval in real or complex Hilbert spaces is the task of recovering a vector, up to an overall unimodular multiplicative constant, from magnitudes of linear measurements. In this paper, we assume that the vector is normalized, but…
Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one…
The problem of finding completely positive matrices with equal cp-rank and rank is considered. We give some easy-to-check sufficient conditions on the entries of a doubly nonnegative matrix for it to be completely positive with equal…
We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously…
A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some…
The Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question…
The model checking problem for various fragments of first-order logic has attracted much attention over the last two decades: in particular, for the primitive positive and the positive Horn fragments, which are better known as the…
Trustworthy classifiers are essential to the adoption of machine learning predictions in many real-world settings. The predicted probability of possible outcomes can inform high-stakes decision making, particularly when assessing the…
We present sufficient condition for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their representation as a series of spherical harmonics. The family analyzed is a…
Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a…
A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e.,…
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive…
We present an overview of a classical theme in analysis and matrix positivity: the question of which functions preserve positive semidefiniteness when applied entrywise. In addition to drawing the attention of experts such as Schoenberg,…
Given a finite family of compact subsets of the complex plane we propose a certificate of mutual non-overlapping with respect to area measure. The criterion is stated as a couple of positivity conditions imposed on a four argument…
Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\…
We consider the matrix completion problem with a deterministic pattern of observed entries. In this setting, we aim to answer the question: under what condition there will be (at least locally) unique solution to the matrix completion…