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In several applications, the data consists of an mn matrixA and it is of interest to find an approximation D of a specified rank k to A where, k is much smaller than m and n.Traditional methods like the Singular Value Decomposition (SVD) help us find the "best" such approximation. However, these methods take time polynomial in m; n which is often too prohibitive.In this paper, we develop an algorithm which is qualitatively faster provided we may sample the entries of the matrix according to a natural probability distribution. Indeed, in the applications such sampling is possible.Our main result is that we can find the description of a matrix D of rank at most k so that jjA , D jj F min D;rankDk jjA , Djj F + "jjAjj F holds with probability at least 1 , . (For any matrix M, jjMjj 2 F denotes the sum of the squares of all the entries of M.) The algorithm takes time polynomial in k;1="; log1= only, independent of m; n.

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Chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedron P and integral ve.:tor w, if max( wx Ix E P, wx integer}= I, then wx"' t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that wx be integer for some other integral vector w. The cutting-plane proofs thus obtained ma) be seen either as an abstraction of Gomory's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvatal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.