Dual polynomials and communication complexity of XOR functions

We show a new duality between the polynomial margin complexity of f and the discrepancy of the function f ◦XOR, called an XOR function. Using this duality, we develop polynomial based techniques for understanding the bounded error (BPP) and the weakly-unbounded error (PP) communication complexities of XOR functions. This enables us to show the following. • A weak form of an interesting conjecture of Zhang and Shi [41] asserts that for symmetric functions f : {0, 1} → {−1, 1}, the weakly unbounded-error complexity of f◦XOR is essentially characterized by the number of points i in the set {0, 1, . . . , n−2} for which Df (i) 6= Df (i+2), where Df is the predicate corresponding to f . The number of such points is called the odd-even degree of f . We observe that a much earlier work of Zhang [40] implies that the PP complexity of f ◦XOR is O(k log n), where k is the odd-even degree of f . We show that the PP complexity of f ◦ XOR is Ω(k/ log(n/k)). • We resolve a conjecture of Zhang [40] characterizing the Threshold of Parity circuit size of symmetric functions in terms of their odd-even degree. • We obtain a new proof of the exponential separation between PP and UPP via an XOR function. • We provide a characterization of the approximate spectral norm of symmetric functions, affirming a conjecture of Ada et al. [2] which has several consequences (cf. [2]). This also provides a new proof of the characterization of the bounded error complexity of symmetric XOR functions due to [41]. Additionally, we prove strong UPP lower bounds for f ◦ XOR, when f is symmetric and periodic with period O(n), for any constant ǫ > 0. More precisely, we show that every such XOR function has unbounded error complexity n, unless f is constant or parity or its complement, in which case the complexity is just O(1). As a direct consequence of this, we derive new exponential lower bounds on the size of depth-2 threshold circuits computing such XOR functions. Our UPP lower bounds do not involve the use of linear programming duality. Partially supported by a Ramanujan fellowship of the DST. [email protected] Supported by a DAE fellowship. [email protected] 1The full conjecture has just been reported to be independently settled by Hatami and Qian [18]. However, their techniques are quite different and are not known to yield many of the results we obtain here. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 62 (2017)