51 Mathematik
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On Least Squares Approximations of Shapley Values and Applications to Interpretable Machine Learning
(2026)
The Shapley value is the predominant point-valued solution concept in cooperative game theory and has recently become a foundational method in interpretable machine learning. In this domain, a prevailing strategy for circumventing the computational intractability of exact Shapley values is to approximate them via a weighted least squares optimization framework. In this paper, we investigate an existing algorithmic framework for weighted least squares Shapley approximation, assessing its feasibility for feature attribution. Methodologically, we conduct a theoretical variance analysis within a Monte Carlo sampling framework, investigate an approach for sample reuse across strata, and establish a relation to Unbiased KernelSHAP. Our analysis reveals three main findings: (i) a structural equivalence between least squares sampling and Unbiased KernelSHAP; (ii) the non-zero covariance between sampled coalitions introduced by reusing samples across strata in one of the existing least squares-based approaches; and (iii) the absence of a universally optimal sampling strategy across tasks. We validate these results empirically on several cooperative games and practical machine learning problems.
Shapley values are the most widely used point-valued solution concept for cooperative games and have recently garnered attention for their applicability in explainable machine learning. Due to the complexity of Shapley value computation, users mostly resort to Monte Carlo approximations for large problems. We take a detailed look at an approximation method grounded in multilinear extensions proposed in 2021 under the name “Owen sampling”. We point out why Owen sampling is biased and propose unbiased alternatives based on combining multilinear extensions with stratified sampling and importance sampling. Finally, we discuss empirical results of the presented algorithms for various cooperative games, including real-world explainability scenarios.
The note presents a compact geometric mass formula for the charged fermions based on regular-heptagon spectral data and a fixed cubic three-root spectrum. The three generations are identified with the roots of the polynomial alpha^3 + alpha^2 - 2 alpha - 1 = 0. Sector-dependent coefficients are generated from a common discrete label assignment for the fermion sectors. The charged-sector reference masses are not fitted independently, but are obtained from the Higgs vacuum expectation value v = 246.22 GeV through an exponential damping law M_ref^(f) = (v/sqrt(2)) exp(-Delta^(f)), with Delta^(f) = k0 - phi r_f - Q_f^2/sqrt(2) and k0 = 6 + 7u - 2u^2, where u = 2 sin(pi/7). For the nine charged fermions, the resulting predictions reproduce the observed hierarchy with a global root-mean-square deviation of 5.7%. This deposition contains the final manuscript and a reference implementation used to generate the reported numerical results.
This paper considers if Artificial Intelligence (AI) tools based on Large Language Models (LLMs) can support math education of engineers. A typical spectrum of engineering math tasks is evaluated with the help of GPT-4 which is also used as a verification tool. Suitable prompting methods compatible with the facilities of students are investigated. A proposal for a zero-shot prompt is made that uses LATEX for the input of mathematical formulas and demands outputs in natural language and Python code as additional output format. For verification purposes the LLM is provided with the natural language and code outputs and the offline computed Python results.
We study the action of the nonlinear mapping G[z] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of G[z] = 0. The results represent local generalizations of the standard implicit or inverse function theorem and of Newton’s Lemma, considering the order of approximation needed to obtain solutions of G[z] = 0. The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2k with k denoting the maximal rank of Jordan chains needed to ensure k-surjectivity of the linear family. Along these lines, the zero set of G[z] in the cone is calculated immediately, agreeing up to the order of k – 1 with the given approximation. Hence, the results may also be interpreted as a version of Tougeron’s implicit function theorem in Banach spaces, essentially restricted to the arc case of a single variable. Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.
We give conditions for local diagonalization of an analytic operator family L (ε) according to L (ε) = ψ (ε)·Δ (ε)·ϕ−1 (ε) with diagonal operator polynomial Δ (ε) and analytic near identity bijections ψ (ε) and ϕ (ε). The family L (ε) is acting between real or complex Banach spaces B and ¯B. The basic assumption is given by stabilization of the Jordan chains at length k in the sense that no root elements with finite rank above k are allowed to exist. Jordan chains with infinite rank may appear. Decompositions of the linear spaces B and ¯B are constructed with corresponding subspaces assumed to be closed. These assumptions ensure finite pole order equal to k of the generalized inverse L−1 (ε) at ε = 0. The Smith form and smooth continuation of kernels and ranges of L (ε) to appropriate limit spaces at ε = 0 arise immediately. An algebraically oriented and self-contained approach is used, based on a recursion that allows for construction of power series solutions of L (ε) · b = 0. The power series solutions are convergent, as soon as analyticity of L (ε) and continuity of related projections are assumed.
We investigate power indices for simple games with precoalitions which distribute power among players in an external and an internal step. We extend an existing approach which uses the Public Good index both on the external level in the quotient game as well as on the internal level for measuring the leverage of players to threaten their peers through departing the precoalition. We replace the Public Good index in that model by five other efficient power indices, i.e., the Shapley–Shubik index, the Deegan–Packel index, the Johnston index and two indices based on null player free winning coalitions. Axiomatizations of the novel power indices with threat partitions are presented. We also propose a slight modification to the existing framework for threat power indices which guarantees that null players are always assigned zero power. Numerical results for all power indices combined with different threat partitions are presented and discussed.
Sum of Ranking Differences (SRD) is a relatively novel, non-para-metric statistical procedure that has become increasingly popular recently. SRD compares solutions via a reference by applying a rank transformation on the input and calculating the distance from the reference in L1 norm. Although the computation of the test statistics is simple, validating the results is cumbersome -- at least by hand. There are two validation steps involved. Comparison of Ranks with Random Numbers, which is a permutation-test, and cross-validation combined with statistical testing. Both options impose computational difficulties albeit different ones. The rSRD package was devised to simplify the validation process by reducing both validation steps into single function calls. In addition, the package provides various useful tools including data preprocessing and plotting. The package makes SRD accessible to a wide audience as there are currently no other software options with such a comprehensive toolkit. This paper aims to serve as a guide for practitioners by offering a detailed presentation of the features.