Related papers: Smoothed analysis of complex conic condition numbe…
We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend…
We perform a smoothed analysis of the componentwise condition numbers for determinant computation, matrix inversion, and linear equations solving for sparse n times n matrices. The bounds we obtain for the ex- pectations of the logarithm of…
We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small…
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs.…
Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While…
Smoothed analysis is a framework for analyzing the complexity of an algorithm, acting as a bridge between average and worst-case behaviour. For example, Quicksort and the Simplex algorithm are widely used in practical applications, despite…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
We perform a smoothed analysis of the condition number of rectangular matrices. We prove that, asymptotically, the expected value of this condition number depends only of the elongation of the matrix, and not on the center and variance of…
We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those…
We develop new techniques for proving lower bounds on the least singular value of random matrices with limited randomness. The matrices we consider have entries that are given by polynomials of a few underlying base random variables. This…
We extend to Gaussian distributions a result providing smoothed analysis estimates for condition numbers given as relativized distances to illposedness. We also introduce a notion of local analysis meant to capture the behavior of these…
Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in unsupervised learning and high-dimensional data analysis. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable…
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of…
In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares problem. By setting specific norms and…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve…
We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new…
We define a new condition number adapted to directionally uniform perturbations. The definitions and theorems can be applied to a large class of problems. We show the relation with the classical condition number, and study some interesting…
In traditional models of supervised learning, the goal of a learner -- given examples from an arbitrary joint distribution on $\mathbb{R}^d \times \{\pm 1\}$ -- is to output a hypothesis that is competitive (to within $\epsilon$) of the…
Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions is studied in plane domains. Necessary and sufficient conditions upon the right-hand side of the problem and nonlocal operators under…