arXiv subjects

Mauricio Velasco

Publications and source records attributed to Mauricio Velasco.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Small Set Expanders and Their Codes

A left-regular bipartite graph $G$ of degree $d$ is called a $(t,\alpha)$-small-set-expander if every subset $X$ of left vertices of size at most $t$ has at least $\alpha |X|$ neighbors. Such a graph is an optimal small-set expander if small subsets have as many neighbors as possible. We characterize optimal expanders combinatorially via girth and prove the existence of $s$-optimal expanders for every $s$. We also prove that $s$-optimality yields new "transfer" lower bounds on the number of neighbors of sets of size $h\geq s$. Finally, as an application, we discuss the use of optimal small-set expanders in building good codes for key exchange protocols in post-quantum cryptography.

math.CO↗

arXiv2026

Mollified Christoffel-Darboux Kernels and Density Recovery on Varieties

We introduce mollified Christoffel-Darboux (CD) kernels on varieties, a systematic regularization of the classical CD kernel associated with a probability measure on a compact domain. The main motivations are twofold: first, to sharpen the classical on/off-support dichotomy of the CD polynomial by replacing linear growth on the support by a uniform bound; second, to obtain consistent and quantitatively controlled recovery of densities from moment data, without the need to know the equilibrium measure of the underlying domain. Our contributions are the following: (i) We introduce families of mollifiers on algebraic varieties. For each measure and degree on such a variety we define a mollified CD kernel, which can be computed from the moments of the underlying measure by linear algebra. (ii) We prove, by elementary arguments, that an improved dichotomy property holds: on the interior of the support the mollified CD polynomial is uniformly bounded in the degree, while outside the support it grows exponentially with the degree. (iii) Assuming Sobolev regularity of the density with respect to a reference measure, we derive explicit convergence rates for density recovery for measures in Euclidean space via mollified CD kernel. (iv) On the unit sphere, we show that suitably chosen algebraic mollifiers, constructed from zonal polynomials, lead to kernels with improved rates, building on classical constructive approximation results.

math.OC↗

arXiv2025

Approximation of starshaped sets using polynomials

We introduce polystar bodies: compact starshaped sets whose gauge or radial functions are expressible by polynomials, enabling tractable computations, such as that of intersection bodies. We prove that polystar bodies are uniformly dense in starshaped sets and obtain asymptotically optimal approximation guarantees. We develop tools for the construction of polystar approximations and illustrate them via several computational examples, including numerical estimations of largest volume slices and widths.

math.OC↗

arXiv2025

Characterization of Logarithmic Fekete Critical Configurations of at Most Six Points in All Dimensions

We consider the logarithmic Fekete problem, which consists of placing a fixed number of points on the unit sphere in $\mathbb{R}^d$, in such a way that the product of all pairs of mutual Euclidean distances is maximized or, equivalently, so that their logarithmic energy is minimized. Using tools from Computational Algebraic Geometry, we find and classify all critical configurations for this problem when considering at most six points in every dimension $d$. In particular, our approach gives new proofs of several key results appearing in the literature, with the benefit of using a unified approach. Furthermore, for seven points in $S^2$, we characterize the global minimizer among critical configurations having at least one pair of antipodal points, and give numerical evidence to support the conjecture that this configuration is also the unrestricted global minimizer.

math.AC↗

arXiv2024

Graph neural networks and non-commuting operators

Graph neural networks (GNNs) provide state-of-the-art results in a wide variety of tasks which typically involve predicting features at the vertices of a graph. They are built from layers of graph convolutions which serve as a powerful inductive bias for describing the flow of information among the vertices. Often, more than one data modality is available. This work considers a setting in which several graphs have the same vertex set and a common vertex-level learning task. This generalizes standard GNN models to GNNs with several graph operators that do not commute. We may call this model graph-tuple neural networks (GtNN). In this work, we develop the mathematical theory to address the stability and transferability of GtNNs using properties of non-commuting non-expansive operators. We develop a limit theory of graphon-tuple neural networks and use it to prove a universal transferability theorem that guarantees that all graph-tuple neural networks are transferable on convergent graph-tuple sequences. In particular, there is no non-transferable energy under the convergence we consider here. Our theoretical results extend well-known transferability theorems for GNNs to the case of several simultaneous graphs (GtNNs) and provide a strict improvement on what is currently known even in the GNN case. We illustrate our theoretical results with simple experiments on synthetic and real-world data. To this end, we derive a training procedure that provably enforces the stability of the resulting model.

stat.ML↗

arXiv2024

Spurious local minima in nonconvex sum-of-squares optimization

We study spurious second-order stationary points and local minima in a nonconvex low-rank formulation of sum-of-squares optimization on a real variety $X$. We reformulate the problem of finding a spurious local minimum in terms of syzygies of the underlying linear series, and also bring in topological tools to study this problem. When the variety $X$ is of minimal degree, there exist spurious second-order stationary points if and only if both the dimension and the codimension of the variety are greater than one, answering a question by Legat, Yuan, and Parrilo. Moreover, for surfaces of minimal degree, we provide sufficient conditions to exclude points from being spurious local minima. In particular, all second-order stationary points associated with infinite Gram matrices on the Veronese surface, corresponding to ternary quartics, lie on the boundary and can be written as a binary quartic, up to a linear change of coordinates, complementing work by Scheiderer on decompositions of ternary quartics as a sum of three squares. For general varieties of higher degree, we give examples and characterizations of spurious second-order stationary points in the interior, together with a restricted path algorithm that avoids such points with controlled step sizes, and numerical experiment results illustrating the empirical successes on plane cubic curves and Veronese varieties.

math.OC↗

arXiv2024

Nonnegativity certificates on real algebraic surfaces

We introduce tools for transferring nonnegativity certificates for global sections between line bundles on real algebraic surfaces. As applications, we improve Hilbert's degree bounds on sum-of-squares multipliers for nonnegative ternary forms, give a complete characterization of nonnegative real forms of del Pezzo surfaces, and establish quadratic upper bounds for the degrees of sum-of-squares multipliers for nonnegative forms on real ruled surfaces.

math.AG↗

arXiv2024

Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming

This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets. A convergent sequence of upper-bounds to the maximal slice volume are computed using the moment-Sum-of-Squares hierarchy of semidefinite programs in increasing size. The computational complexity of this scheme can be reduced by utilizing topological structure (in dimensions 2, 3, 4, 8) and symmetry. This numerical convergence can be accelerated through the introduction of redundant Stokes-based constraints. Demonstrations of slice-volume calculation are performed on example sets.

math.OC↗

arXiv2023

Estimation of large covariance matrices via free deconvolution: computational and statistical aspects

The estimation of large covariance matrices has a high dimensional bias. Correcting for this bias can be reformulated via the tool of Free Probability Theory as a free deconvolution. The goal of this work is a computational and statistical resolution of this problem. Our approach is based on complex-analytic methods methods to invert $S$-transforms. In particular, one needs a theoretical understanding of the Riemann surfaces where multivalued $S$ transforms live and an efficient computational scheme.

math.PR↗

arXiv2022

Harmonic Hierarchies for Polynomial Optimization

We introduce novel polyhedral approximation hierarchies for the cone of nonnegative forms on the unit sphere in $\mathbb{R}^n$ and for its (dual) cone of moments. We prove computable quantitative bounds on the speed of convergence of such hierarchies. We also introduce a novel optimization-free algorithm for building converging sequences of lower bounds for polynomial minimization problems on spheres. Finally some computational results are discussed, showcasing our implementation of these hierarchies in the programming language Julia.

math.OC↗

arXiv2021

On random walks and switched random walks on homogeneous spaces

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$. We introduce the switched random walk determined by a finite set of probability distributions on $G$, prove that its long-term behavior is determined by the Fourier joint spectral radius of the distributions and give hermitian sum-of-squares algorithms for the effective estimation of this quantity.

math.PR↗

arXiv2021

Minimum cross-entropy distributions on Wasserstein balls and their applications

Given a prior probability density $p$ on a compact set $K$ we characterize the probability distribution $q_δ^*$ on $K$ contained in a Wasserstein ball $B_δ(μ)$ centered in a given discrete measure $μ$ for which the relative-entropy $H(q,p)$ achieves its minimum. This characterization gives us an algorithm for computing such distributions efficiently

math.OC↗

arXiv2021

Moment-SOS hierarchy and exit time of stochastic processes

The moment sum of squares (moment-SOS) hierarchy produces sequences of upper and lower bounds on functionals of the exit time solution of a polynomial stochastic differential equation with polynomial constraints, at the price of solving semidefinite optimization problems of increasing size. In this note we use standard results from elliptic partial differential equation analysis to prove convergence of the bounds produced by the hierarchy. We also use elementary convex analysis to describe a super- and sub-solution interpretation dual to a linear formulation on occupation measures. The practical relevance of the hierarchy is illustrated with numerical examples.

math.OC↗

arXiv2021

Sums of Squares: A Real Projective Story

The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety. We survey the rich history of this area and focus on recent developments highlighting unexpected links to complex algebraic geometry and commutative algebra.

math.AG↗

arXiv2020

Constructing Partial MDS Codes from Reducible Curves

We propose reducible algebraic curves as a mechanism to construct Partial MDS (PMDS) codes geometrically. We obtain new general existence results, new explicit constructions and improved estimates on the smallest field sizes over which such codes can exist. Our results are obtained by combining ideas from projective algebraic geometry, combinatorics and probability theory.

cs.IT↗

arXiv2020

Sums of Squares and Quadratic Persistence on Real Projective Varieties

We bound the Pythagoras number of a real projective subvariety: the smallest positive integer $r$ such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most $r$ squares. Enhancing existing methods, we exhibit three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant of a projective subvariety called the quadratic persistence. Defined by projecting away from points, this numerical invariant is closely related to the linear syzygies of the variety. In addition, we classify the projective subvarieties of maximal and almost-maximal quadratic persistence, and determine their Pythagoras numbers.

math.AG↗

arXiv2020

Approximate super-resolution of positive measures in all dimensions

We study the problem of reconstructing a positive discrete measure on a compact set $K \subseteq \mathbb{R}^n$ from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.

math.OC↗

arXiv2018

Sharp Degree Bounds for Sum-of-Squares Certificates on Projective Curves

Given a real projective curve with homogeneous coordinate ring R and a nonnegative homogeneous element f in R, we bound the degree of a nonzero homogeneous sum-of-squares g in R such that the product fg is again a sum of squares. Better yet, our degree bounds only depend on geometric invariants of the curve and we show that there exist smooth curves and nonnegative elements for which our bounds are sharp. We deduce the existence of a multiplier g from a new Bertini Theorem in convex algebraic geometry and prove sharpness by deforming rational Harnack curves on toric surfaces. Our techniques also yield similar bounds for multipliers on surfaces of minimal degree, generalizing Hilbert's work on ternary forms.

math.AG↗