Partially Symmetric Functions are Efficiently Isomorphism-Testable

Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.Comment: 22 page

Increasing flow in the actor's work

This paper is an account of preparation I undertook to play the roles of Eric, Venturewell, and Barberosa, in Tim Askew's adaptation of Francis Beaumont's The Knight of the Burning Pestle. Primary to approaching this track I have addressed my artistic challenge of increasing flow in my work as an actor. Starting with Mihaly Csikszentmihalyi concept :of flow I began to investigate blocks during class work, rehearsals, performance and show development. I have applied theory and techniques as described in the writings of Declan Donnellan, Robert Triplett, Stephen Nachmanovitch, Eugen Herrigel and Shunryu Suzuki. Practical studio work revolved around mask work and improvisation as a framework through which explore: block and flow. Other research included historical and critical surveys of Francis Beaumont and The Knight of The Burning Pestle. The paper also includes journal entries from the rehearsal process of the York production and a conclusion of my findings

Distribution Testing Lower Bounds via Reductions from Communication Complexity

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef (Computational Complexity, 2012), we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds. Our main result is concerned with testing identity to a specific distribution p, given as a parameter. In a recent and influential work, Valiant and Valiant (FOCS, 2014) showed that the sample complexity of the aforementioned problem is closely related to the 2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre\u27s K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction

On Testing and Robust Characterizations of Convexity

A body K ? ?? is convex if and only if the line segment between any two points in K is completely contained within K or, equivalently, if and only if the convex hull of a set of points in K is contained within K. We show that neither of those characterizations of convexity are robust: there are bodies in ?? that are far from convex - in the sense that the volume of the symmetric difference between the set K and any convex set C is a constant fraction of the volume of K - for which a line segment between two randomly chosen points x,y ? K or the convex hull of a random set X of points in K is completely contained within K except with exponentially small probability. These results show that any algorithms for testing convexity based on the natural line segment and convex hull tests have exponential query complexity

Testing Submodularity and Other Properties of Valuation Functions

We show that for any constant epsilon > 0 and p ge 1, it is possible to distinguish functions f : {0,1}^n to [0,1] that are submodular from those that are epsilon-far from every submodular function in ell_p distance with a constant number of queries. More generally, we extend the testing-by-implicit-learning framework of Diakonikolas et al.(2007) to show that every property of real-valued functions that is well-approximated in ell_2 distance by a class of k-juntas for some k = O(1) can be tested in the ell_p-testing model with a constant number of queries. This result, combined with a recent junta theorem of Feldman and Vondrak (2016), yields the constant-query testability of submodularity. It also yields constant-query testing algorithms for a variety of other natural properties of valuation functions, including fractionally additive (XOS) functions, OXS functions, unit demand functions, coverage functions, and self-bounding functions

Optimal Separation and Strong Direct Sum for Randomized Query Complexity

We establish two results regarding the query complexity of bounded-error randomized algorithms. * Bounded-error separation theorem. There exists a total function $f : \{0,1\}^n \to \{0,1\}$ whose $\epsilon$-error randomized query complexity satisfies $\overline{\mathrm{R}}_\epsilon(f) = \Omega( \mathrm{R}(f) \cdot \log\frac1\epsilon)$. * Strong direct sum theorem. For every function $f$ and every $k \ge 2$, the randomized query complexity of computing $k$ instances of $f$ simultaneously satisfies $\overline{\mathrm{R}}_\epsilon(f^k) = \Theta(k \cdot \overline{\mathrm{R}}_{\frac\epsilon k}(f))$. As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function $f$ for which $\mathrm{R}(f^k) = \Theta( k \log k \cdot \mathrm{R}(f))$. This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of G\"o\"os, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies $\mathrm{R}^{\mathrm{cc}} (f^k) = \Theta( k \log k \cdot \mathrm{R}^{\mathrm{cc}}(f))$, answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).Comment: 15 pages, 2 figures, CCC 201