Title: The Solvability of Interpretability Evaluation Metrics
Abstract: Feature attribution methods are popular for explaining neural network predictions, and they are often evaluated on metrics such as comprehensiveness and sufficiency. In this paper, we highlight an intriguing property of these metrics: their solvability. Concretely, we can define the problem of optimizing an explanation for a metric, which can be solved by beam search. This observation leads to the obvious yet unaddressed question: why do we use explainers (e.g., LIME) not based on solving the target metric, if the metric value represents explanation quality? We present a series of investigations showing strong performance of this beam search explainer and discuss its broader implication: a definition-evaluation duality of interpretability concepts. We implement the explainer and release the Python solvex package for models of text, image and tabular domains.


Title: Hadwiger's conjecture and topological bounds
Abstract: The Odd Hadwiger's conjecture, formulated by Gerards and Seymour in 1995, is a substantial strengthening of Hadwiger's famous coloring conjecture from 1943. We investigate whether the hierarchy of topological lower bounds on the chromatic number, introduced by Matou\v{s}ek and Ziegler (2003) and refined recently by Daneshpajouh and Meunier (2023), forms a potential avenue to a disproof of Hadwiger's conjecture or its odd-minor variant. In this direction, we prove that, in a very general sense, every graph $G$ that admits a topological lower bound of $t$ on its chromatic number, contains $K_{\lfloor t/2\rfloor +1}$ as an odd-minor. This solves a problem posed by Simonyi and Zsb\'{a}n [European Journal of Combinatorics, 31(8), 2110--2119 (2010)]. We also prove that if for a graph $G$ the Dol'nikov-K\v{r}\'{i}\v{z} lower bound on the chromatic number (one of the lower bounds in the aforementioned hierarchy) attains a value of at least $t$, then $G$ contains $K_t$ as a minor. Finally, extending results by Simonyi and Zsb\'{a}n, we show that the Odd Hadwiger's conjecture holds for Schrijver and Kneser graphs for any choice of the parameters. The latter are canonical examples of graphs for which topological lower bounds on the chromatic number are tight.


Title: Joint distribution in residue classes of families of polynomially-defined multiplicative functions
Abstract: We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, when the multiplicative functions can be controlled by the values of polynomials at the first few prime powers. We obtain complete uniform extensions of a criterion of Narkiewicz for families of such multiplicative functions, thus also generalizing and improving upon previous work for a single such function and establishing results that are optimal in most parameters and hypotheses. As a special case of our results, for any fixed $\epsilon>0$, the Euler totient function $\phi(n)$ and sum of divisors function $\sigma(n)$ are jointly asymptotically equidistributed among the reduced residue classes to moduli $q$ coprime to $6$ varying uniformly up to $(\log x)^{(1-\epsilon)\alpha(q)}$, where $\alpha(q) = \prod_{\ell \mid q} (\ell-3)/(\ell-1)$; furthermore, the coprimality restriction is necessary and the range of $q$ is essentially optimal.


Title: The extension problem in free harmonic analysis
Abstract: This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of 'relative energy' which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szego parameters.