Small-Sample Statistical Estimates for Matrix Norms
SIAM Journal on Matrix Analysis and Applications
Some large-scale matrix computation problems
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Statistical Condition Estimation for Linear Systems
SIAM Journal on Scientific Computing
Database-friendly random projections
PODS '01 Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
An estimator for the diagonal of a matrix
Applied Numerical Mathematics
Fast Counting of Triangles in Large Real Networks without Counting: Algorithms and Laws
ICDM '08 Proceedings of the 2008 Eighth IEEE International Conference on Data Mining
Nonlinear estimators and tail bounds for dimension reduction in l1 using Cauchy random projections
COLT'07 Proceedings of the 20th annual conference on Learning theory
Difference Filter Preconditioning for Large Covariance Matrices
SIAM Journal on Matrix Analysis and Applications
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We analyze the convergence of randomized trace estimators. Starting at 1989, several algorithms have been proposed for estimating the trace of a matrix by 1/M∑i=1M ziT Azi, where the zi are random vectors; different estimators use different distributions for the zis, all of which lead to E(1/M∑i=1M ziT Azi) = trace(A). These algorithms are useful in applications in which there is no explicit representation of A but rather an efficient method compute zTAz given z. Existing results only analyze the variance of the different estimators. In contrast, we analyze the number of samples M required to guarantee that with probability at least 1−δ, the relative error in the estimate is at most &epsis;. We argue that such bounds are much more useful in applications than the variance. We found that these bounds rank the estimators differently than the variance; this suggests that minimum-variance estimators may not be the best. We also make two additional contributions to this area. The first is a specialized bound for projection matrices, whose trace (rank) needs to be computed in electronic structure calculations. The second is a new estimator that uses less randomness than all the existing estimators.