Aiming to unify known results about clustering mixtures of distributions under separation conditions, Kumar and Kannan [KK10] introduced a deterministic condition for clustering datasets. They showed that this single deterministic condition encompasses many previously studied clustering assumptions. More specifically, their proximity condition requires that in the target k-clustering, the proje...

We study the class of polynomial threshold functions using harmonic analysis and apply the results to derive lower bounds related to ACo functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). Our main result is that the class of polynomial threshold functions can be characteriz...

In this paper we get the explicit condition number formulas for the W–weighted Drazin inverse of a rectangular matrix using the Schur decomposition and the spectral norm. We characterize the spectral norm and the Frobenius norm of the relative condition number of the W– weighted Drazin inverse, and the level-2 condition number of the W– weighted Drazin inverse. The sensitivity for the W–weighte...

In this note are proved two conjectures of Daubechies and Lagarias. The first asserts that if Z is a bounded set of matrices such that all left infinite products converge, then 8 generates a bounded semigroup. The second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices. One definition involves the conventional spectral radius, and one involves t...

This paper establishes the following results concerning the estimation of the power spectrum of a single, deterministic, infinitely long signal. a) If ŝx is the signal's power spectral density, correlogram spectral estimates obtained from increasingly longer signal segments tend to ŝx * ŵ/2π in the L-norm, where ŵ is the Fourier transform of the window used to generate the estimates. b) The L-n...

We consider the problem of selecting the “best” subset of exactly k columns from an m× n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn2,m2n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-...