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M\"obius inversion and Shapley values are two mathematical tools for characterizing and decomposing higher-order structure in complex systems. The former defines higher-order interactions as discrete derivatives over a partial order; the…
We consider the problem of exactly learning an $s$-sparse real-valued Boolean polynomial of degree $d$ of the form $f:\{ 0,1\}^n \rightarrow \mathbb{R}$. This problem corresponds to decomposing functions in the AND basis and is known as…
Attention mechanisms and Transformer architectures have revolutionized Natural Language Processing (NLP) by enabling exceptional modeling of long-range dependencies and capturing intricate linguistic patterns. However, their inherent…
M\"obius transformations play an important role in both geometry and spherical image processing - they are the group of conformal automorphisms of 2D surfaces and the spherical equivalent of homographies. Here we present a novel,…
Building on work of Charton, we train small transformer models to calculate the M\"{o}bius function $\mu(n)$ and the squarefree indicator function $\mu^2(n)$. The models attain nontrivial predictive power. We apply a mixture of additional…
Relating macroscopic observables to microscopic interactions is a central challenge in the study of complex systems. While current approaches often focus on pairwise interactions, a complete understanding requires going beyond these to…
We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the…
The partial information decomposition (PID) and its extension integrated information decomposition ($\Phi$ID) are promising frameworks to investigate information phenomena involving multiple variables. An important limitation of these…
Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to…
Information related measures are useful tools for multi variable data analysis, as measures of dependence among variables, and as descriptions of order in biological and physical systems. Information related measures, like marginal…
Shapley effects are attracting increasing attention as sensitivity measures. When the value function is the conditional variance, they account for the individual and higher order effects of a model input. They are also well defined under…
We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N, compute their subset convolution f*g, defined for all S\subseteq N by (f * g)(S) = \sum_{T…
Estimating nonlinear functionals of probability distributions from samples is a fundamental statistical problem. The "plug-in" estimator obtained by applying the target functional to the empirical distribution of samples is biased.…
We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications…
Machine learning models have shown exceptional prowess in solving complex issues across various domains. However, these models can sometimes exhibit biased decision-making, resulting in unequal treatment of different groups. Despite…
Classification has been a major task for building intelligent systems as it enables decision-making under uncertainty. Classifier design aims at building models from training data for representing feature-label distributions--either…
Graph-based representations play a key role in machine learning. The fundamental step in these representations is the association of a graph structure to a dataset. In this paper, we propose a method that aims at finding a block sparse…
In this paper we associate with every (directed) graph G a transformation called the Mobius transformation of the graph G. The Mobius transformation of the graph (O) is of major significance for Dempster-Shafer theory of evidence. However,…
In this paper we describe spectral transformation algorithms for the computation of eigenvalues with positive real part of sparse nonsymmetric matrix pencils $(J,L)$, where $L$ is of the form $\pmatrix{M&0\cr 0&0}$. For this we define a…
SHAP is a popular method for measuring variable importance in machine learning models. In this paper, we study the algorithm used to estimate SHAP scores and outline its connection to the functional ANOVA decomposition. We use this…