Related papers: A panorama of positivity
Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of block matrices, and especially their positivity properties. This is done by focussing on one major problem:…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…
Physics relies on mathematical spaces carefully matched to the phenomena under study. Phase space in classical mechanics, Hilbert space in quantum theory, configuration spaces in field theory all provide representations in which physical…
We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New…
In this paper, I propose a project of enlisting quantum information science as a source of task-oriented axioms for use in the investigation of operational theories in a general framework capable of encompassing quantum mechanics, classical…
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown…
We show that, for positive definite kernels, if specific forms of regularity (continuity, Sn-differentiability or holomorphy) hold locally on the diagonal, then they must hold globally on the whole domain of positive-definiteness. This…
This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn's problem, the…
Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical…
Inversion of various inclusions, that characterize continuity in topological spaces, results in numerous variants of quotient and perfect maps. In the framework of convergences, the said inclusions are no longer equivalent, and each of them…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
Two dimensional matrices with binary (0/1) entries are a common data structure in many research fields. Examples include ecology, economics, mathematics, physics, psychometrics and others. Because the columns and rows of these matrices…
The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix…
Positivity, the assumption that every unique combination of confounding variables that occurs in a population has a non-zero probability of an action, can be further delineated as deterministic positivity and stochastic positivity. Here, we…
Symmetric Positive Definite (SPD) matrices have become popular to encode image information. Accounting for the geometry of the Riemannian manifold of SPD matrices has proven key to the success of many algorithms. However, most existing…
The tensorial principal component analysis is a generalization of ordinary principal component analysis, focusing on data which are suitably described by tensors rather than matrices. This paper aims at giving the nonperturbative…
Spectral kernel methods are techniques for transforming data into a coordinate system that efficiently reveals the geometric structure - in particular, the "connectivity" - of the data. These methods depend on certain tuning parameters. We…