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1. Weighted Minwise Hashing Beats Linear Sketching for Inner Product Estimation

   https://doi.org/10.1145/3584372.3588679  

   Bessa, Aline; Daliri, Majid; Freire, Juliana; Musco, Cameron; Musco, Christopher; Santos, Aécio; Zhang, Haoxiang (June 2023, Proceedings of the 42nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems)

   We present a new approach for independently 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 exactly matches 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, as well as in increasingly popular dataset search applications, where inner products 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. 

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2. Improved Frequency Estimation Algorithms with and without Predictions

   Aamand, Anders; Chen, Justin Y.; Nguyen, Huy; Silwal, Sandeep; Vakilian, Ali (September 2023, Advances in Neural Information Processing Systems)

   Estimating frequencies of elements appearing in a data stream is a key task in large-scale data analysis. Popular sketching approaches to this problem (e.g., CountMin and CountSketch) come with worst-case guarantees that probabilistically bound the error of the estimated frequencies for any possible input. The work of Hsu et al.~(2019) introduced the idea of using machine learning to tailor sketching algorithms to the specific data distribution they are being run on. In particular, their learning-augmented frequency estimation algorithm uses a learned heavy-hitter oracle which predicts which elements will appear many times in the stream. We give a novel algorithm, which in some parameter regimes, already theoretically outperforms the learning based algorithm of Hsu et al. without the use of any predictions. Augmenting our algorithm with heavy-hitter predictions further reduces the error and improves upon the state of the art. Empirically, our algorithms achieve superior performance in all experiments compared to prior approaches. 

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3. Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning

   Dereziński, Michał; Musco, Christopher; Yang, Jiaming (January 2025, ACM-SIAM Symposium on Discrete Algorithms)

   We present a new class of preconditioned iterative methods for solving linear systems of the form Ax = b. Our methods are based on constructing a low-rank Nyström approximation to A using sparse random matrix sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of A, which improves as the rank of the Nyström approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any n × n linear system that is well-conditioned except for k outlying large singular values in Õ (n2.065 + kω) time, improving on a recent result of [Derezmski, Yang, STOC 2024] for all k ≳ n0.78. 2. We give the first Õ (n2 + dλω) time algorithm for solving a regularized linear system (A+λΙ)x = b, where A is positive semidefinite with effective dimension dλ = tr(A(A + λΙ)-1). This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten p-norms and other matrix norms. For example, for the Schatten 1-norm (nuclear norm), we give an algorithm that runs in Õ (n2.11) time, improving on an Õ (n2.18) method of [Musco et al., ITCS 2018]. All results are proven in the real RAM model of computation. Interestingly, previous state-of-the-art algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching. 

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4. A nearly-optimal bound for fast regression with ℓ∞ guarantee

   Song, Zhao; Ye, Mingquan; Yin, Junze; Zhang, Lichen (July 2023, Proceedings of Machine Learning Research)

   Given a matrix A ∈ ℝn\texttimes{}d and a vector b ∈ ℝn, we consider the regression problem with ℓ∞ guarantees: finding a vector x′ ∈ ℝd such that $$||x'-x^* ||_infty leq frac{epsilon}{sqrt{d}}cdot ||Ax^*-b||_2cdot ||A^dagger||$$, where x* = arg minx∈Rd ||Ax – b||2. One popular approach for solving such ℓ2 regression problem is via sketching: picking a structured random matrix S ∈ ℝm\texttimes{}n with m < n and S A can be quickly computed, solve the "sketched" regression problem arg minx∈ℝd ||S Ax – Sb||2. In this paper, we show that in order to obtain such ℓ∞ guarantee for ℓ2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m = ε-2d log3(n/δ) such that solving the sketched regression problem gives the ℓ∞ guarantee, with probability at least 1 – δ. Moreover, the matrix S A can be computed in time O(nd log n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in (Price et al., 2017), in which a superlinear in d rows, m = Ω(ε-2d1+γ) for γ ∈ (0, 1) is required. Moreover, we develop a novel analytical framework for ℓ∞ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in (Song \& Yu, 2021). Our analysis is much simpler and more general than that of (Price et al., 2017). Leveraging this framework, we extend the ℓ∞ guarantee regression result to dense sketching matrices for computing the fast tensor product of vectors. 

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5. Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning

   Derezinski, M; Musco, C; Yang, J (January 2025, ACM-SIAM Symposium on Discrete Algorithms (SODA 2025))

   We present a new class of preconditioned iterative methods for solving linear systems of the form Ax=b. Our methods are based on constructing a low-rank Nyström approximation to A using sparse random matrix sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of A, which improves as the rank of the Nyström approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any n×n linear system that is well-conditioned except for k outlying large singular values in O~(n^2.065+k^ω) time, improving on a recent result of [Dereziński, Yang, STOC 2024] for all k≳n^0.78. 2. We give the first O~(n^2+d_λ^ω) time algorithm for solving a regularized linear system (A+λI)x=b, where A is positive semidefinite with effective dimension d_λ=tr(A(A+λI)^{−1}). This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten p-norms and other matrix norms. For example, for the Schatten 1-norm (nuclear norm), we give an algorithm that runs in O~(n ^{2.11}) time, improving on an O~(n ^{2.18}) method of [Musco et al., ITCS 2018]. All results are proven in the real RAM model of computation. Interestingly, previous state-of-the-art algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching. 

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