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Approximate Matrix Multiplication through Joint Leverage Score Sampling
Neophytos Charalambides, Mert Pilanci, Alfred O. Hero III
TL;DR
Repeated matrix multiplication can bottleneck large-scale computation, and this paper addresses approximate matrix multiplication using a sampling distribution based on both matrices’ leverage scores. It states a high-probability spectral-error guarantee for the algorithm, with sampling probabilities assumed to be approximate.
Problem
The paper addresses approximate matrix multiplication, where repeated matrix products can bottleneck large-scale computations and general CR−MM guarantees for A ≠ B are quantified in Frobenius rather than spectral norm.
Source: §1 · pp. 1–2Method
The algorithm samples row pairs using a non-uniform distribution based on the left leverage scores of both matrices.
Source: §1 · p. 1Results
The paper states a high-probability spectral-norm approximation guarantee for Algorithm 1, unlike the general CR−MM Frobenius-norm guarantee.
Source: §2.4 · pp. 7, 10Takeaways & Limitations
The proposed algorithm produces a low-rank approximation that the paper says is more efficient to work with in storage and computation.
Source: §4 · p. 15The guarantees assume an approximate sampling distribution, and the paper says high joint leverage scores must not be underestimated too much.
Source: §2.2 · pp. 5–6
Abstract
from arXiv · showhide
In this paper, we investigate approximate matrix multiplication through random sampling. Specifically, we propose using a non-uniform sampling distribution constructed from the leverage scores of the two matrices, and characterize the associated approximation error in terms of the matrix spectral norm; which is well-known to better characterize the downstream error when the matrix product operator is applied to an input vector. We provide the probability error bounds for the proposed algorithm are also given, and discuss connections to data-oblivious approximate matrix multiplication, and graph sparsification algorithms.
1 Introduction
Repeated matrix multiplication can bottleneck large-scale computation, while faster exact algorithms remain impractical because of large hidden constants. The paper introduces joint leverage-score sampling for approximate matrix multiplication and motivates its spectral-error guarantee.
1 Introduction
- Large hidden constants make faster exact matrix multiplication algorithms impractical despite their lower asymptotic operation counts.Naive multiplication takes O(N^3) operations; the cited faster algorithms have exponents below 2.81.Source: §1 · p. 1
- Joint leverage scores define a non-uniform sampling distribution from both matrices’ left leverage scores, identifying influential row pairs.The distribution generalizes matrix leverage scores.Source: §1 · pp. 1–2
- Unlike general CR−MM guarantees stated in Frobenius norm, the proposed leverage-score sampling supports a spectral-norm error bound.The paper also notes that a spectral bound for CR−MM when A ≠ B may require very high sampling complexity.Source: §1 · p. 2
- The approach gives an explicit approximate matrix multiplication algorithm, whereas related OSE work cited by the paper does not provide an explicit AMM algorithm.The paper describes its approach as directly generalizing the ℓ2-subspace embedding property.Source: §1 · p. 2
- Leverage-score sampling is not data-oblivious because the sampling distribution is determined using information from the matrices.The paper contrasts this with sketches that do not directly extract information from the data to determine the sketching matrix.Source: §1 · p. 2
- Matrix sketching is also used in applications including matrix and tensor algorithms, graph algorithms, and approximate matrix multiplication.The introduction also connects leverage scores to graph sparsification and lists applications including block Kaczmarz, low-rank approximation, and k-means clustering.Source: §1 · pp. 2–3
2 Joint Leverage Score Multiplication
This section develops same-index joint leverage score sampling for approximate matrix multiplication, building on norm-based row-pair sampling and characterizing the sketch’s spectral-norm error and guarantees under stated assumptions, with oversampling when probabilities are approximate. It also presents a guarantee for products of distinct matrices, with the Gram-matrix case recovering the leverage-score sketch’s ℓ2-subspace-embedding guarantee.
2.1 Preliminaries
- CR-MM samples row pairs according to row norms; its estimator is unbiased, its probabilities minimize variance, and it gives a high-probability multiplicative approximation.For general A and B, the recalled approximation has Frobenius error O(∥A∥F∥B∥F/√r).Source: §2.1 · p. 3
2.2 Joint Leverage Score Sampling
- The method samples only same-index row pairs using normalized geometric means of the corresponding leverage scores, then sketches both matrices with a shared sampling matrix.Approximate probabilities are permitted if they do not underestimate each true probability by more than the factor β; this requires increasing the sampling complexity by 1/β.Source: §2.2 · pp. 4–5
2.3 Spectral Characterization for AMM
- The spectral characterization gives a sufficient sketching condition for approximating A⊤B, and any sketch satisfying it can also produce an ℓ2-subspace embedding for A.When A and B are orthonormal, Algorithm 1 produces the same output as CR-MM.Source: §2.3 · p. 6
2.4 Approximation Guarantees
- The proposed guarantee bounds approximation error in spectral norm with high probability, under a matrix assumption, using r = Θ(µ log(2µ/δ)/(βϵ²)) samples.The analysis notes that the relative sizes L and M also affect the results; approximate joint leverage probabilities require oversampling by 1/β.Source: §2.4 · pp. 7, 9, 10
- Under ∥A∥2, ∥B∥2 ≤ 1 and matched sampling complexities, the section compares Algorithm 1’s sampling complexity and error probability with CR-MM’s guarantee.The comparison notes that the preceding sampling complexities depend on the largest singular values of both matrices.Source: §2.4 · p. 11
- Theorem 11 applies to products of any distinct matrices A and B, whereas Theorem 2 considers only the Gram matrix of A.The section identifies the leverage-score sampling sketch’s ℓ2-subspace-embedding guarantee as a corollary when B = A.Source: §2.4 · p. 11
- Setting B = A recovers the leverage-score sampling sketch’s ℓ2-subspace-embedding guarantee, stated as Theorem 14 ([Woo14]).The theorem assumes approximate sampling probabilities satisfy ˜ℓi ≥ βℓi for β ∈ (0, 1].Source: §2.4 · p. 11
3 Implications on other Approximation Algorithms
This section extends the joint-leverage-score analysis to CR−MM, an oblivious AMM route, and graph spectral sparsification. It also identifies a computational cost for calculating joint leverage scores and describes when transformed uniform sampling can satisfy the AMM guarantee.
3.1 CR−MM through Approximate Joint Leverage Scores
- Calculating joint leverage scores from singular-value or QR decompositions costs O(N^2(L + M)) operations.The paper notes that leverage-score approximation methods can avoid directly computing each left orthonormal basis.Source: §3.1 · pp. 11–12
- Proposition 16 quantifies a spectral AMM bound for CR−MM using its sampling distribution.The proposition specifies a sample-size condition involving the misestimation factor, accuracy, and failure probability.Source: §3.1 · p. 12
3.2 Data-Oblivious AMM through Joint Leverage Scores
- After a random transform flattens both leverage-score profiles, uniform sampling of transformed row pairs yields an estimator satisfying (2.10) with overwhelming probability.The SRHT is one such transform; Gaussian or orthonormal alternatives are slower than the SRHT, which sketches in O(NL log N) time.Source: §3.2 · pp. 12–13
3.3 Graph Spectral Sparsifiers
- Joint-leverage-score AMM generalizes effective-resistance sampling and can sparsify an intersection graph without explicitly constructing it.The paper connects this approach to approximating the intersection graph's Laplacian and to joint effective resistances across the two graphs.Source: §3.3 · p. 14
- The sampling procedure produces an ε-spectral sparsifier of the intersection graph H.The proof applies prior sparsification results to H.Source: §3.3 · p. 15
4 Concluding Remarks and Future Work
The paper introduces generalized leverage scores and an AMM algorithm with a spectral guarantee, and connects the method to graph sparsification. It identifies further possible applications and theoretical directions for joint-leverage-score sampling.
4 Concluding Remarks and Future Work
- The paper defines generalized leverage scores, develops an AMM algorithm with a spectral guarantee, and proposes an AMM condition similar to the ℓ2-subspace embedding property.The algorithm produces a low-rank approximation that is more efficient in storage and computation.Source: §4 · p. 15
- The paper connects joint-leverage-score AMM to graph sparsifiers and introduces a generalized notion of effective resistances for further investigation..Source: §4 · p. 15
- Potential future directions include joint-score methods for tensor products, distributed least-squares regression, and studying leverage scores through information theory and entropy.The distributed setting described has two parties with distinct dataset features that they do not share.Source: §4 · p. 15