arXiv subjects

Christopher Musco

Publications and source records attributed to Christopher Musco.

At least 55 records · Page 3Linked to original sources

arXiv2023

Moments, Random Walks, and Limits for Spectrum Approximation

We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $ε$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $(1\pm2^{-Ω(1/ε)})$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $2^{O(1/ε)}$ random walks initiated at uniformly random nodes in the graph. As a strengthening of our main result, we show that improving the dependence on $1/ε$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $ε$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $2^{Ω(1/ε)}$ random walks of length $2^{Ω(1/ε)}$ started at random nodes.

cs.DS↗

arXiv2023

Dimensionality Reduction for General KDE Mode Finding

Finding the mode of a high dimensional probability distribution $D$ is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when $D$ is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy $(1-ε)$ for any $ε> 0$. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless $\mathit{P} = \mathit{NP}$. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

cs.LG↗

arXiv2023

Low-memory Krylov subspace methods for optimal rational matrix function approximation

We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.

math.NA↗

arXiv2023

Weighted Minwise Hashing Beats Linear Sketching for Inner Product Estimation

We present a new approach for computing compact sketches that can be used to approximate the inner product between pairs of high-dimensional vectors. Based on the Weighted MinHash algorithm, our approach admits strong accuracy guarantees that improve on the guarantees of popular linear sketching approaches for inner product estimation, such as CountSketch and Johnson-Lindenstrauss projection. Specifically, while our method admits guarantees that exactly match linear sketching for dense vectors, it yields significantly lower error for sparse vectors with limited overlap between non-zero entries. Such vectors arise in many applications involving sparse data. They are also important in increasingly popular dataset search applications, where inner product sketches are used to estimate data covariance, conditional means, and other quantities involving columns in unjoined tables. We complement our theoretical results by showing that our approach empirically outperforms existing linear sketches and unweighted hashing-based sketches for sparse vectors.

cs.DB↗

arXiv2023

Near Optimal Linear Algebra in the Online and Sliding Window Models

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in a stream form the underlying data set. We first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on "reverse online" versions of offline sampling distributions such as (ridge) leverage scores, $\ell_1$ sensitivities, and Lewis weights to quantify both the importance and the recency of a row. Our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et. al.,(FOCS 2019). We also give the first online algorithm for $\ell_1$-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model that use nearly optimal space.

cs.DS↗

arXiv2022

Near-Linear Sample Complexity for $L_p$ Polynomial Regression

We study $L_p$ polynomial regression. Given query access to a function $f:[-1,1] \rightarrow \mathbb{R}$, the goal is to find a degree $d$ polynomial $\hat{q}$ such that, for a given parameter $\varepsilon > 0$, $$ \|\hat{q}-f\|_p\le (1+\varepsilon) \cdot \min_{q:\text{deg}(q)\le d}\|q-f\|_p. $$ Here $\|\cdot\|_p$ is the $L_p$ norm, $\|g\|_p = (\int_{-1}^1 |g(t)|^p dt)^{1/p}$. We show that querying $f$ at points randomly drawn from the Chebyshev measure on $[-1,1]$ is a near-optimal strategy for polynomial regression in all $L_p$ norms. In particular, to find $\hat q$, it suffices to sample $O(d\, \frac{\text{polylog}\,d}{\text{poly}\,\varepsilon})$ points from $[-1,1]$ with probabilities proportional to this measure. While the optimal sample complexity for polynomial regression was well understood for $L_2$ and $L_\infty$, our result is the first that achieves sample complexity linear in $d$ and error $(1+\varepsilon)$ for other values of $p$ without any assumptions. Our result requires two main technical contributions. The first concerns $p\leq 2$, for which we provide explicit bounds on the $L_p$ Lewis weight function of the infinite linear operator underlying polynomial regression. Using tools from the orthogonal polynomial literature, we show that this function is bounded by the Chebyshev density. Our second key contribution is to take advantage of the structure of polynomials to reduce the $p>2$ case to the $p\leq 2$ case. By doing so, we obtain a better sample complexity than what is possible for general $p$-norm linear regression problems, for which $Ω(d^{p/2})$ samples are required.

cs.DS↗

arXiv2022

A Tight Analysis of Hutchinson's Diagonal Estimator

Let $\mathbf{A}\in \mathbb{R}^{n\times n}$ be a matrix with diagonal $\text{diag}(\mathbf{A})$ and let $\bar{\mathbf{A}}$ be $\mathbf{A}$ with its diagonal set to all zeros. We show that Hutchinson's estimator run for $m$ iterations returns a diagonal estimate $\tilde{d}\in \mathbb{R}^n$ such that with probability $(1-δ)$, $$\|\tilde{d} - \text{diag}(\mathbf{A})\|_2 \leq c\sqrt{\frac{\log(2/δ)}{m}}\|\bar{\mathbf{A}}\|_F,$$ where $c$ is a fixed constant independent of all other parameters. This results improves on a recent result of [Baston and Nakatsukasa, 2022] by a $\log(n)$ factor, yielding a bound that is independent of the matrix dimension $n$.

cs.DS↗

arXiv2022

Active Linear Regression for $\ell_p$ Norms and Beyond

We study active sampling algorithms for linear regression, which aim to query only a few entries of a target vector $b\in\mathbb R^n$ and output a near minimizer to $\min_{x\in\mathbb R^d} \|Ax-b\|$, for a design matrix $A\in\mathbb R^{n \times d}$ and loss $\|\cdot\|$. For $p$ norm regression for any $0<p<\infty$, we give an algorithm based on Lewis weight sampling outputting a $(1+ε)$-approximate solution using just $\tilde O(d/ε^2)$ queries to $b$ for $p\in(0,1)$, $\tilde{O}(d/ε)$ queries for $1<p<2$, and $\tilde{O}(d^{p/2}/ε^p)$ queries for $2<p<\infty$. For $0<p<2$, our bounds are optimal up to log factors, settling the query complexity for this range. For $2<p<\infty$, our dependence on $d$ is optimal, while our dependence on $ε$ is off by at most $ε$, up to log factors. Our result resolves an open question of [CD21], who gave near optimal bounds for the $1$ norm, but required $d^2/ε^2$ samples for $\ell_p$ regression with $1<p<2$, and gave no bounds for $2<p<\infty$ or $0<p<1$. We also give the first total sensitivity bound of $O(d^{\max\{1,p/2\}}\log^2n)$ for loss functions of degree $p$ polynomial growth, improving a result of [TMF20]. By combining this with our techniques for $\ell_p$ regression, we obtain an active regression algorithm making $\tilde O(d^{1+\max\{1,p/2\}}/\mathrm{poly}(ε))$ queries for such loss functions, including the Tukey and Huber losses, answering another question of [CD21]. For the Huber loss, we further improve our bound to $\tilde O(d^{4-2\sqrt2}/\mathrm{poly}(ε))$ samples. Our sensitivity bounds also have many applications, including Orlicz norm subspace embeddings, robust subspace approximation, and dimension reduction for smoothed $p$-norms. Finally, our active sampling results give the first sublinear time algorithms for Kronecker product regression under every $p$ norm.

cs.LG↗

arXiv2022

Error bounds for Lanczos-based matrix function approximation

We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given vector. Assuming that $f : \mathbb{C} \rightarrow \mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive a priori and a posteriori error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $\mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $\mathbf{b}^\textsf{H} f(\mathbf{A}) \mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments.

math.NA↗

arXiv2022

Sublinear Time Spectral Density Estimation

We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $ε$ accuracy in the Wasserstein-1 distance in $O(n\cdot \text{poly}(1/ε))$ time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of $n$, but exponential in $1/ε$. We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian $A$, this moment matching method returns an $ε$ approximation to the spectral density using just $O({1}/ε)$ matrix-vector products with $A$. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used \emph{kernel polynomial method} (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.

cs.DS↗

arXiv2022

Fast Regression for Structured Inputs

We study the $\ell_p$ regression problem, which requires finding $\mathbf{x}\in\mathbb R^{d}$ that minimizes $\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$ for a matrix $\mathbf{A}\in\mathbb R^{n \times d}$ and response vector $\mathbf{b}\in\mathbb R^{n}$. There has been recent interest in developing subsampling methods for this problem that can outperform standard techniques when $n$ is very large. However, all known subsampling approaches have run time that depends exponentially on $p$, typically, $d^{\mathcal{O}(p)}$, which can be prohibitively expensive. We improve on this work by showing that for a large class of common \emph{structured matrices}, such as combinations of low-rank matrices, sparse matrices, and Vandermonde matrices, there are subsampling based methods for $\ell_p$ regression that depend polynomially on $p$. For example, we give an algorithm for $\ell_p$ regression on Vandermonde matrices that runs in time $\mathcal{O}(n\log^3 n+(dp^2)^{0.5+ω}\cdot\text{polylog}\,n)$, where $ω$ is the exponent of matrix multiplication. The polynomial dependence on $p$ crucially allows our algorithms to extend naturally to efficient algorithms for $\ell_\infty$ regression, via approximation of $\ell_\infty$ by $\ell_{\mathcal{O}(\log n)}$. Of practical interest, we also develop a new subsampling algorithm for $\ell_p$ regression for arbitrary matrices, which is simpler than previous approaches for $p \ge 4$.

cs.DS↗

arXiv2021

Fast and Near-Optimal Diagonal Preconditioning

The convergence rates of iterative methods for solving a linear system $\mathbf{A} x = b$ typically depend on the condition number of the matrix $\mathbf{A}$. Preconditioning is a common way of speeding up these methods by reducing that condition number in a computationally inexpensive way. In this paper, we revisit the decades-old problem of how to best improve $\mathbf{A}$'s condition number by left or right diagonal rescaling. We make progress on this problem in several directions. First, we provide new bounds for the classic heuristic of scaling $\mathbf{A}$ by its diagonal values (a.k.a. Jacobi preconditioning). We prove that this approach reduces $\mathbf{A}$'s condition number to within a quadratic factor of the best possible scaling. Second, we give a solver for structured mixed packing and covering semidefinite programs (MPC SDPs) which computes a constant-factor optimal scaling for $\mathbf{A}$ in $\widetilde{O}(\text{nnz}(\mathbf{A}) \cdot \text{poly}(κ^\star))$ time; this matches the cost of solving the linear system after scaling up to a $\widetilde{O}(\text{poly}(κ^\star))$ factor. Third, we demonstrate that a sufficiently general width-independent MPC SDP solver would imply near-optimal runtimes for the scaling problems we consider, and natural variants concerned with measures of average conditioning. Finally, we highlight connections of our preconditioning techniques to semi-random noise models, as well as applications in reducing risk in several statistical regression models.

math.OC↗

arXiv2021

How to Quantify Polarization in Models of Opinion Dynamics

It is widely believed that society is becoming increasingly polarized around important issues, a dynamic that does not align with common mathematical models of opinion formation in social networks. In particular, measures of polarization based on opinion variance are known to decrease over time in frameworks such as the popular DeGroot model. Complementing recent work that seeks to resolve this apparent inconsistency by modifying opinion models, we instead resolve the inconsistency by proposing changes to how polarization is quantified. We present a natural class of group-based polarization measures that capture the extent to which opinions are clustered into distinct groups. Using theoretical arguments and empirical evidence, we show that these group-based measures display interesting, non-monotonic dynamics, even in the simple DeGroot model. In particular, for many natural social networks, group-based metrics can increase over time, and thereby correctly capture perceptions of increasing polarization. Our results build on work by DeMarzo et al., who introduced a group-based polarization metric based on ideological alignment. We show that a central tool from that work, a limit analysis of individual opinions under the DeGroot model, can be extended to the dynamics of other group-based polarization measures, including established statistical measures like bimodality. We also consider local measures of polarization that operationalize how polarization is perceived in a network setting. In conjunction with evidence from prior work that group-based measures better align with real-world perceptions of polarization, our work provides formal support for the use of these measures in place of variance-based polarization in future studies of opinion dynamics.

cs.SI↗

arXiv2021

Dynamic Trace Estimation

We study a dynamic version of the implicit trace estimation problem. Given access to an oracle for computing matrix-vector multiplications with a dynamically changing matrix A, our goal is to maintain an accurate approximation to A's trace using as few multiplications as possible. We present a practical algorithm for solving this problem and prove that, in a natural setting, its complexity is quadratically better than the standard solution of repeatedly applying Hutchinson's stochastic trace estimator. We also provide an improved algorithm assuming slightly stronger assumptions on the dynamic matrix A. We support our theory with empirical results, showing significant computational improvements on three applications in machine learning and network science: tracking moments of the Hessian spectral density during neural network optimization, counting triangles, and estimating natural connectivity in a dynamically changing graph.

cs.DS↗

arXiv2021

Fourier Sparse Leverage Scores and Approximate Kernel Learning

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study $s$-sparse functions of the form $f(x) = \sum_{j=1}^s a_j e^{i λ_j x}$ for coefficients $a_j \in \mathbb{C}$ and frequencies $λ_j \in \mathbb{R}$. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the $statistical\ dimension$ of the approximated kernel matrix. It is the first "oblivious sketching method" with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.

cs.DS↗

arXiv2021

Hutch++: Optimal Stochastic Trace Estimation

We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 \pm ε)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/ε)$ matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires $O(1/ε^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory mainly requires $A$ to be positive semidefinite, we provide generalized guarantees for general square matrices, and show empirical gains in such applications.

cs.DS↗

arXiv2021

Correlation Sketches for Approximate Join-Correlation Queries

The increasing availability of structured datasets, from Web tables and open-data portals to enterprise data, opens up opportunities~to enrich analytics and improve machine learning models through relational data augmentation. In this paper, we introduce a new class of data augmentation queries: join-correlation queries. Given a column $Q$ and a join column $K_Q$ from a query table $\mathcal{T}_Q$, retrieve tables $\mathcal{T}_X$ in a dataset collection such that $\mathcal{T}_X$ is joinable with $\mathcal{T}_Q$ on $K_Q$ and there is a column $C \in \mathcal{T}_X$ such that $Q$ is correlated with $C$. A naïve approach to evaluate these queries, which first finds joinable tables and then explicitly joins and computes correlations between $Q$ and all columns of the discovered tables, is prohibitively expensive. To efficiently support correlated column discovery, we 1) propose a sketching method that enables the construction of an index for a large number of tables and that provides accurate estimates for join-correlation queries, and 2) explore different scoring strategies that effectively rank the query results based on how well the columns are correlated with the query. We carry out a detailed experimental evaluation, using both synthetic and real data, which shows that our sketches attain high accuracy and the scoring strategies lead to high-quality rankings.

cs.DB↗

arXiv2021

Public Transport Planning: When Transit Network Connectivity Meets Commuting Demand

In this paper, we make a first attempt to incorporate both commuting demand and transit network connectivity in bus route planning (CT-Bus), and formulate it as a constrained optimization problem: planning a new bus route with k edges over an existing transit network without building new bus stops to maximize a linear aggregation of commuting demand and connectivity of the transit network. We prove the NP-hardness of CT-Bus and propose an expansion-based greedy algorithm that iteratively scans potential candidate paths in the network. To boost the efficiency of computing the connectivity of new networks with candidate paths, we convert it to a matrix trace estimation problem and employ a Lanczos method to estimate the natural connectivity of the transit network with a guaranteed error bound. Furthermore, we derive upper bounds on the objective values and use them to greedily select candidates for expansion. Our experiments conducted on real-world transit networks in New York City and Chicago verify the efficiency, effectiveness, and scalability of our algorithms.

cs.DB↗