arXiv subjects

Álvaro Samperio

Publications and source records attributed to Álvaro Samperio.

4 recordsLinked to original sources

arXiv2026

Simultaneous recovery of a sparse topology and the admittance of an electrical network

We show that the problem of recovering the topology and admittance of an electrical network from power and voltage data at all vertices is often ill-posed, and sometimes it even has multiple solutions. We reformulate the problem to seek for a sparse network, i.e., with few edges, which fits the data up to a given tolerance. We propose an algorithm to solve this reformulated problem. It combines, in an iterative procedure, the resolution of non-negative linear regression problems, and techniques of spectral graph sparsification. The algorithm is based on original results bounding the fitting error of a sparse approximation of a network. We illustrate our techniques with several experimental results in which we are able to recover a sparse network.

math.OC↗

arXiv2025

Estimation of Algebraic Sets: Extending PCA Beyond Linearity

An algebraic set is defined as the zero locus of a system of real polynomial equations. In this paper we address the problem of recovering an unknown algebraic set $\mathcal{A}$ from noisy observations of latent points lying on $\mathcal{A}$ -- a task that extends principal component analysis, which corresponds to the purely linear case. Our procedure consists of three steps: (i) constructing the {\it moment matrix} from the Vandermonde matrix associated with the data set and the degree of the fitted polynomials, (ii) debiasing this moment matrix to remove the noise-induced bias, (iii) extracting its kernel via an eigenvalue decomposition of the debiased moment matrix. These steps yield $n^{-1/2}$-consistent estimators of the coefficients of a set of generators for the ideal of polynomials vanishing on $\mathcal{A}$. To reconstruct $\mathcal{A}$ itself, we propose three complementary strategies: (a) compute the zero set of the fitted polynomials; (b) build a semi-algebraic approximation that encloses $\mathcal{A}$; (c) when structural prior information is available, project the estimated coefficients onto the corresponding constrained space. We prove (nearly) parametric asymptotic error bounds and show that each approach recovers $\mathcal{A}$ under mild regularity conditions.

math.ST↗

arXiv2025

A Spectral Approach to Kemeny's Constant

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, providing insight into the global behavior of a Markov chain. We present a novel eigenvector-based formula for computing Kemeny's constant. Moreover, we analyze the impact of network structure on Kemeny's constant. In particular, we use various spectral techniques, such as spectral sparsification of graphs and eigenvalue interlacing, and show that they are particularly useful in this context for deriving approximations and sharp bounds for Kemeny's constant

math.CO↗

arXiv2023

Stable recovery of piecewise constant conductance on spider networks

We address the discrete inverse conductance problem for well-connected spider networks; that is, to recover the conductance function on a well-connected spider network from the Dirichlet-to-Neumann map. It is well-known that this inverse problem is exponentially ill-posed, requiring the implementation of a regularization strategy for numerical solutions. Our focus lies in exploring whether prior knowledge of the conductance being piecewise constant within a partition of the edge set comprising few subsets enables stable conductance recovery. To achieve this, we propose formulating the problem as a polynomial optimization one, incorporating a regularization term that accounts for the piecewise constant hypothesis. We show several experimental examples in which the stable conductance recovery under the aforementioned hypothesis is feasible.

math.CO↗