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Carlos Beltrán

Publications and source records attributed to Carlos Beltrán.

At least 19 recordsLinked to original sources
arXiv2026

A set of points with optimal $L^2$ spherical cap discrepancy

We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the {\em deterministic Diamond points}. For every $0<α<2$ we prove that there exists a constant $C_α>0$ such that \[ 0\leq \frac{2^{α+1}}{α+2}N^2-\sum_{x,y\in\cP_N}|x-y|^α\le C_αN^{1-α/2}. \] This result shows that $\cP_N\subset\mathbb S^2$ has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for $α=1$, Stolarsky's invariance principle implies that $\cP_N\subset\mathbb S^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order \[ D_{L^2}^C(\mathcal P_N)\asymp N^{-3/4}, \] making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set $\cP_N$ has Sobolev $H^{s}(\Sph)$ worst-case error of optimal order for all $1<s<2$.

math.CA↗
arXiv2026

Discrepancy of determinantal point processes on compact, connected two-point homogeneous spaces

We study the $L^{\infty}$ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces, namely spheres and projective spaces. Using concentration inequalities and variance estimates for the number of points in metric balls, we derive general upper bounds for the discrepancy of homogeneous determinantal point processes. In the particular case of the harmonic ensemble, we show that the discrepancy of $N$ points is $O((N^{1-1/D})^{1/2}\log N)$ with high probability, where $D$ denotes the real dimension of the manifold. For the projective ensemble on $\mathbb{CP}^d$, we obtain the sharper bound $O((N^{1-1/D}\log N)^{1/2})$. These results extend previously known discrepancy estimates for determinantal point processes on the sphere to all compact, connected two-point homogeneous spaces.

math.CA↗
arXiv2026

Implementation of the hybrid exchange-correlation functionals in the SIESTA code

We present an efficient and accurate implementation of hybrid exchange-correlation (XC) functionals in the SIESTA code, enabling large-scale simulations based on Hartree-Fock-type exact exchange combined with strictly localized numerical atomic orbitals (NAOs). Our approach exploits a fitted representation of the NAOs in terms of Gaussian-type orbitals (GTOs), which allows for the analytical evaluation of four-center electron repulsion integrals (ERIs) via the LIBINT library. This framework is seamlessly integrated with SIESTA's real-space grid and sparse-matrix infrastructure, and is combined with multiple screening techniques to control the computational complexity. We also introduce a fully analytical formulation of hybrid-functional forces and a dynamic parallel distribution scheme that ensures excellent scalability. We validate our implementation through benchmark calculations on a broad set of systems (including semiconductors, insulators, and two-dimensional materials) and demonstrate that the HSE06 functional significantly improves the prediction of band gaps compared to PBE, in close agreement with G0W0 and experimental data. We analyze in detail the trade-offs between accuracy and computational efficiency as a function of the number of Gaussians, basis set range, and integral screening thresholds. Our results confirm that hybrid functional calculations in SIESTA are now feasible for large extended systems, making accurate first-principles predictions of electronic and structural properties accessible at scale.

cond-mat.mtrl-sci↗
arXiv2026

The manifold of unitary and symmetric matrices: characterization, Riemannian optimization and application to BD-RIS design

This paper proposes and analyzes Riemannian optimization algorithms on the manifold of unitary and symmetric matrices, denoted ${\cal {U}}_s$, which naturally models the scattering matrices of passive and reciprocal devices such as beyond-diagonal reconfigurable intelligent surfaces (BD-RISs). Despite its relevance, the geometry of ${\cal {U}}_s$ has remained largely unexplored, and existing BD-RIS optimization methods either ignore the symmetry constraint or rely on costly Takagi-based parameterizations. We first provide a rigorous geometric characterization of ${\cal {U}}_s$, deriving its tangent space, a simple retraction, and closed-form expressions for geodesics. Building on these results, we develop two Riemannian manifold optimization (MO) algorithms tailored to ${\cal {U}}_s$: a line-search (LS) based scheme and a phase-optimization (PO) update along geodesics. We then apply the proposed framework to BD-RIS-assisted multiple-input multiple-output (MIMO) links, addressing sum-gain maximization, rate maximization, and minimum mean-square error problems, where they outperform existing approaches. Furthermore, we show that when the number of BD-RIS elements exceeds the total number of antennas, the optimal scattering matrix is low-rank, which motivates and enables efficient low-rank variants of the proposed algorithms.

eess.SP↗
arXiv2025

Univariate amenable functions

The concepts of amenable and compatible functions have been introduced in a recent work, in order to state precise mathematical theorems that guarantee that a backward stable algorithm is also forward stable, and that the composition of two stable algorithms results in an stable algorithm. In this work, we elaborate in this theory for univariate real analytic functions, providing simple tests for both concepts and producing tables for a number of elementary functions which are or fail to be amenable.

math.NA↗
arXiv2025

Points on $\operatorname{SO}(3)$ with low logarithmic energy

We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$ spherical points, which come from the zeros of a randomly chosen degree $r$ polynomial, and considering at each of these points a set of $s$ evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of $n=rs$ rotation matrices.

math.CA↗
arXiv2023

Measure-preserving mappings from the unit cube to some symmetric spaces

We construct measure-preserving mappings from the $d$-dimensional unit cube to the $d$-dimensional unit ball and the compact rank one symmetric spaces, namely the $d$-dimensional sphere, the real, complex, and quaternionic projective spaces, and the Cayley plane. We also give a procedure to generate measure-preserving mappings from the $d$-dimensional unit cube to product spaces and fiber bundles under certain conditions.

math.CA↗
arXiv2022

Statistical characterization of the chordal product determinant of Grassmannian codes

We consider the chordal product determinant, a measure of the distance between two subspaces of the same dimension. In information theory, collections of elements in the complex Grassmannian are searched with the property that their pairwise chordal products are as large as possible. We characterize this function from an statistical perspective, which allows us to obtain bounds for the minimal chordal product and related energy of such collections.

cs.IT↗
arXiv2022

The average condition number of most tensor rank decomposition problems is infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks $r\geq 3$ as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.

math.NA↗
arXiv2022

Constellations on the Sphere with Efficient Encoding-Decoding for Noncoherent Communications

In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray coding. Simulation results show that Grass-Lattice has symbol and bit error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature and a coherent pilot-based scheme.

cs.IT↗
arXiv2022

Constrained Riemannian Noncoherent Constellations for the MIMO Multiple Access Channel

We consider the design of multiuser constellations for a multiple access channel (MAC) with K users, with M antennas each, that transmit simultaneously to a receiver equipped with N antennas through a Rayleigh block-fading channel, when no channel state information (CSI) is available to either the transmitter or the receiver. In full-diversity scenarios where the coherence time is at least T>= (K+1)M, the proposed constellation design criterion is based on the asymptotic expression of the multiuser pairwise error probability (PEP) derived by Brehler and Varanasi. In non-full diversity scenarios, for which the previous PEP expression is no longer valid, the proposed design criteria are based on proxies of the PEP recently proposed by Ngo and Yang. Although both the PEP expression and its bounds or proxies were previously considered intractable for optimization, in this work we derive their respective unconstrained gradients. These gradients are in turn used in the optimization of the proposed cost functions in different Riemannian manifolds representing different power constraints. In particular, in addition to the standard unitary space-time modulation (USTM) leading to optimization on the Grassmann manifold, we consider a more relaxed per-codeword power constraint leading to optimization on the so-called oblique manifold, and an average power constraint leading to optimization on the so-called trace manifold. Equipped with these theoretical tools, we design multiuser constellations for the MIMO MAC in full-diversity and non-full-diversity scenarios with state-of-the-art performance in terms of symbol error rate (SER).

cs.IT↗
arXiv2022

Low energy points on the sphere and the real projective plane

We present a generalization of a family of points on $\mathbb{S}^2$, the Diamond ensemble, containing collections of $N$ points on $\mathbb{S}^2$ with very small logarithmic energy for all $N\in\mathbb{N}$. We extend this construction to the real projective plane $\mathbb{RP}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.

math.CA↗
arXiv2022

When can forward stable algorithms be composed stably?

We state some widely satisfied hypotheses, depending only on two functions $g$ and $h$, under which the composition of a stable algorithm for $g$ and a stable algorithm for $h$ is a stable algorithm for the composition $g \circ h$.

math.NA↗
arXiv2021

On Gegenbauer Point Processes on the unit interval

In this note we compute the logarithmic energy of points in the unit interval $[-1,1]$ chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expresion for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.

math.CA↗
arXiv2021

How well-conditioned can the eigenvalue problem be?

The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with exact first order asymptotic.

math.NA↗
arXiv2020

On the minimum value of the condition number of polynomials

In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Beltán, U. Etayo, J. Marzo and J. Ortega-Cerdà, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer.

math.CV↗