We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. As we show, our bounds compare favorably with previously known bounds. Aside from the new results that we derive, our method yields new and more transparent proofs of some known results as well. Among our new results we extend some known lower bounds to the realm of quantum communication complexity with entanglement. 1

A halfspace matrix is a Boolean matrix A with rows indexed by linear threshold functions f, columns indexed by inputs x ∈ {−1,1} n, and the entries given by A f,x = f (x). We demonstrate the potential of halfspace matrices as tools to answer nontrivial open questions. 1. (Communication complexity) We exhibit a Boolean function f with discrepancy Ω(1/n 4) under every product distribution but O ( √ n/2 n/4) under a certain non-product distribution. This partially solves an open problem of Kushilevitz and Nisan [25]. 2. (Complexity of sign matrices) We construct a matrix A ∈ {−1,1} N×NlogN with dimension complexity logN but margin complexity Ω(N 1/4 / √ logN). This gap is an exponential improvement over previous work. As an application to circuit complexity, we prove an Ω(2n/4 /(d √ n)) circuit lower bound for computing halfspaces by a majority of an arbitrary set of d gates. This complements a result of Goldmann, H˚astad, and Razborov [15]. In addition, we prove new results on the complexity measures of sign matrices, complementing recent work by Linial et al. [27–29]. 3. (Learning theory) We give a short and simple proof that the statistical-query (SQ) dimension of halfspaces in n dimensions is less than 2(n + 1) 2 under all distributions (with n + 1 being a trivial lower bound). This improves on the n O(1) estimate from the fundamental paper of Blum et al. [5]. Finally, we motivate our learning-theoretic result for the complexity community by showing that SQ dimension estimates for natural classes of Boolean functions can resolve major open problems in complexity theory. Specifically, we show that an exp(2 (logn)o(1) ) upper bound on the SQ dimension of AC 0 would imply an explicit language in PSPACE cc \ PH cc. 1

This paper has two main focal points. We first consider an important class of machine learning algorithms- large margin classifiers (Support Vector Machines belong to this class). The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas. In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity. 1

A kernel over the Boolean domain is said to be reflection-invariant, if its value does not change when we flip the same bit in both arguments. (Many popular kernels have this property.) We study the geometric margins that can be achieved when we represent a specific Boolean function f by a classifier that employs a reflectioninvariant kernel. It turns out ‖ ˆ f‖ ∞ is an upper bound on the average margin. Furthermore, ‖ ˆ f‖−1 ∞ is a lower bound on the smallest dimension of a feature space associated with a reflection-invariant kernel that allows for a correct representation of f. This is, to the best of our knowledge, the first paper that exhibits margin and dimension bounds for specific functions (as opposed to function families). Several generalizations are considered as well. The main mathematical results are presented in a setting with arbitrary finite domains and a quite general notion of invariance. 1