Related papers: Complexity Measures on the Symmetric Group and Bey…
In the decision tree computation model for Boolean functions, the depth corresponds to query complexity, and size corresponds to storage space. The depth measure is the most well-studied one, and is known to be polynomially related to…
Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of $f$ is defined in terms of a monotone, submodular utility function…
We present a new approach to study measures on ensembles of contours, polymers or other objects interacting by some sort of exclusion condition. For concreteness we develop it here for the case of Peierls contours. Unlike existing methods,…
We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a…
Coherence measures and their operational interpretations lay the cornerstone of coherence theory. In this paper, we introduce a class of coherence measures with $\alpha$-affinity, say $\alpha$-affinity of coherence for $\alpha \in (0, 1)$.…
Given a graph of interactions, a module (also called a community or cluster) is a subset of nodes whose fitness is a function of the statistical significance of the pairwise interactions of nodes in the module. The topic of this paper is a…
We study sharp detection thresholds for degree corrections in Stochastic Block Models in the context of a goodness of fit problem, and explore the effect of the unknown community assignment (a high dimensional nuisance parameter) and the…
Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…
We compare several complexity measures for self-mappings of finite fields. In particular, we show that Carlitz rank and additive index cannot be small simultaneously up to trivial exceptions. That is, these two measures detect cryptographic…
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to…
The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…
We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…
When epidemiologic studies are conducted in a subset of the population, selection bias can threaten the validity of causal inference. This bias can occur whether or not that selected population is the target population, and can occur even…
Modularity is a very widely used measure of the level of clustering or community structure in networks. Here we consider a recent generalisation of the definition of modularity to temporal graphs, whose edge-sets change over discrete…
We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, Lopez-Ruiz, Mancini, and Calbet [Phys. Lett. A 209 (1995) 321] introduced another…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
Unmeasured confounding remains a critical challenge in causal inference for the social sciences. This paper proposes a sensitivity analysis framework to systematically evaluate how unmeasured confounders influence statistical inference in…
Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in…
We show that there is a largely unexplored class of functions (positive polymatroids) that can define proper discrete metrics over pairs of binary vectors and that are fairly tractable to optimize over. By exploiting submodularity, we are…
We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For…