Sample complexity bounds for localized sketching

We consider sketched approximate matrix multiplication and ridge regression in the novel setting of localized sketching, where at any given point, only part of the data matrix is available. This corresponds to a block diagonal structure on the sketching matrix. We show that, under mild conditions, block diagonal sketching matrices require only O(sr/ ) and O(sdλ/ ) total sample complexity for matrix multiplication and ridge regression, respectively. This matches the state-of-the-art bounds that are obtained using global sketching matrices. The localized nature of sketching considered allows for different parts of the data matrix to be sketched independently and hence is more amenable to computation in distributed and streaming settings and results in a smaller memory and computational footprint.

[1]  Shusen Wang,et al.  Sketched Ridge Regression: Optimization Perspective, Statistical Perspective, and Model Averaging , 2017, ICML.

[2]  Brian McWilliams,et al.  LOCO: Distributing Ridge Regression with Random Projections , 2014, 1406.3469.

[3]  David P. Woodruff.  Sketching as a Tool for Numerical Linear Algebra , 2014, Found. Trends Theor. Comput. Sci..

[4]  David P. Woodruff,et al.  Low rank approximation and regression in input sparsity time , 2012, STOC '13.

[5]  Michael B. Wakin,et al.  The Restricted Isometry Property for Random Block Diagonal Matrices , 2012, ArXiv.

[6]  Yang Liu,et al.  Fast Relative-Error Approximation Algorithm for Ridge Regression , 2015, UAI.

[7]  Bernard Chazelle,et al.  Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform , 2006, STOC '06.

[8]  Petros Drineas,et al.  An Iterative, Sketching-based Framework for Ridge Regression , 2018, ICML.

[9]  Holger Rauhut,et al.  Suprema of Chaos Processes and the Restricted Isometry Property , 2012, ArXiv.

[10]  David P. Woodruff,et al.  Sharper Bounds for Regularized Data Fitting , 2016, APPROX-RANDOM.

[11]  David P. Woodruff,et al.  Faster Kernel Ridge Regression Using Sketching and Preconditioning , 2016, SIAM J. Matrix Anal. Appl..

[12]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[13]  David P. Woodruff,et al.  Optimal Approximate Matrix Product in Terms of Stable Rank , 2015, ICALP.

[14]  Petros Drineas,et al.  Feature Selection for Ridge Regression with Provable Guarantees , 2016, Neural Computation.

[15]  Michael W. Mahoney,et al.  Implementing Randomized Matrix Algorithms in Parallel and Distributed Environments , 2015, Proceedings of the IEEE.