We introduce a new Approximate Cholesky factorization algorithm and software for solving linear equations in symmetric diagonally dominant matrices with nonpositive off-diagonal entries (SDDM matrices), including Laplacian matrices. We use preconditioned conjugate gradient to solve these systems. Our preconditioner is a variant of the Approximate Cholesky factorization of Kyng and Sachdeva (FOCS 2016). Our factorization approach is simple: we eliminate matrix rows/columns one at a time and use sampling to update the remaining entries of the matrix, approximating complete Cholesky factorization. Unlike Kyng-Sachdeva, we do not prove theoretical guarantees about our algorithm, but we observe experimentally that our algorithm runs much faster. Our algorithm broadly resembles a common approach of using Incomplete Cholesky factorization as a preconditioner in conjugate gradient (ICCG). Unlike earlier approaches such as Kyng-Sachdeva and Incomplete Cholesky variants, our sampled entries always maintain a connected support graph on the neighbors of the eliminated variable. We implement our solver algorithm in Julia and experimentally evaluate its performance when using 1 or 2 samples for each original entry. We compare our single-threaded performance to that of state-of-the-art solvers. We evaluate performance across many problem classes, including all large SDDM matrices from the SuiteSparse collection, and a broad array of programmatically generated instances, with tests ranging up to 200 million nonzeros per system of linear equations. We hope that others will adopt this suite of tests as a benchmark, which we refer to as SDDM2023. Our experiments suggest that our solvers are much more robust than existing solvers for SDDM linear equations, while retaining good performance across all instances. Our code is public, and we detail precisely the tests we run and provide a tutorial on how to replicate them.

Bang (1976) proved the inequality for matrix permanents per^2(A) ≥ 2^-2nper(A ⊗ J_2), where J_2 is the 2 × 2 all-ones matrix and A is any n × n matrix with non-negative entries. We show that, if A is any n × n real-valued matrix with rank at most two (possibly having negative entries), this inequality can be sharpened, replacing the constant 2^-2n by 1 / 2nn = (n!)^2 / (2n)! > 2^-2n. We then show that this sharpened inequality also implies new inequalities for finite free convolutions of polynomials: if p and q are monic real-rooted polynomials of degree n, then (p ⊞_n q)(x)^2 ≥ (p^2 ⊞_2n q^2)(x) and (p ⊠_n q)(x)^2 ≥ (p^2 ⊠_2n q^2)(x) for all x ∈ℝ, for ⊞_n and ⊠_n the finite free additive and multiplicative convolution operations, respectively, on polynomials of degree n.

We study the problem of detecting or recovering a planted ranked subgraph from a directed graph, an analog for directed graphs of the well-studied planted dense subgraph model. We suppose that, among a set of.. items, there is a subset.. of.. items having a latent ranking in the form of a permutation.. of.., and that we observe a fraction.. of pairwise orderings between elements of.. which agree with.. with probability 12 +.. and otherwise are uniformly random. Unlike in the planted dense subgraph and planted clique problems where the community.. is distinguished by its unusual density of edges, here the community is only distinguished by the unusual consistency of its pairwise orderings. We establish computational and statistical thresholds for both detecting and recovering such a ranked community. In the log-density setting where..,.., and.. all scale as powers of.., we establish the exact thresholds in the associated exponents at which detection and recovery become statistically and computationally feasible. These regimes include a rich variety of behaviors, exhibiting both statistical-computational and detection-recovery gaps. We also give finer-grained results for two extreme cases: (1).. = 1,.. =.., and.. small, where a full tournament is observed that is weakly correlated with a global ranking, and (2).. = 1,.. = 12, and.. small, where a small "ordered clique" (totally ordered directed subgraph) is planted in a random tournament.

The design of experiments involves a compromise between covariate balance and robustness. This paper provides a formalization of this trade-off and describes an experimental design that allows experimenters to navigate it. The design is specified by a robustness parameter that bounds the worst-case mean squared error of an estimator of the average treatment effect. Subject to the experimenter's desired level of robustness, the design aims to simultaneously balance all linear functions of potentially many covariates. Less robustness allows for more balance. We show that the mean squared error of the estimator is bounded in finite samples by the minimum of the loss function of an implicit ridge regression of the potential outcomes on the covariates. Asymptotically, the design perfectly balances all linear functions of a growing number of covariates with a diminishing reduction in robustness, effectively allowing experimenters to escape the compromise between balance and robustness in large samples. Finally, we describe conditions that ensure asymptotic normality and provide a conservative variance estimator, which facilitate the construction of asymptotically valid confidence intervals.

We consider the problem of inferring an unknown ranking of n items from a random tournament on n vertices whose edge directions are correlated with the ranking. We establish, in terms of the strength of these correlations, the computational and statistical thresholds for detection (deciding whether an observed tournament is purely random or drawn correlated with a hidden ranking) and recovery (estimating the hidden ranking with small error in Spearman's footrule or Kendall's tau metric on permutations). Notably, we find that this problem provides a new instance of a detection-recovery gap: solving the detection problem requires much weaker correlations than solving the recovery problem. In establishing these thresholds, we also identify simple algorithms for detection (thresholding a degree 2 polynomial) and recovery (outputting a ranking by the number of "wins" of a tournament vertex, i.e., the out-degree) that achieve optimal performance up to constants in the correlation strength. For detection, we find that the above low-degree polynomial algorithm is superior to a natural spectral algorithm. We also find that, whenever it is possible to achieve strong recovery (i.e., to estimate with vanishing error in the above metrics) of the hidden ranking, then the above "Ranking By Wins" algorithm not only does so, but also outputs a close approximation of the maximum likelihood estimator, a task that is NP-hard in the worst case.

We introduce a new algorithm and software for solving linear equations in symmetric diagonally dominant matrices with non-positive off-diagonal entries (SDDM matrices), including Laplacian matrices. We use pre-conditioned conjugate gradient (PCG) to solve the system of linear equations. Our preconditioner is a variant of the Approximate Cholesky factorization of Kyng and Sachdeva (FOCS 2016). Our factorization approach is simple: we eliminate matrix rows/columns one at a time and update the remaining matrix using sampling to approximate the outcome of complete Cholesky factorization. Unlike earlier approaches, our sampling always maintains a connectivity in the remaining non-zero structure. Our algorithm comes with a tuning parameter that upper bounds the number of samples made per original entry. We implement our algorithm in Julia, providing two versions, AC and AC2, that respectively use 1 and 2 samples per original entry. We compare their single-threaded performance to that of current state-of-the-art solvers Combinatorial Multigrid (CMG), BoomerAMG-preconditioned Krylov solvers from HyPre and PETSc, Lean Algebraic Multigrid (LAMG), and MATLAB's with Incomplete Cholesky Factorization (ICC). Our evaluation uses a broad class of problems, including all large SDDM matrices from the SuiteSparse collection and diverse programmatically generated instances. Our experiments suggest that our algorithm attains a level of robustness and reliability not seen before in SDDM solvers, while retaining good performance across all instances. Our code and data are public, and we provide a tutorial on how to replicate our tests. We hope that others will adopt this suite of tests as a benchmark, which we refer to as SDDM2023. Our solver code is available at: https://github.com/danspielman/Laplacians.jl/ Our benchmarking data and tutorial are available at: https://rjkyng.github.io/SDDM2023/

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.

Marcus, Spielman and Srivastava [MSS15] solved the Kadison–Singer Problem by proving a strong form of Weaver’s conjecture: they showed that for all α > 0 and all lists of vectors of norm at most √ α whose outer products sum to the identity, there exists a signed sum of those outer products with operator norm at most √ 8 α + 2 α. We prove that it is NP-hard to distinguish such a list of vectors for which there is a signed sum that equals the zero matrix from those in which every signed sum has operator norm at least κ √ α , for some absolute constant κ > 0 . Thus, it is NP-hard to construct a signing that is a constant factor better than that guaranteed to exist. For α = 1 / 4, we prove that it is NP-hard to distinguish whether there is a signed sum that equals the zero matrix from the case in which every signed sum has operator norm at least 1 / 4.

We use the method of interlacing families of polynomials to derive a simple proof of Bourgain and Tzafriri's Restricted Invertibility Principle, and then to sharpen the result in two ways. We show that the stable rank can be replaced by the Schatten 4-norm stable rank and that tighter bounds hold when the number of columns in the matrix under consideration does not greatly exceed its number of rows. Our bounds are derived from an analysis of the smallest zeros of Jacobi and associated Laguerre polynomials.

The performance of the Rapidly-exploring Random Tree (RRT) algorithm to find low-stretch spanning trees is analyzed on a special family of graphs. The construction of this graph family is described in the report, and we show that it is possible to force a group of edges to be removed with high probability. On this graph family, running the RRT algorithm gives a spanning tree with one group of edges that has expected stretch O(nc), c < 1. Adding multi-edges to the graph will lead to a counter example for the RRT algorithm because there exists at least one graph family such that running the RRT algorithm on this graph family gives a spanning tree of average stretch O(nc). Several ways to modify the algorithm are proposed. The idea of using random walk in the algorithm was proposed in Fall 2017 and a generalization of a random-walk step is investigated in this project. However, experimental evidence shows that a generalized random walk version of Rapidly-exploring Random Tree algorithm does not give trees with average stretch lower than O(log n). Another idea for changing the RRT algorithm by sampling vertices based on weighted degree also does not give trees with low average stretch. We also present another modified version of the RRT algorithm that may lead to a promising approach to find low-stretch trees. The new algorithm relies on the interpretation of graphs as systems of resistors or as flow networks, and the Laplacian of the graph is used to determine the sampling distribution on vertices.

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions introduced recently by the authors.

This survey accompanies the Josiah Williard Gibbs Lecture that I gave at the 2016 Joint Mathematics Meetings. It provides an introduction to three topics: algebraic and spectral graph theory, the sparsification of graphs, and the recent resolution of the Kadison–Singer Problem. The first and third are connected by the second.

We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process. We use these new algorithms to derive the first nearly linear time algorithms for solving systems of equations in connection Laplacians, a generalization of Laplacian matrices that arise in many problems in image and signal processing. We also prove that every connection Laplacian has a linear sized approximate inverse. This is an LU factorization with a linear number of nonzero entries that is a strong approximation of the original matrix. Using such a factorization one can solve systems of equations in a connection Laplacian in linear time. Such a factorization was unknown even for ordinary graph Laplacians.

We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large $p$ of $p$-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time $\widetilde{O} (m n)$. The latter algorithm has variants that seem to run much faster in practice. These extensions are particularly amenable to regularization: we can perform $l_{0}$-regularization on the given values in polynomial time and $l_{1}$-regularization on the initial function values and on graph edge weights in time $\widetilde{O} (m^{3/2})$.