We consider a generalization of the classic linear regression problem to the case when the loss is an Orlicz norm. An Orlicz norm is parameterized by a non-negative convex function G: R_+ - > R_+ with G(0) = 0: the Orlicz norm of a n-dimensional vector x is defined as |x|_G = inf{ alpha > 0 | sum_{i = 1}^n G( |x_i| / alpha ) < = 1 }. We consider the cases where the function G grows subquadratically. Our main result is based on a new oblivious embedding which embeds the column space of a given nxd matrix A with Orlicz norm into a lower dimensional space with L2 norm. Specifically, we show how to efficiently find an mxn embedding matrix S (m < n), such that for every d-dimensional vector x, we have Omega(1/(d log n)) |Ax|_G < = |SAx|_2 < = O(d^2 log n) |Ax|_G. By applying this subspace embedding technique, we show an approximation algorithm for the regression problem min_x |Ax-b|_G, up to a O( d log^2 n ) factor. As a further application of our techniques, we show how to also use them to improve on the algorithm for the Lp low rank matrix approximation problem for 1 < = p < 2.

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for f(x and y) and f(x xor y) with monotone functions f, where and and xor are bit-wise AND and XOR , respectively. In this paper, we extend these results to functions f which alternate values for a relatively small number of times on any monotone path from 0^n to 1^n. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.

An edge cover of a graph is a set of edges such that every vertex has at least an adjacent edge in it. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for counting the number of edge covers for any graph. Previously, approximation algorithm is only known for 3 regular graphs and it is randomized [3]. Our main technique is correlation decay, which is a powerful tool to design FPTAS for counting problems. In order to get FPTAS for general graphs without degree bound, we make use of a stronger notion called computationally efficient correlation decay, which was introduced in [19].