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We prove that for any real-valued matrix X ∈ R m×n , and positive integers r k, there is a subset of r columns of X such that projecting X onto their span gives a r+1 r−k+1 -approximation to best rank-k approximation of X in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in O(rnm ω log m) arithmetic operations where ω is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in O(rnm 2 ) arithmetic operations.

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NP-hardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of a matrix in Frobenius norm using columns of the matrix.

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We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any ε, δ ∈ (0, 1), given cost and demand graphs with edge weights C, D : V 2 → R+ respectively, we can find a set T ⊆ V with C(T,V \T ) D(T,V \T ) at most 1+ε δ times the optimal non-uniform sparsest cut value, in time 2 r/(δε) poly(n) provided λr ≥ Φ * /(1 − δ). Here λr is the r'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs;is the weight of edges crossing the (T, V \ T ) cut in cost (resp. demand) graph and Φ * is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the r'th smallest eigenvalue of the normalized Laplacian playing the role of λr(G) in the latter two cases.Our proof is based on an 1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting 1-embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [9] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut.