Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency – given n input points, most kernel-based algorithms need to materialize the full n × n kernel matrix before performing any subsequent computation, thus incurring Ω(n^2) runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts.
We break the quadratic barrier and obtain subquadratic time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving lin- ear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recently developed Kernel Density Estimation framework, which (after preprocessing in time subquadratic in n) can return estimates of row/column sums of the kernel matrix. In particular, we de- velop efficient reductions from weighted vertex and weighted edge sampling on kernel graphs, simulating random walks on kernel graphs, and importance sampling on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in sublinear (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsi- fication, where we observe a 9x decrease in the number of kernel evaluations over baselines for LRA and a 41x reduction in the graph size for spectral sparsification.
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Improved Dynamic Graph Learning through Fault-Tolerant Sparsification
Graph sparsification has been used to improve the computational cost of learning over graphs, e.g., Laplacian-regularized estimation and graph semi-supervised learning (SSL). However, when graphs vary over time, repeated sparsification requires polynomial order computational cost per update. We propose a new type of graph sparsification namely fault-tolerant (FT) sparsification to significantly reduce the cost to only a constant. Then the computational cost of subsequent graph learning tasks can be significantly improved with limited loss in their accuracy. In particular, we give theoretical analyze to upper bound the loss in the accuracy of the subsequent Laplacian-regularized estimation and graph SSL, due to the FT sparsification. In addition, FT spectral sparsification can be generalized to FT cut sparsification, for cut-based graph learning. Extensive experiments have confirmed the computational efficiencies and accuracies of the proposed methods for learning on dynamic graphs.
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- Award ID(s):
- 1718738
- NSF-PAR ID:
- 10107148
- Date Published:
- Journal Name:
- Proceedings of the 36th International Conference on Machine Learning
- Volume:
- 97
- Page Range / eLocation ID:
- 7624-7633
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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