No abstract

We consider the problem of computing the rank of an m × n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns inÕ(|A| + r ω ) field operations, where |A| denotes the number of nonzero entries in A and ω < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with running time O(mnr ω−2 ). Our algorithm is faster when r < max{m, n}, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank inÕ(mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in numerical linear algebra, combinatorial optimization and dynamic data structure.

The Paulsen problem is a basic open problem in operator theory: Given vectors u 1 , . . . , u n ∈ R d that are ǫ-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors v 1 , . . . , v n ∈ R d that exactly satisfy the Parseval's condition and the equal norm condition? Given u 1 , . . . , u n , the squared distance (to the set of exact solutions) is defined as inf v * University of Waterloo.

We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V, E) and a parameter k ≤ |E|, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph partitioning algorithms, we obtain the following bicriteria approximation algorithms for the small sparsest cut problem:• If there is a set U ⊆ V with conductance φ and vol(U ) ≤ k, then there is a polynomial time algorithm to find a set S with conductance O( φ/ǫ) and vol(S) ≤ k 1+ǫ for any ǫ > 1/k.• If there is a set U ⊆ V with conductance φ and vol(U ) ≤ k, then there is a polynomial time algorithm to find a set S with conductance O( φ log k/ǫ) and vol(S) ≤ (1 + ǫ)k for any ǫ > 2 ln k/k.These algorithms can be implemented locally using truncated random walk, with running time almost linear to the output size. This provides a local graph partitioning algorithm with a better conductance guarantee when k is sublinear. * This is independent from the work [15] which obtained similar results.

We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G,denotes the robust vertex expansion of G and φ(G) denotes the edge expansion of G. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of G, for all k ≥ 2, * É cole polytechnique fédérale de Lausanne,Theorem 4. For any (unknown target) set S ⊆ V , there is a polynomial time randomized algorithm to find a set S with 1. φ(S ′ ) = O(kφ(S)/(ǫφ k (G))) and |S ′ | = O(|S| 1+ǫ ), 2. φ(S ′ ) = O(kφ(S) log(|S|)/φ k (G)) and |S ′ | = O(|S|),