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05/17/2023
The Complexity of Diagonalization
We survey recent progress on efficient algorithms for approximately diag...
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12/14/2022
On Eigenvalue Gaps of Integer Matrices
Given an n× n matrix with integer entries in the range [-h,h], how close...
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05/13/2022
Global Convergence of Hessenberg Shifted QR II: Numerical Stability
We develop a framework for proving rapid convergence of shifted QR algor...
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05/13/2022
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 i...
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11/15/2021
Global Convergence of Hessenberg Shifted QR I: Dynamics
Rapid convergence of the shifted QR algorithm on symmetric matrices was ...
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09/28/2021
Bit Complexity of Jordan Normal Form and Spectral Factorization
We study the bit complexity of two related fundamental computational pro...
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09/23/2021
Many nodal domains in random regular graphs
Let G be a random d-regular graph. We prove that for every constant α > ...
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01/28/2021
A Spectral Approach to Polytope Diameter
We prove upper bounds on the graph diameters of polytopes in two setting...
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07/25/2020
Support of Closed Walks and Second Eigenvalue Multiplicity of the Normalized Adjacency Matrix
We show that the multiplicity of the second normalized adjacency matrix ...
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05/18/2020
Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations
Let G_n be an n × n matrix with real i.i.d. N(0,1/n) entries, let A be a...
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12/18/2019
Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time
We exhibit a randomized algorithm which given a square n× n complex matr...
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08/10/2019
High-girth near-Ramanujan graphs with localized eigenvectors
We show that for every prime d and α∈ (0,1/6), there is an infinite sequ...
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06/27/2019
Gaussian Regularization of the Pseudospectrum and Davies' Conjecture
A matrix A∈C^n× n is diagonalizable if it has a basis of linearly indepe...
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03/21/2018
On Non-localization of Eigenvectors of High Girth Graphs
We prove improved bounds on how localized an eigenvector of a high girth...
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12/29/2017
Optimal Lower Bounds for Sketching Graph Cuts
We study the space complexity of sketching cuts and Laplacian quadratic ...
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11/30/2017