Low-degree tests at large distances

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave ``randomly'' with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius

Testing Properties of Multiple Distributions with Few Samples

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from $s$ distributions, $p_1, p_2, \ldots, p_s$, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the $p_i$'s are uniform or $\epsilon$-far from being uniform in $\ell_1$-distance (2) Identity Testing: Testing whether all the $p_i$'s are equal to an explicitly given distribution $q$ or $\epsilon$-far from $q$ in $\ell_1$-distance, and (3) Closeness Testing: Testing whether all the $p_i$'s are equal to a distribution $q$ which we have sample access to, or $\epsilon$-far from $q$ in $\ell_1$-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.Comment: ITCS 202

Approximate reasoning for real-time probabilistic processes

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.Comment: Preliminary version appeared in QEST 0