arXiv subjects

Lap Chi Lau

Publications and source records attributed to Lap Chi Lau.

At least 19 recordsLinked to original sources

arXiv2026

Derandomizing Matrix Concentration Inequalities from Free Probability

Recently, sharp matrix concentration inequalities~\cite{BBvH23,BvH24} were developed using the theory of free probability. In this work, we design polynomial time deterministic algorithms to construct outcomes that satisfy the guarantees of these inequalities. As direct consequences, we obtain polynomial time deterministic algorithms for the matrix Spencer problem~\cite{BJM23} and for constructing near-Ramanujan graphs. Our proofs show that the concepts and techniques in free probability are useful not only for mathematical analyses but also for efficient computations.

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arXiv2025

A Combinatorial Characterization of Constant Mixing Time

Classical spectral graph theory characterizes graphs with logarithmic mixing time. In this work, we present a combinatorial characterization of graphs with constant mixing time. The combinatorial characterization is based on the small-set bipartite density condition, which is weaker than having near-optimal spectral radius and is stronger than having near-optimal small-set vertex expansion.

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arXiv2025

Optimal Bounds for Tyler's M-Estimator for Elliptical Distributions

A fundamental problem in statistics is estimating the shape matrix of an Elliptical distribution. This generalizes the familiar problem of Gaussian covariance estimation, for which the sample covariance achieves optimal estimation error. For Elliptical distributions, Tyler proposed a natural M-estimator and showed strong statistical properties in the asymptotic regime, independent of the underlying distribution. Numerical experiments show that this estimator performs very well, and that Tyler's iterative procedure converges quickly to the estimator. Franks and Moitra recently provided the first distribution-free error bounds in the finite sample setting, as well as the first rigorous convergence analysis of Tyler's iterative procedure. However, their results exceed the sample complexity of the Gaussian setting by a $\log^{2} d$ factor. We close this gap by proving optimal sample threshold and error bounds for Tyler's M-estimator for all Elliptical distributions, fully matching the Gaussian result. Moreover, we recover the algorithmic convergence even at this lower sample threshold. Our approach builds on the operator scaling connection of Franks and Moitra by introducing a novel pseudorandom condition, which we call $\infty$-expansion. We show that Elliptical distributions satisfy $\infty$-expansion at the optimal sample threshold, and then prove a novel scaling result for inputs satisfying this condition.

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arXiv2024

Streaming and Communication Complexity of Load-Balancing via Matching Contractors

In the load-balancing problem, we have an $n$-vertex bipartite graph $G=(L, R, E)$ between a set of clients and servers. The goal is to find an assignment of all clients to the servers, while minimizing the maximum load on each server, where load of a server is the number of clients assigned to it. We study load-balancing in the one-way communication model: the edges of the input graph are partitioned between Alice and Bob, and Alice needs to send a message to Bob for him to output the solution. We show that settling the one-way communication complexity of load-balancing is equivalent to a natural sparsification problem for load-balancing. We then prove a dual interpretation of this sparsifier, showing that the minimum density of a sparsifier is effectively the same as the maximum density one can achieve for an extremal graph family that is new to this paper, called Matching-Contractors; these graphs are intimately connected to the well-known Ruzsa-Szemeredi graphs and generalize them in certain aspects. Our chain of equivalences thus shows that the one-way communication complexity of load-balancing can be reduced to a purely graph theoretic question: what is the maximum density of a Matching-Contractor on $n$ vertices? Finally, we present a novel combinatorial construction of some-what dense Matching-Contractors, which implies a strong one-way communication lower bound for load-balancing: any one-way protocol (even randomized) with $\tilde{O}(n)$ communication cannot achieve a better than $n^{\frac14-o(1)}$-approximation. Previously, no non-trivial lower bounds were known for protocols with even $O(n\log{n})$ bits of communication. Our result also implies the first non-trivial lower bounds for semi-streaming load-balancing in the edge-arrival model, ruling out $n^{\frac14-o(1)}$-approximation in a single-pass.

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arXiv2024

Experimental Design Using Interlacing Polynomials

We present a unified deterministic approach for experimental design problems using the method of interlacing polynomials. Our framework recovers the best-known approximation guarantees for the well-studied D/A/E-design problems with simple analysis. Furthermore, we obtain improved non-trivial approximation guarantee for E-design in the challenging small budget regime. Additionally, our approach provides an optimal approximation guarantee for a generalized ratio objective that generalizes both D-design and A-design.

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arXiv2024

Spectral Sparsification by Deterministic Discrepancy Walk

Spectral sparsification and discrepancy minimization are two well-studied areas that are closely related. Building on recent connections between these two areas, we generalize the "deterministic discrepancy walk" framework by Pesenti and Vladu [SODA~23] for vector discrepancy to matrix discrepancy, and use it to give a simpler proof of the matrix partial coloring theorem of Reis and Rothvoss [SODA~20]. Moreover, we show that this matrix discrepancy framework provides a unified approach for various spectral sparsification problems, from stronger notions including unit-circle approximation and singular-value approximation to weaker notions including graphical spectral sketching and effective resistance sparsification. In all of these applications, our framework produces improved results with a simpler and deterministic analysis.

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arXiv2024

On the Houdré-Tetali conjecture about an isoperimetric constant of graphs

Houdré and Tetali defined a class of isoperimetric constants $φ_p$ of graphs for $0 \leq p \leq 1$, and conjectured a Cheeger-type inequality for $φ_\frac12$ of the form $$λ_2 \lesssim φ_\frac12 \lesssim \sqrt{λ_2}$$ where $λ_2$ is the second smallest eigenvalue of the normalized Laplacian matrix. If true, the conjecture would be a strengthening of the hard direction of the classical Cheeger's inequality. Morris and Peres proved Houdré and Tetali's conjecture up to an additional log factor, using techniques from evolving sets. We present the following related results on this conjecture. - We provide a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed. - We match Morris and Peres's bound using standard spectral arguments. - We prove that Houdré and Tetali's conjecture is true for any constant $p$ strictly bigger than $\frac12$, which is also a strengthening of the hard direction of Cheeger's inequality. Furthermore, our results can be extended to directed graphs using Chung's definition of eigenvalues for directed graphs.

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arXiv2023

Fast Algorithms for Directed Graph Partitioning Using Flows and Reweighted Eigenvalues

We consider a new semidefinite programming relaxation for directed edge expansion, which is obtained by adding triangle inequalities to the reweighted eigenvalue formulation. Applying the matrix multiplicative weight update method to this relaxation, we derive almost linear-time algorithms to achieve $O(\sqrt{\log{n}})$-approximation and Cheeger-type guarantee for directed edge expansion, as well as an improved cut-matching game for directed graphs. This provides a primal-dual flow-based framework to obtain the best known algorithms for directed graph partitioning. The same approach also works for vertex expansion and for hypergraphs, providing a simple and unified approach to achieve the best known results for different expansion problems and different algorithmic techniques.

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arXiv2023

Experimental Design for Any $p$-Norm

We consider a general $p$-norm objective for experimental design problems that captures some well-studied objectives (D/A/E-design) as special cases. We prove that a randomized local search approach provides a unified algorithm to solve this problem for all $p$. This provides the first approximation algorithm for the general $p$-norm objective, and a nice interpolation of the best known bounds of the special cases.

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arXiv2022

Cheeger Inequalities for Directed Graphs and Hypergraphs Using Reweighted Eigenvalues

We derive Cheeger inequalities for directed graphs and hypergraphs using the reweighted eigenvalue approach that was recently developed for vertex expansion in undirected graphs [OZ22,KLT22,JPV22]. The goal is to develop a new spectral theory for directed graphs and an alternative spectral theory for hypergraphs. The first main result is a Cheeger inequality relating the vertex expansion $\vecψ(G)$ of a directed graph $G$ to the vertex-capacitated maximum reweighted second eigenvalue $\vecλ_2^{v*}$: \[ \vecλ_2^{v*} \lesssim \vecψ(G) \lesssim \sqrt{\vecλ_2^{v*} \cdot \log (Δ/\vecλ_2^{v*})}. \] This provides a combinatorial characterization of the fastest mixing time of a directed graph by vertex expansion, and builds a new connection between reweighted eigenvalued, vertex expansion, and fastest mixing time for directed graphs. The second main result is a stronger Cheeger inequality relating the edge conductance $\vecϕ(G)$ of a directed graph $G$ to the edge-capacitated maximum reweighted second eigenvalue $\vecλ_2^{e*}$: \[ \vecλ_2^{e*} \lesssim \vecϕ(G) \lesssim \sqrt{\vecλ_2^{e*} \cdot \log (1/\vecλ_2^{e*})}. \] This provides a certificate for a directed graph to be an expander and a spectral algorithm to find a sparse cut in a directed graph, playing a similar role as Cheeger's inequality in certifying graph expansion and in the spectral partitioning algorithm for undirected graphs. We also use this reweighted eigenvalue approach to derive the improved Cheeger inequality for directed graphs, and furthermore to derive several Cheeger inequalities for hypergraphs that match and improve the existing results in [Lou15,CLTZ18]. These are supporting results that this provides a unifying approach to lift the spectral theory for undirected graphs to more general settings.

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arXiv2022

Cheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues

The classical Cheeger's inequality relates the edge conductance $ϕ$ of a graph and the second smallest eigenvalue $λ_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $ψ^2 / \log |V| \lesssim λ_2^* \lesssim ψ$ connecting the vertex expansion $ψ$ of a graph $G=(V,E)$ and the maximum reweighted second smallest eigenvalue $λ_2^*$ of the Laplacian matrix. In this work, we first improve their result to $ψ^2 / \log d \lesssim λ_2^* \lesssim ψ$ where $d$ is the maximum degree in $G$, which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analog in relating vertex expansions and reweighted eigenvalues. These include an analog of Trevisan's result on bipartiteness, an analog of higher order Cheeger's inequality, and an analog of improved Cheeger's inequality. Finally, inspired by this connection, we present negative evidence to the $0/1$-polytope edge expansion conjecture by Mihail and Vazirani. We construct $0/1$-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these $0/1$-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of $0/1$-polytopes.

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arXiv2020

A Local Search Framework for Experimental Design

We present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov's exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design.

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arXiv2020

A Spectral Approach to Network Design

We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional non-negative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for the classical survivable network design problem, and partially answers a question of Bansal about survivable network design with concentration property. We also show many other applications of the spectral rounding results, including weighted experimental design and additive spectral sparsification.

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arXiv2020

Improved Analysis of Higher Order Random Walks and Applications

The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second eigenvalue of the down-up walk on a pure simplicial complex, in terms of the second eigenvalues of its links. We show some applications of this result in analyzing mixing times of Markov chains, including sampling independent sets of a graph and sampling common independent sets of two partition matroids.

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arXiv2019

Spectral analysis of matrix scaling and operator scaling

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the capacity of the matrix and operator. These bounds can be used in various applications of scaling problems, including matrix scaling on expander graphs, permanent lower bounds on random matrices, the Paulsen problem on random frames, and Brascamp-Lieb constants on random operators. In some applications, the inputs of interest satisfy the spectral condition and we prove significantly stronger bounds than the worst case bounds.

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arXiv2019

Network design for s-t effective resistance

We consider a new problem of designing a network with small $s$-$t$ effective resistance. In this problem, we are given an undirected graph $G=(V,E)$, two designated vertices $s,t \in V$, and a budget $k$. The goal is to choose a subgraph of $G$ with at most $k$ edges to minimize the $s$-$t$ effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.

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arXiv2017

Approximating Unique Games Using Low Diameter Graph Decomposition

We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the ${\sf Max2Lin}_k$ problem in $K_r$-minor free graphs, when there is an assignment satisfying $1-\varepsilon$ fraction of constraints, we present an algorithm that produces an assignment satisfying $1-O(r\varepsilon)$ fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the ${\sf MaxCut}$ problem for $K_r$-minor free graphs. For general Unique Games in $K_r$-minor free graphs, we provide another algorithm that produces an assignment satisfying $1-O(r \sqrt{\varepsilon})$ fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus $g$, and we get similar results with $r$ replaced by $\log g$ in the ${\sf Max2Lin}_k$ problem and by $\sqrt{\log g}$ in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the ${\sf Max2Lin}_k$ case, and the latter result generalizes the result of Trevisan for general Unique Games.

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arXiv2017

Graph Clustering using Effective Resistance

$ \def\vecc#1{\boldsymbol{#1}} $We design a polynomial time algorithm that for any weighted undirected graph $G = (V, E,\vecc w)$ and sufficiently large $δ> 1$, partitions $V$ into subsets $V_1, \ldots, V_h$ for some $h\geq 1$, such that $\bullet$ at most $δ^{-1}$ fraction of the weights are between clusters, i.e. \[ w(E - \cup_{i = 1}^h E(V_i)) \lesssim \frac{w(E)}δ;\] $\bullet$ the effective resistance diameter of each of the induced subgraphs $G[V_i]$ is at most $δ^3$ times the average weighted degree, i.e. \[ \max_{u, v \in V_i} \mathsf{Reff}_{G[V_i]}(u, v) \lesssim δ^3 \cdot \frac{|V|}{w(E)} \quad \text{ for all } i=1, \ldots, h.\] In particular, it is possible to remove one percent of weight of edges of any given graph such that each of the resulting connected components has effective resistance diameter at most the inverse of the average weighted degree. Our proof is based on a new connection between effective resistance and low conductance sets. We show that if the effective resistance between two vertices $u$ and $v$ is large, then there must be a low conductance cut separating $u$ from $v$. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.

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