arXiv subjects

Jorge Garza-Vargas

Publications and source records attributed to Jorge Garza-Vargas.

13 recordsLinked to original sources

arXiv2026

Finite Free Information Inequalities

We develop finite free information theory for real-rooted polynomials, establishing finite free analogues of entropy and Fisher information monotonicity, as well as the Stam and entropy power inequalities. These results resolve conjectures by Shlyakhtenko and Gribinski and recover inequalities in free probability in the large-degree limit. Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators which preserve real-rootedness. Our proofs leverage a new connection between score vectors and Jacobians of root maps, combined with convexity results for hyperbolic polynomials. We further characterize the equality cases in our inequalities, which arise from Hermite polynomials.

math.PR↗

arXiv2026

A new approach to strong convergence II. The classical ensembles

The first paper in this series introduced a new approach to strong convergence of random matrices that is based primarily on soft arguments. This method was applied to achieve a refined qualitative and quantitative understanding of strong convergence of random permutation matrices and of more general representations of the symmetric group. In this paper, we introduce new ideas that make it possible to achieve stronger quantitative results and that facilitate the application of the method to new models. When applied to the Gaussian GUE/GOE/GSE ensembles of dimension $N$, these methods achieve strong convergence for noncommutative polynomials with matrix coefficients of dimension $\exp(o(N))$. This provides a sharp form of a result of Pisier on strong convergence with coefficients in a subexponential operator space. Analogous results up to logarithmic factors are obtained for Haar-distributed random matrices in $\mathrm{U}(N)/\mathrm{O}(N)/\mathrm{Sp}(N)$. We further illustrate the methods of this paper in the following applications. 1. We obtain improved rates for strong convergence of random permutations. 2. We obtain a quantitative form of strong convergence of the model introduced by Hayes for the solution of the Peterson-Thom conjecture. 3. We prove strong convergence of tensor GUE models of $Γ$-independence. 4. We prove strong convergence of irreducible representations of $\mathrm{U}(N)$ of dimension up to $\exp(N^{1/3-δ})$, improving a result of Magee and de la Salle.

math.PR↗

arXiv2025

A new approach to strong convergence

A family of random matrices $\boldsymbol{X}^N=(X_1^N,\ldots,X_d^N)$ is said to converge strongly to a family of bounded operators $\boldsymbol{x}=(x_1,\ldots,x_d)$ when $\|P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})\|\to\|P(\boldsymbol{x}, \boldsymbol{x}^*)\|$ for every noncommutative polynomial $P$. This phenomenon plays a key role in several recent breakthroughs on random graphs, geometry, and operator algebras. However, proofs of strong convergence are notoriously delicate and have relied largely on problem-specific methods. In this paper, we develop a new approach to strong convergence that uses only soft arguments. Our method exploits the fact that for many natural models, the expected trace of $P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})$ is a rational function of $\frac{1}{N}$ whose lowest order asymptotics are easily understood. We develop a general technique to deduce strong convergence directly from these inputs using the inequality of A. and V. Markov for univariate polynomials and elementary Fourier analysis. To illustrate the method, we develop the following applications. 1. We give a short proof of the result of Friedman that random regular graphs have a near-optimal spectral gap, and obtain a sharp understanding of the large deviations probabilities of the second eigenvalue. 2. We prove a strong quantitative form of the strong convergence property of random permutation matrices due to Bordenave and Collins. 3. We extend the above to any stable representation of the symmetric group, providing many new examples of the strong convergence phenomenon.

math.PR↗

arXiv2024

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that $\boxtimes_d$ converges to $\boxtimes$ as $d$ goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building off our results we give a new short and conceptual proof of a recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] that connects root distributions of polynomial derivatives with free fractional convolution powers.

math.CO↗

arXiv2023

Global Convergence of Hessenberg Shifted QR I: Exact Arithmetic

Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than fifty years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of the shifted QR algorithm on nonsymmetric matrices has remained elusive. We introduce a new family of shifting strategies for the Hessenberg shifted QR algorithm. We prove that when the input is a diagonalizable Hessenberg matrix $H$ of bounded eigenvector condition number $κ_V(H)$ -- defined as the minimum condition number of $V$ over all diagonalizations $VDV^{-1}$ of $H$ -- then the shifted QR algorithm with a certain strategy from our family is guaranteed to converge rapidly to a Hessenberg matrix with a zero subdiagonal entry, in exact arithmetic. Our convergence result is nonasymptotic, showing that the geometric mean of certain subdiagonal entries of $H$ decays by a fixed constant in every $QR$ iteration. The arithmetic cost of implementing each iteration of our strategy scales roughly logarithmically in the eigenvector condition number $κ_V(H)$, which is a measure of the nonnormality of $H$. The key ideas in the design and analysis of our strategy are: (1) We are able to precisely characterize when a certain shifting strategy based on Ritz values stagnates. We use this information to design certain ``exceptional shifts'' which are guaranteed to escape stagnation whenever it occurs. (2) We use higher degree shifts (of degree roughly $\log κ_V(H)$) to dampen transient effects due to nonnormality, allowing us to treat nonnormal matrices in a manner similar to normal matrices.

math.NA↗

arXiv2023

Spectral asymptotics for contracted tensor ensembles

Let $\mathbf{T}_{d, N}: Ω\to \mathbb{R}^{N^d}$ be a random real symmetric Wigner-type tensor. For unit vectors $(u_N^{(i, j)})_{i \in I, j \in [d-2]} \subset \mathbb{S}^{N-1}$, we study the contracted tensor ensemble \[ \left(\frac{1}{\sqrt{N}}\mathbf{T}_{d, N}\left[u_N^{(i, 1)} \otimes \cdots \otimes u_N^{(i, d-2)}\right]\right)_{i \in I}. \] For large $N$, we show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family $(s_i)_{i \in I}$ whose covariance $(\mathbf{K}_{i, i'}^{(N)})_{i, i'\in I}$ is given by the rescaled overlaps of the corresponding symmetrized contractions \[ \mathbf{K}_{i, i'}^{(N)} = \frac{1}{d(d-1)}\langle u_N^{(i, 1)} \odot \cdots \odot u_N^{(i, d-2)}, u_N^{(i', 1)} \odot \cdots \odot u_N^{(i', d-2)} \rangle, \] which is the true covariance of the ensemble up to a $O_d(N^{-1})$ correction. We further characterize the extreme cases of the variance $\mathbf{K}_{i, i}^{(N)} \in [\frac{1}{d!}, \frac{1}{d(d-1)}]$. Our analysis relies on a tensorial extension of the usual graphical calculus for moment method calculations in random matrix theory, allowing us to access the independence in our random tensor ensemble.

math.PR↗

arXiv2022

Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time

We exhibit a randomized algorithm which given a matrix $A\in \mathbb{C}^{n\times n}$ with $\|A\|\le 1$ and $δ>0$, computes with high probability an invertible $V$ and diagonal $D$ such that $\|A-VDV^{-1}\|\le δ$ using $O(T_{MM}(n)\log^2(n/δ))$ arithmetic operations, in finite arithmetic with $O(\log^4(n/δ)\log n)$ bits of precision. Here $T_{MM}(n)$ is the number of arithmetic operations required to multiply two $n\times n$ complex matrices numerically stably, known to satisfy $T_{MM}(n)=O(n^{ω+η})$ for every $η>0$ where $ω$ is the exponent of matrix multiplication (Demmel et al., Numer. Math., 2007). Our result significantly improves the previously best known provable running times of $O(n^{10}/δ^2)$ arithmetic operations for diagonalization of general matrices (Armentano et al., J. Eur. Math. Soc., 2018), and (with regards to the dependence on $n$) $O(n^3)$ arithmetic operations for Hermitian matrices (Dekker and Traub, Lin. Alg. Appl., 1971). It is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic), thereby matching the complexity of other dense linear algebra operations such as inversion and $QR$ factorization up to polylogarithmic factors. The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to any matrix splits its pseudospectrum into $n$ small well-separated components. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts' Newton iteration method (Roberts, Int. J. Control, 1980) for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986.

math.NA↗

arXiv2022

Spectra of infinite graphs via freeness with amalgamation

We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability. With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the non-commutative probability context, it corresponds to the notion of freeness with amalgamation. We show that Cayley graphs of amalgamated free products of groups, as well as universal covering trees, can be constructed using our graph product.

math.CO↗

arXiv2022

Global Convergence of Hessenberg Shifted QR III: Approximate Ritz Values via Shifted Inverse Iteration

We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $δ\|M\|$ in $O(n^4+n^3\log^2(n/δ)+\log(n/δ)^2\log\log(n/δ))$ floating point operations using $O(\log^2(n/δ))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-Vargas, Srivastava, 2022).

math.NA↗

arXiv2022

Global Convergence of Hessenberg Shifted QR II: Numerical Stability

We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap.

math.NA↗

arXiv2020

Point Spectrum of Periodic Operators on Universal Covering Trees

For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree $\mathcal{T}$, we completely characterize the point spectrum of operators $A_{\mathcal{T}}$ on $\mathcal{T}$ arising as pull-backs of local, self-adjoint operators $A_{G}$ on $G$. This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum he derived in (Aomoto, 1991). Our result gives a finite time algorithm to compute the point spectrum of $A_{\mathcal{T}}$ from the graph $G$, and additionally allows us to show that this point spectrum is contained in the spectrum of $A_{G}$. Finally, we prove that typical pull-back operators have a spectral delocalization property: the set of edge weight and vertex potential parameters of $A_{G}$ giving rise to $A_{\mathcal{T}}$ with purely absolutely continuous spectrum is open and its complement has large codimension.

math.SP↗

arXiv2020

The Lanczos Algorithm Under Few Iterations: Concentration and Location of the Output

We study the Lanczos algorithm where the initial vector is sampled uniformly from $\mathbb{S}^{n-1}$. Let $A$ be an $n \times n$ Hermitian matrix. We show that when run for few iterations, the output of Lanczos on $A$ is almost deterministic. More precisely, we show that for any $ \varepsilon \in (0, 1)$ there exists $c >0$ depending only on $\varepsilon$ and a certain global property of the spectrum of $A$ (in particular, not depending on $n$) such that when Lanczos is run for at most $c \log n$ iterations, the output Jacobi coefficients deviate from their medians by $t$ with probability at most $\exp(-n^\varepsilon t^2)$ for $t<\Vert A \Vert$. We directly obtain a similar result for the Ritz values and vectors. Our techniques also yield asymptotic results: Suppose one runs Lanczos on a sequence of Hermitian matrices $A_n \in M_n(\mathbb{C})$ whose spectral distributions converge in Kolmogorov distance with rate $O(n^{-\varepsilon})$ to a density $μ$ for some $\varepsilon > 0$. Then we show that for large enough $n$, and for $k=O(\sqrt{\log n})$, the Jacobi coefficients output after $k$ iterations concentrate around those for $μ$. The asymptotic setting is relevant since Lanczos is often used to approximate the spectral density of an infinite-dimensional operator by way of the Jacobi coefficients; our result provides some theoretical justification for this approach. In a different direction, we show that Lanczos fails with high probability to identify outliers of the spectrum when run for at most $c' \log n$ iterations, where again $c'$ depends only on the same global property of the spectrum of $A$. Classical results imply that the bound $c' \log n$ is tight up to a constant factor.

math.NA↗

arXiv2016

Polytopality of Maniplexes

Given an abstract polytope $\cal P$, its flag graph is the edge-coloured graph whose vertices are the flags of $\cal P$ and the $i$-edges correspond to $i$-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, given a maniplex $\cal M$, on can define a poset $\cal P_M$ by means of the non empty intersection of its faces. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex $\cal M$ in order for $\cal P_M$ to be an abstract polytope. Moreover, in such case, we show that $\cal M$ is isomorphic to the flag graph of $\cal P_M$. This in turn gives necessary and sufficient conditions for a maniplex to be (isomorphic to) the flag graph of a polytope.

math.CO↗