arXiv subjects

Ioana Dumitriu

Publications and source records attributed to Ioana Dumitriu.

At least 19 recordsLinked to original sources

arXiv2026

Fast and Stable Arbitrary Precision Approximate Matrix Multiplication

We present several new and surprising results on the theory and practice of Arbitrary Precision Approximate (APA) matrix multiplication, which refers to algorithms that output merely approximations to matrix multiplication even in exact arithmetic. While APA algorithms have been around for nearly half a century, and despite the fact that they have potentially lower (asymptotic) cost than many fast exact matrix multiplication schemes, they have not been studied as closely, to the extent that relatively little is known about their numerical properties. In this paper, we shine a spotlight on these algorithms by presenting three important results. First, we provide a new search algorithm for APA routines that is easy to use, automatically generates encoding matrices with functional entries, and can match or beat the best known APA algorithms for several sizes. In particular, we improve the upper bound on the border rank of $\langle 2,4,5 \rangle$ matrix multiplication. Second, we combine this insight with alternative bases to minimize the number of additions in an APA algorithm, thereby improving its leading coefficient. Finally, we prove the same kind of stability that characterizes fast exact matrix multiplication for an algorithm of Bini [Calcolo 1980] that computes an exact product of two matrices using a modified APA scheme (with or without alternative bases).

math.NA↗

arXiv2026

Majority Dynamics on Resampled Sparse Erdős--Rényi Graphs: Gaussian Winner Selection in Collaborative Agent Networks

Collective decisions in agent networks emerge from repeated local interactions, as agents update their opinions (colors) based on their neighbors, potentially leading to consensus. In sparse communication graphs, repeated interactions can amplify both the initial majority preference and the random fluctuations caused by individual agents. Understanding when the initial majority determines the final preference and how quickly an agreement emerges is, therefore, a basic question in collective decision-making. We study binary majority dynamics as an idealized model of this process. At each round, the interaction graph is independently resampled from the Erdős--Rényi model $\mathbb{G}(N,p)$, with $p=b\log N/N$ and fixed $b>1$. Each agent adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. For a fixed initial configuration, let $Δ_0$ denote the initial difference between the number of blue agents and the number of red agents. We identify three regimes for the consensus time, determined by $|Δ_0|$. First, when $Δ_0\sqrt p=O(1)$, the probability of blue unanimity is $$ Φ\left(Δ_0\sqrt{2p/π}\right) + o(1), $$ where $Φ$ is the cumulative distribution function of a standard Gaussian. Moreover, consensus is reached after $(1+o(1))\log N/\log\log N$ rounds with high probability. Second, in the intermediate regime $1\llΔ_0 \sqrt{p}\lesssim \sqrt{N}$, we establish explicit high probability upper and lower bounds on the time to blue-unanimity. Lastly, when the initial blue advantage is above an explicit constant multiple of $N/\sqrt{\log N}$, blue unanimity is guaranteed within two rounds with high probability. Our findings provide a complete description of how the initial advantage influences winner selection and the speed of consensus in this type of majority dynamics with sparse interactions.

math.PR↗

arXiv2026

Structured Divide-and-Conquer for the Definite Generalized Eigenvalue Problem

This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils $(A,B)$ in which $A$ and $B$ are Hermitian and the Crawford number $γ(A,B) = \min_{\|x\|_2 = 1} |x^H(A+iB)x|$ is positive. Adapted from the fastest known method for diagonalizing arbitrary matrix pencils [Foundations of Computational Mathematics 2024], the algorithm is both inverse-free and highly parallel. As in the general case, randomization takes the form of perturbations applied to the input matrices, which regularize the problem for compatibility with fast, divide-and-conquer eigensolvers -- i.e., the now well-established phenomenon of pseudospectral shattering. We demonstrate that this high-level approach to diagonalization can be executed in a structure-aware fashion by (1) extending pseudospectral shattering to definite pencils under structured perturbations (either random diagonal or sampled from the Gaussian unitary ensemble) and (2) formulating the divide-and-conquer procedure in a way that maintains definiteness. The result is a specialized solver whose complexity, when applied to definite pencils, is provably lower than that of general divide-and-conquer.

math.NA↗

arXiv2026

Majority Dynamics on Assortative Sparse Stochastic Block Models

Majority dynamics is a two-opinion process in which each vertex repeatedly updates to the majority opinion among its neighbors. We study this process on a resampled sparse binary stochastic block model in the assortative regime. At each time step, a graph is sampled from the current opinion partition: vertices with the same opinion are joined with probability $α=a\log N/N$, while vertices with differing opinions are joined with probability $β=b\log N/N$, where $a>b>1$. Let $B_t$ and $R_t$ denote the blue and red camps at time $t$. We show that the weighted advantage $\widetildeΔ_t =b|B_t|-a|R_t|$, rather than the unweighted advantage $Δ_t=|B_t|-|R_t|$ alone, governs the pace to unanimity. Our results, which hold with high probability as $N\to\infty$, identify three regimes for blue unanimity under the initial blue advantage, i.e., $Δ_0>0$: constant time, subpolynomial time, and polynomial time. First, when $\widetildeΔ_0 \gtrsim -N/\sqrt{\log N}$, blue unanimity occurs within three updates. Second, when $\widetildeΔ_0 < 0$ and $|\widetildeΔ_0| = o(N)$, blue unanimity occurs within $N^{o(1)}$ updates. Furthermore, when $\widetildeΔ_0 < 0$, $|\widetildeΔ_0| = O(N)$, and $Δ_0\gg\sqrt{N/\log N}$, blue unanimity still occurs within $N^{I_0+o(1)}$ updates, where \[ I_0= \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}}-\sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] and $\mathbf{ReLU}(x)=\max\{x,0\}$. Conversely, away from the weighted threshold, when $|B_0|/|R_0|\le a/b-κ$ and $Δ_0>0$, $N^{I_0 - o(1)}$ updates are necessary for blue unanimity. Our analysis relies on detailed estimates for one-vertex flip probabilities in sparse binomial differences, which could be of independent interest.

math.PR↗

arXiv2026

BBP Phase Transition for a Doubly Sparse Deformed Model

We prove the equivalent of the Baik, Ben Arous, Péché (2004) phenomenon for a novel, doubly sparse model where both the Wigner noise matrix and signal vector(s) are sparse. Specifically, we consider a deformed sub-Gaussian sparse Wigner ensemble with a fixed number of sub-Gaussian spike vectors of the same-order sparsity added. We show that spike vectors with signals greater than one are correlated with the top eigenvectors of the deformed ensemble and that each spike vector of signal greater than one induces an outlier eigenvalue. Notably, our results hold in the supercritical sparsity regime for the Wigner matrix ($q \gg \frac{\log n}{n}$) and for any sparse spike vector with an unbounded number of entries ($np\to \infty$). No further relationship between the sparsities of the noise matrix ($q$) and spike vectors ($p$) is necessary. This generalizes the work of Benaych-Georges and Nadakuditi (2010) and Péché (2005).

math.PR↗

arXiv2026

Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $γ$ fraction of the vertices classified correctly, where $γ\in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati (2017) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

math.ST↗

arXiv2026

Optimal and exact recovery on the general nonuniform Hypergraph Stochastic Block Model

Consider the community detection problem in random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), where each hyperedge appears independently with some given probability depending only on the labels of its vertices. We establish, for the first time in the literature, a sharp threshold for exact recovery under this non-uniform case, subject to minor constraints; in particular, we consider the model with multiple communities. One crucial point here is that by aggregating information from all the uniform layers, we may obtain exact recovery even in cases when this may appear impossible if each layer were considered alone. Besides that, we prove a wide-ranging, information-theoretic lower bound on the number of misclassified vertices \emph{for any algorithm}, depending on a \emph{generalized Chernoff-Hellinger} divergence involving model parameters. We provide two efficient algorithms which successfully achieve exact recovery when above the threshold, and attain the lowest possible mismatch ratio when the exact recovery is impossible, proved to be optimal. The theoretical analysis of our algorithms relies on the concentration and regularization of the adjacency matrix for non-uniform random hypergraphs, which could be of independent interest. We also address some open problems regarding parameter knowledge and estimation.

math.ST↗

arXiv2026

Rank One Completion for Higher Order Tensors

We study the rank one completion problem for tensors of arbitrary orders. The notion of rank one determinable tensors is introduced. We explore its properties and propose a recursive algorithm for computing rank one tensor completion. This algorithm only requires solving linear systems and computing singular vectors. In the absence of noise, it produces a unique rank one completion under some assumptions. In the presence of noise, we show that the computed rank one tensor completion is close to the exact one when the noise is sufficiently small. Numerical experiments demonstrate the efficiency and accuracy of the proposed method.

math.NA↗

arXiv2026

Fast and Inverse-Free Algorithms for Deflating Subspaces

This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil $(A,B)$, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to computing these projectors, which combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. In an appendix, we demonstrate that recent, randomized divide-and-conquer eigensolvers -- which are built on fast methods for individual projectors -- can be adapted to produce the generalized Schur form of any matrix pencil in nearly matrix multiplication time.

math.NA↗

arXiv2025

Singular values of sparse random rectangular matrices: Emergence of outliers at criticality

Consider the random bipartite Erdős-Rényi graph $\mathbb{G}(n, m, p)$, where each edge with one vertex in $V_{1}=[n]$ and the other vertex in $V_{2} =[m]$ is connected with probability $p$, and $n=\lfloor γm\rfloor$ for a constant aspect ratio $γ\geq 1$. It is well known that the empirical spectral measure of its centered and normalized adjacency matrix converges to the Marčenko-Pastur (MP) distribution. However, largest and smallest singular values may not converge to the right and left edges, respectively, especially when $p = o(1)$. Notably, it was proved by Dumitriu and Zhu (2024) that there are almost surely no singular value outside the compact support of the MP law when $np = ω(\log(n))$. In this paper, we consider the critical sparsity regime where $p = b\log(n)/\sqrt{mn}$ for some constant $b>0$. We quantitatively characterize the emergence of outlier singular values as follows. For explicit $b_{*}$ and $b^{*}$ as functions of $γ$, we prove that when $b > b_{*}$, there is no outlier outside the bulk; when $b^{*}< b < b_{*}$, outliers are present only outside the right edge of the MP law; and when $b < b^{*}$, outliers are present on both sides, all with high probability. Moreover, the locations of those outliers are precisely characterized by a function depending on the largest and smallest degree vertices of the random graph. We estimate the number of outliers as well. Our results follow the path forged by Alt, Ducatez and Knowles (2021), and can be extended to sparse random rectangular matrices with bounded entries.

math.PR↗

arXiv2024

Extreme singular values of inhomogeneous sparse random rectangular matrices

We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a bipartite block structure. We obtain probabilistic upper (respectively, lower) bounds for the largest (respectively, smallest) singular values of a large rectangular random matrix $X$. These bounds are given in terms of the maximal and minimal $\ell_2$-norms of the rows and columns of the variance profile of $X$. The proofs involve finding probabilistic upper bounds on the spectral radius of an associated non-backtracking matrix $B$. The two-sided bounds can be applied to the centered adjacency matrix of sparse inhomogeneous Erdős-Rényi bipartite graphs for a wide range of sparsity, down to criticality. In particular, for Erdős-Rényi bipartite graphs $G(n,m,p)$ with $p=ω(\log n)/n$, and $m/n\to y \in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Marčenko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.

math.PR↗

arXiv2024

Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling $(A,B)$ regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of $(A,B)$ in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible $S,T$ and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - ST^{-1}||_2 \leq \varepsilon$ in at most $O \left(\log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where $T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact arithmetic matrix pencil diagonalization.

math.NA↗

arXiv2024

Faithful and Efficient Explanations for Neural Networks via Neural Tangent Kernel Surrogate Models

A recent trend in explainable AI research has focused on surrogate modeling, where neural networks are approximated as simpler ML algorithms such as kernel machines. A second trend has been to utilize kernel functions in various explain-by-example or data attribution tasks. In this work, we combine these two trends to analyze approximate empirical neural tangent kernels (eNTK) for data attribution. Approximation is critical for eNTK analysis due to the high computational cost to compute the eNTK. We define new approximate eNTK and perform novel analysis on how well the resulting kernel machine surrogate models correlate with the underlying neural network. We introduce two new random projection variants of approximate eNTK which allow users to tune the time and memory complexity of their calculation. We conclude that kernel machines using approximate neural tangent kernel as the kernel function are effective surrogate models, with the introduced trace NTK the most consistent performer. Open source software allowing users to efficiently calculate kernel functions in the PyTorch framework is available (https://github.com/pnnl/projection\_ntk).

cs.LG↗

arXiv2023

Spectral Evolution and Invariance in Linear-width Neural Networks

We investigate the spectral properties of linear-width feed-forward neural networks, where the sample size is asymptotically proportional to network width. Empirically, we show that the spectra of weight in this high dimensional regime are invariant when trained by gradient descent for small constant learning rates; we provide a theoretical justification for this observation and prove the invariance of the bulk spectra for both conjugate and neural tangent kernels. We demonstrate similar characteristics when training with stochastic gradient descent with small learning rates. When the learning rate is large, we exhibit the emergence of an outlier whose corresponding eigenvector is aligned with the training data structure. We also show that after adaptive gradient training, where a lower test error and feature learning emerge, both weight and kernel matrices exhibit heavy tail behavior. Simple examples are provided to explain when heavy tails can have better generalizations. We exhibit different spectral properties such as invariant bulk, spike, and heavy-tailed distribution from a two-layer neural network using different training strategies, and then correlate them to the feature learning. Analogous phenomena also appear when we train conventional neural networks with real-world data. We conclude that monitoring the evolution of the spectra during training is an essential step toward understanding the training dynamics and feature learning.

cs.LG↗

arXiv2023

Global eigenvalue fluctuations of random biregular bipartite graphs

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent interest. We also prove a semicircle law for random $(d_1,d_2)$-biregular bipartite graphs when $\frac{d_1}{d_2}\to\infty$. As an application, we translate the results to adjacency matrices of uniformly distributed random regular hypergraphs.

math.PR↗

arXiv2022

Spectra of Overlapping Wishart Matrices and the Gaussian Free Field

Consider a doubly-infinite array of iid centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in work of Borodin, Borodin and Gorin, and Johnson and Pal can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.

math.PR↗

arXiv2021

Spectra of random regular hypergraphs

In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996). Our main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs. We then relate the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hypergraphs. We also prove the spectral gap for the non-backtracking operator of a random regular hypergraph introduced in Angelini et al. (2015). Finally, we obtain the convergence of the empirical spectral distribution (ESD) for random regular hypergraphs in different regimes. Under certain conditions, we can show a local law for the ESD.

math.CO↗

arXiv2021

Spectral gap in random bipartite biregular graphs and applications

We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara-Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. A byproduct of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full-rank with high probability. Finally, we illustrate applications to community detection, coding theory, and deterministic matrix completion.

math.PR↗