We study the basic problem of distinguishing between two symmetric probability distributions over n bits by observing k bits of a sample, subject to the constraint that all (k-1)-wise marginal distributions of the two distributions are identical to each other. Previous works of Bogdanov et al. [Bogdanov et al., 2019] and of Huang and Viola [Huang and Viola, 2019] have established approximately tight results on the maximal possible statistical distance between the k-wise marginals of such distributions when k is at most a small constant fraction of n. Naor and Shamir [Naor and Shamir, 1994] gave a tight bound for all k in the special case k = n and when distinguishing with the OR function; they also derived a non-tight result for general k and n. Krause and Simon [Krause and Simon, 2000] gave improved upper and lower bounds for general k and n when distinguishing with the OR function, but these bounds are exponentially far apart when k = Ω(n). In this work we provide sharp upper and lower bounds on the maximal statistical distance that hold for all k and n. Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials. This is the first work to construct suitable non-uniform approximations for this purpose; the sharpness and wider applicability of our result stems from this non-uniformity.

The $ε$-approximate degree $deg_ε(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $ε$. The approximate degree of $f$ is at least $k$ iff there exists a pair of probability distributions, also known as a dual polynomial, that are perfectly $k$-wise indistinguishable, but are distinguishable by $f$ with advantage $1 - ε$. Our contributions are: We give a simple new construction of a dual polynomial for the AND function, certifying that $deg_ε(f) \geq Ω(\sqrt{n \log 1/ε})$. This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the $1/3$-approximate degree of any read-once DNF is $Ω(\sqrt{n})$. We show that any pair of symmetric distributions on $n$-bit strings that are perfectly $k$-wise indistinguishable are also statistically $K$-wise indistinguishable with error at most $K^{3/2} \cdot \exp(-Ω(k^2/K))$ for all $k < K < n/64$. This implies that any symmetric function $f$ is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-$K$ coalitions with statistical error $K^{3/2} \exp(-Ω(deg_{1/3}(f)^2/K))$ for all values of $K$ up to $n/64$ simultaneously. Previous secret sharing schemes required that $K$ be determined in advance, and only worked for $f=$ AND. Our analyses draw new connections between approximate degree and concentration phenomena. As a corollary, we show that for any $d < n/64$, any degree $d$ polynomial approximating a symmetric function $f$ to error $1/3$ must have $\ell_1$-norm at least $K^{-3/2} \exp({Ω(deg_{1/3}(f)^2/d)})$, which we also show to be tight for any $d > deg_{1/3}(f)$. These upper and lower bounds were also previously only known in the case $f=$ AND.

Two distributions over n-bit strings are (k,delta)-wise indistinguishable if no statistical test that observes k of the n bits can tell the two distributions apart with advantage better than delta. Motivated by secret sharing and cryptographic leakage resilience, we study the existence of pairs of distributions that are (k, delta)-wise indistinguishable, but can be distinguished by some function f of suitably low complexity. We prove bounds tight up to constants when f is the OR function, and tight up to logarithmic factors when f is a read-once uniform AND \circ OR formula, extending previous works that address the perfect indistinguishability case delta = 0. We also give an elementary proof of the following result in approximation theory: If p is a univariate degree-k polynomial such that |p(x)| <= 1 for all |x| <= 1 and p(1) = 1, then l (p) >= 2^{Omega(p'(1)/k)}, where lˆ (p) is the sum of the absolute values of p’s coefficients. A more general 1 statement was proved by Servedio, Tan, and Thaler (2012) using complex-analytic methods. As a secondary contribution, we derive new threshold weight lower bounds for bounded depth AND-OR formulas.

We designed, implemented and evaluated a new concept for direct manipulation of databases, called dynamic queries, that allows users to formulate queries with graphical widgets, such as sliders. By providing a graphical visualization of the database and search results, users can find trends and exceptions easily. Eighteen undergraduate chemistry students performed statistically significantly faster using a dynamic queries interface compared to two interfaces both providing form fill-in as input method, one with graphical visualization output and one with all-textual output. The interfaces were used to explore the periodic table of elements and search on their properties.

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We designed, implemented, and evaluated a new concept for visualizing and searching databases utilizing direct manipulation called dynamic queries. Dynamic queries allow users to formulate queries by adjusting graphical widgets, such as sliders, and see the results immediately. By providing a graphical visualization of the database and search results, users can find trends and exceptions easily. User testing was done with eighteen undergraduate students who performed significantly faster using a dynamic queries interface compared to both a natural language system and paper printouts. The interfaces were used to explore a real-estate database and find homes meeting specific search criteria.