Email: lastname at berkeley dot edu
Across a wide range of applications in the physical and data sciences, the computational
study of Markov chains over large state spaces motivates practitioners to distill the dynamics
into a reduced model over a smaller state space, with a view toward preserving the key long-timescale
properties of the original chain. Motivated by the challenge of building a solid theoretical framework
for such Markov state models, our recent work introduces a rigorous and comprehensive
approximation theory for the compression of a general reversible Markov chain, bounding the
compression error with respect to the full dynamical evolution. This work connects with our advances in
column subset selection which offer strong a priori guarantees for the compression.
Low-rank structure plays a fundamental role in scientific computation. Recent directions include column subset selection for matrix as well as
tensor decompositions, constructions of quantized tensor trains and related operators, a sketching-based approach to generative
modeling with tensor networks, and calculation of committor functions for high-dimensional stochastic processes. Even our recent efforts in developing adaptive multigrid solvers,
which flexibly adapt to a variety of numerical PDE,
are grounded in randomized low-rank numerical linear algebra.
One recent work introduces entropy regularization as a general strategy to obtain
nearly-optimal scaling solvers for SDP, without any rank assumptions, by using the von Neumann
entropy as a regularizer. The resulting dual problem
can be approached using matrix-free randomized trace estimation techniques, combining
tools from numerical analysis and randomized numerical linear algebra. See this paper for a general
convergence analysis and this paper for
detailed application to the hard problem of permutation synchronization.
Monte Carlo sampling is the most flexible and widely-used tool for estimating sums and integrals in high dimensions. I am
interested in developing generic sampling techniques such as this ensemble MCMC sampler that can circumvent slow
mixing due to metastability,
as well as specialized method development in quantum Monte Carlo, e.g.,
scale-invariant HMC for condensed matter field theories.
Motivated by the need for quantum chemistry basis sets that adapt to problem geometry while enabling fast computation, we
have introduced several approaches to structured discretization for electronic structure, including nested gausslets and
adaptive grid deformation. More recent efforts define a new discontinuous Galerkin framework
for electronic structure, coupled to fast multigrid solvers. In fact, parallel efforts in developing adaptive multigrid techniques point to
flexible automatic approaches for computational PDEs more broadly.
I have written a text on algorithms for continuous optimization, tying together various perspectives from the convex analysis, machine learning,
theoretical computer science, and scientific computing communities. I hope you find it useful!
Column subset selection and Markov chain compression
Fast adaptive discontinuous basis sets for electronic structure
UC Berkeley
Here is a list of what I read in graduate school, which may suggest
some interesting resources.
Two views on optimal transport and its numerical solution (2015).
My undergraduate thesis (supervised by Yanir Rubinstein
and Rafe Mazzeo),
which presented two new formulations of optimal transport problems leading to two corresponding methods for numerically solving them.
This thesis won the Kennedy Thesis Prize for the top undergraduate thesis in the natural sciences at Stanford.
Michael Lindsey
Office: 1089 Evans Hall
Background: I am an Assistant Professor in the Department of Mathematics at UC Berkeley, as well as a
Faculty Scientist in the Mathematics Group at Lawrence Berkeley National Laboratory.
Here is my CV (4/2026).
Research interests: I work on fast algorithms for scientific computing, driven by numerical linear algebra, optimization, and randomization, with a special focus on
high-dimensional scientific computing problems. Such challenges include quantum many-body
problems arising in quantum chemistry and condensed matter physics, as well the analysis of stochastic processes with many degrees of freedom. Approaches draw on a wide variety of
techniques including low-rank decomposition, semidefinite relaxation, Monte Carlo sampling, and optimization over parametric function classes such as tensor networks and neural networks.
Selected research directions
Click titles to expand.
Analysis of Markov processes
Rank-structured matrix and tensor computations
Entropy-regularized convex programming
These directions also connect with recent work on stochastic density functional theory, which puts forward
a theoretically justified algorithmic framework for DFT with optimal computational scaling.
Monte Carlo sampling
Fast linear solvers and adaptive discretization of quantum chemistry
Pedagogical materials
For an introduction to quantum many-body physics written for an applied math audience, see
my dissertation (8/2019) and in particular the first part.
Also see this talk for a tutorial on many-body perturbation theory and Green's
function methods.
Selected presentations
Expand
Brown Probability Seminar (March 2026)
[slides]
A mathematical analysis of stochastic density functional theory
Telluride Science Workshop (June 2025)
[slides]
Adaptive diagonal basis sets for electronic structure
SciCADE (July 2024)
[slides]
Direct interpolative construction of quantized tensor trains
Tensor4All Meeting (April 2024)
[slides]
Interacting ensemble MCMC and fast entropically regularized SDP
Mathematics of Data and Decisions at UC Davis (MADDD) Seminar (November 2023)
[slides]
Efficient sampling for lattice QMC
Julian Schwinger Foundation Workshop on the Fermion Sign Problem (July 2023)
[slides]
Scalable variational embedding for quantum many-body problems
SIAM Conference on Mathematical Aspects of Materials Science (May 2021)
[slides]
Optimization for variational Monte Carlo with neural quantum states
Modeling and Simulation Group Seminar: Machine Learning in Science at NYU (April 2021)
[slides]
Optimal transport via a Monge-Ampère optimization problem
SIAM Conference on the Mathematics of Data Science (May 2020)
[slides]
Toward sharp error analysis of extended Langrangian molecular dynamics for polarizable force field simulation
Ki-Net Young Researchers Workshop (October 2019)
[slides]
Semidefinite relaxation of multi-marginal optimal transport, with application to strictly correlated electrons in second quantization
ICIAM (July 2019)
[slides]
A classical statistical mechanics approach to understanding Green's function methods and the Luttinger-Ward formalism
Oberwolfach (March 2018)
[slides]
Adaptive compression for Hartree-Fock-like equations
SIAM Conference on Applied Linear Algebra (May 2018)
[slides]
Publications and preprints
Yulong Pan and Michael Lindsey
[arXiv:2510.21213]
An approximation theory for Markov chain compression
with Mark Fornace
[arXiv:2506.22918]
Fast entropy-regularized SDP relaxations for permutation synchronization
with Yunpeng Shi
[arXiv:2506.20191]
Non-Euclidean dual gradient ascent for entropically regularized linear and semidefinite programming
with Yuhang Cai
[arXiv:2506.09711]
[code]
Improved energies and local energies with weighted variational Monte Carlo
Huan Zhang, Robert J. Webber, Michael Lindsey, Timothy C. Berkelbach, and Jonathan Weare
Phys. Rev. Research 8, 013213 (2026)
[journal] [arXiv:2507.01905]
Toward optimal-scaling DFT: stochastic Hartree theory in the thermodynamic and complete basis set limits at arbitrary temperature
with Yuhang Cai
[arXiv:2504.15816]
[code]
Geometric adaptive smoothed aggregation multigrid for discontinuous Galerkin discretisations
Yulong Pan, Michael Lindsey, and Per-Olof Persson
[arXiv:2504.13373]
[code]
Fast and accurate interpolative decompositions for general, sparse, and structured tensors
Yifan Zhang, Mark Fornace, and Michael Lindsey
[arXiv:2503.18921]
[code]
Implicit bias of gradient descent for non-homogeneous deep networks
Yuhang Cai, Kangjie Zhou, Jingfeng Wu, Song Mei, Michael Lindsey, and Peter L. Bartlett
ICML 2025
[arXiv:2502.16075]
MNE: overparametrized neural evolution with applications to diffusion processes and sampling
Michael Lindsey
[arXiv:2502.03645]
[code]
A gradient-based and determinant-free framework for fully Bayesian Gaussian process regression
with P. Michael Kielstra
SIAM/ASA JUQ 13, 1709 (2025)
[journal] [arXiv:2412.20884]
[code]
Fast and spectrally accurate construction of adaptive diagonal basis sets for electronic structure
with Sandeep Sharma
J. Chem. Phys. 161, 214111 (2024)
[journal]
[arXiv:2407.06171]
Gaussian process regression with log-linear scaling for common non-stationary kernels
with P. Michael Kielstra
Appl. Comput. Harmon. Anal. 79, 101792 (2025)
[journal]
[arXiv:2407.03608]
[code]
Column and row subset selection using nuclear scores: algorithms and theory for Nyström approximation, CUR decomposition, and graph Laplacian reduction
with Mark Fornace
[arXiv:2407.01698]
[code]
Large stepsize gradient descent for non-homogeneous two-layer networks: margin improvement and fast optimization
Yuhang Cai, Jingfeng Wu, Song Mei, Michael Lindsey, and Peter L. Bartlett
NeurIPS 2024
[conference] [arXiv:2406.08654]
Direct interpolative construction of the discrete Fourier transform as a matrix product operator
with Jielun Chen
Appl. Comput. Harmon. Anal. 81, 101817 (2026)
[journal] [arXiv:2404.03182]
Simple diagonal designs with reconfigurable real-time circuits
Yizhi Shen, Katherine Klymko, Eran Rabani, Norm M. Tubman, Daan Camps, Roel Van Beeumen, and Michael Lindsey
Phys. Rev. Research 8, 013042 (2026)
[journal] [arXiv:2401.04176]
Multiscale interpolative construction of quantized tensor trains
Michael Lindsey
[arXiv:2311.12554]
Multimarginal generative modeling with stochastic interpolants
with Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden
ICLR 2024
[conference] [arXiv:2310.03695]
Nested gausslet basis sets
Steven R. White and Michael Lindsey
J. Chem. Phys. 159, 234112 (2023)
[journal]
[arXiv:2309.10704]
Fast randomized entropically regularized semidefinite programming
Michael Lindsey
Accepted SIAM J. Sci. Comput.
[arXiv:2303.12133]
Understanding and eliminating spurious modes in variational Monte Carlo using collective variables
Huan Zhang, Robert J. Webber, Michael Lindsey, Timothy C. Berkelbach, and Jonathan Weare
Phys. Rev. Research 5, 023101 (2023)
[journal]
[arXiv:2211.09767]
Non-Hertz-Millis scaling of the antiferromagnetic quantum critical metal via scalable Hybrid Monte Carlo
Peter Lunts, Michael S. Albergo, and Michael Lindsey
Nat. Commun. 14, 2547 (2023)
[journal]
[arXiv:2204.14241]
Generative modeling via tensor train sketching
YoonHaeng Hur, Jeremy G. Hoskins, Michael Lindsey, E.M. Stoudenmire, and Yuehaw Khoo
Appl. Comput. Harmon. Anal. 67, 101575 (2023)
[journal]
[arXiv:2202.11788]
Committor functions via tensor networks
with Yian Chen, Jeremy Hoskins, and Yuehaw Khoo
J. Comput. Phys. 472, 111646 (2023)
[journal]
[arXiv:2106.12515]
Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo
Robert J. Webber and Michael Lindsey
Phys. Rev. Research 4, 033099 (2022)
[journal]
[arXiv:2106.10558]
Ensemble Markov chain Monte Carlo with teleporting walkers
with Jonathan Weare and Anna Zhang
SIAM/ASA JUQ 10, 860 (2022)
[journal] [arXiv:2106.02686]
Scalable semidefinite programming approach to variational embedding for quantum many-body problems
with Yuehaw Khoo
J. Comput. Phys. 510, 113041 (2024)
[journal] [arXiv:2106.02682]
Multiscale semidefinite programming approach to positioning problems with pairwise structure
with Yian Chen and Yuehaw Khoo
J. Sci. Comput. 101:42 (2024)
[journal] [arXiv:2012.10046]
Towards sharp error analysis of extended Lagrangian molecular dynamics
with Dong An and Lin Lin
J. Comput. Phys. 466, 111403 (2022)
[journal]
[arXiv:2010.07508]
Enhancing robustness and efficiency of density matrix embedding theory via semidefinite programming and local correlation potential fitting
Xiaojie Wu, Michael Lindsey, Tiangang Zhou, Yu Tong, and Lin Lin
Phys. Rev. B 102, 085123 (2020)
[journal]
[arXiv:2003.00873]
(Note: Editor's Suggestion.)
Variational embedding for quantum many-body problems
with Lin Lin
Comm. Pure Appl. Math. 75, 2033 (2022)
[journal] [arXiv:1910.00560]
Efficient hybridization fitting for dynamical mean-field theory via semi-definite relaxation
Carlos Mejuto-Zaera, Leonardo Zepeda-Núñez, Michael Lindsey, Norm Tubman, Birgitta Whaley, and Lin Lin
Phys. Rev. B 101, 035143 (2020)
[journal]
[arXiv:1907.07191]
Semidefinite relaxation of multi-marginal optimal transport for strictly correlated electrons in second quantization
with Yuehaw Khoo, Lin Lin, and Lexing Ying
SIAM J. Sci. Comput. 42, B1462 (2020)
[journal]
[arXiv:1905.08322]
Projected density matrix embedding theory with applications to the two-dimensional Hubbard model
Xiaojie Wu, Zhi-Hao Cui, Yu Tong, Michael Lindsey, Garnet Kin-Lic Chan, and Lin Lin
J. Chem. Phys. 151, 064108 (2019)
[journal]
[arXiv:1905.00886]
Sparsity pattern of the self-energy for classical and quantum impurity problems
with Lin Lin
Ann. Henri Poincaré 21, 2219 (2020)
[journal]
[arXiv:1902.04796]
Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part II: Non-perturbative analysis
with Lin Lin
Arch. Ration. Mech. Anal. 242, 527 (2021)
[journal]
[arXiv:1809.02901]
Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part I: Perturbative approach
with Lin Lin
Arch. Ration. Mech. Anal. 242, 581 (2021)
[journal]
[arXiv:1809.02900]
Variational structure of Luttinger-Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory
with Lin Lin
Proc. Natl. Acad. Sci. 115, 2282 (2018)
[journal]
[.pdf] [supporting info]
Convergence of adaptive compression methods for Hartree-Fock-like equations
with Lin Lin
Comm. Pure Appl. Math. 72, 451 (2019)
[journal] [arXiv:1703.05441]
Optimal transport via a Monge-Ampère optimization problem
with Yanir A. Rubinstein
SIAM J. Math. Anal. 49, 3073 (2017)
[journal] [.pdf]
(Note: recognized with the
SIAM Student Paper Prize.)
On discontinuity of planar optimal transport maps
with Otis Chodosh, Vishesh Jain, Lyuboslav Panchev, and Yanir A. Rubinstein
Journal of Topology and Analysis 07, 239 (2015)
[journal] [arXiv:1312.2929]
(Note: research carried out during
SURIM 2012.)
Infrared imagery of streak formation in a breaking wave
Ivan Savelyev, Robert A. Handler, and Michael Lindsey
Physics of Fluids 24, 121701 (2012)
[journal]
Teaching
Expand
Fall 2025: Math 170 and Math 228A
Spring 2024:Math 128B
Fall 2023: Math 228A
Spring 2023: Math 128B
Fall 2022: Math 228A
New York University
Spring 2022: Linear Algebra I (graduate)
Fall 2021: Mathematical Statistics
Spring 2021: Linear Algebra I (graduate)
Fall 2020: Calculus I
UC Berkeley (GSI)
Spring 2018: Math 54, Alexander Paulin [section page]
Spring 2016: Math 53, Denis Auroux [quizzes]
Fall 2015: Math 1B, Ole Hald
Miscellanea
Reading list
Undergraduate research
Expand
[.pdf]
Asymptotics of Hermite polynomials (2015). A largely expository paper
(for a course on orthogonal polynomials) about asymptotics of Hermite polynomials and the Gaussian Unitary Ensemble (GUE).
Presents a result about the stationary states for the quantum harmonic oscillator, which,
though likely nothing new, I think is fairly cool.
[.pdf]
Spectral methods for neural computation (2013/14). A presentation for the
Brains in Silicon
lab outlining some ideas about how favorable Fourier-domain properties of certain neural
tuning curves are naturally suited (in an idealized setting) for the computation of simple functions.
[slides]
3D Shape Understanding Using Machine Learning (2013). A presentation about my
work on using a deep learning framework to perform labeled segmentation of discrete
surfaces and extract multiscale learned shape descriptors. To help with training,
I introduced a new set of shape descriptors based on conformal maps.
[slides]
(Note: research carried out during CURIS
2013 in Stanford's Geometric Computation Group.)
The k-discs algorithm and its kernelization (2012). Introduces and
analyzes an extension of k-means allowing cluster centroids to be discs of
arbitrary dimension, capable of recovering more diverse cluster geometries.
[.pdf]
(Note: class project for CS 229 at Stanford.)