Sparsifying Sums of Norms

For any norms N-1, ... , N-m on R-n and N(x) := N-1(x) + ... + N-m(x), we show there is a sparsified norm (N) over tilde (x) = w(1)N(1)(x) + ... + w(m)N(m)(x) such that vertical bar N(x)-(N) over tilde (x)vertical bar <= epsilon N(x) for all x is an element of R-n, where w(1), ... , w(m) are non-negative weights, of which only O(epsilon(-2) n log(n/epsilon)(log n)(2.5)) are non-zero. Additionally, we show that such weights can be found with high probability in time O(m(log n)(O(1)) + poly(n))T, where T is the time required to evaluate a norm N-i(x), assuming that N(x) is poly(n)-equivalent to the Euclidean norm. This immediately yields analogous statements for sparsifying sums of symmetric submodular functions. More generally, we show how to sparsify sums of pth powers of norms when the sum is p-uniformly smooth. (1)