Applying the superfast divide-and-conquer MBA algorithm for generally singular n × n Toeplitz-like or Hankel-like integer input matrices, we perform computations in the ring of integers modulo a power of a fixed prime, especially power of 2. This is practically faster than computing modulo a random prime but requires additional care to avoid degeneration, particularly at the stages of compressi...

In this paper, we focus on developing randomized algorithms for the computation of low multilinear rank approximations tensors based random projection and singular value decomposition. Following theory values sub-Gaussian matrices, make a probabilistic analysis error bounds algorithm. We demonstrate effectiveness proposed via several numerical examples.

Motivated by the definition of P†-matrix ([9]), another generalization of a P -matrix for square singular matrices called PD-matrix is proposed first. Then the uniqueness of solution of Linear Complementarity Problems for square singular matrices is proved using PD-matrices. Finally some results which are true for P -matrices are extended to PD-matrices.

This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups UN and the general linear groups GLN , for N ∈ N. It establishes the strongest known convergence results for the empirical eigenvalues in the UN case, and the first known almost sure convergence results for the eigenvalues and singular values in ...

These notes were prepared for three guest lectures I gave at Stanford in Spring quarter of 2017 as part of Amir Dembo’s graduate-level course on Large Deviations and Random Matrices. My primary sources were: (1) Sections 2.3 and 2.7 of Tao’s text on random matrix theory [Tao12] (also available on his blog). (2) The ICM lecture notes of Rudelson and Vershynin [RV10]. (3) The paper [TV10a] of Tao...

We present a general method to detect and extract from a finite time sample statistically meaningful correlations between input and output variables of large dimensionality. Our central result is derived from the theory of free random matrices, and gives an explicit expression for the interval where singular values are expected in the absence of any true correlations between the variables under...

In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2 × 2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the ...