A probability distribution over {-1, 1}^n is (epsilon, k)-wise uniform if, roughly, it is epsilon-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (epsilon, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (epsilon, k)-wise uniform distribution is O(n^{k/2}epsilon)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (epsilon, k)-wise uniform distribution that is Omega(n^{k/2}epsilon)-far from any k-wise uniform distribution in total variation distance. For k=1, we get a better upper bound of O(epsilon), which is also optimal. One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^{k}/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^{k/2}/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2. Our results improve upon the best known bounds from [Alon et al., 2007], and have simpler proofs.

Let f(x) = f(x_1, ..., x_n) = sum_{|S|<=k} a_S prod_{i in S} x_i be an n-variate real multilinear polynomial of degree at most k, where S subseteq [n] = {1, 2, ..., n}. For its one-block decoupled version, vf(y,z) = sum_{abs(S)<=k} a_S sum_{i in S}} y_i prod_{j in S\{i}} z_j, we show tail-bound comparisons of the form Pr(abs(vf)(y,z)) > C_k t} <= D_k Pr(abs(f(x)) > t). Our constants C_k, D_k are significantly better than those known for "full decoupling". For example, when x, y, z are independent Gaussians we obtain C_k = D_k = O(k); when x, by, z are +/-1 random variables we obtain C_k = O(k^2), D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is known in these settings. We describe consequences of these results for query complexity (related to conjectures of Aaronson and Ambainis) and for analysis of Boolean functions (including an optimal sharpening of the DFKO Inequality).

In this work, we study the parity complexity measures pCmin[f] and PDT[f]. Pcmin[f] is the parity kill number of f, the fewest number of parities on the input variables one has to fix in order to "kill" f, i.e. To make it constant. PDT[f] is the depth of the shortest emph{parity decision tree} which computes f. These complexity measures have in recent years become increasingly important in the fields of communication complexity [1], [2], [3], [4] and pseudorandomness [5], [6], [7]. Our main result is a composition theorem for pCmin. The k-th power of f, denoted f^{circ k}, is the function which results from composing f with itself k times. We prove that if f is not a parity function, then pCmin[f^{circ k}] geq Omega(Cmin[f]^{k}). In other words, the parity kill number of f is essentially super multiplicative in the normal kill number of f (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of sort^{circ k} and HI^{circ k}. Here sort is the sort function due to Ambainis [8], and HI is Kushilevitz's hemi-icosahedron function [9]. In doing so, we disprove a conjecture of Montanaro and Osborne [2] which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of [2], [3] and [4].