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Alternative Methods to SHAP Derived from Properties of Kernels: A Note on Theoretical Analysis

Kazuhiro Hiraki, Shinichi Ishihara, Junnosuke Shino

TL;DR

This work establishes a general analytical linkage between Additive Feature Attribution (AFA) and the kernel used in LIME, deriving expressions that map any AFA to its kernel and enabling construction of new AFAs with desirable kernel properties. It shows SHAP arises from a specific kernel and demonstrates how alternative kernels yield variants such as LN Kernel (LnK), ExK, Cncav, and others aligned with LS prenucleolus or LIME-like weightings. By presenting explicit closed-form solutions for unconstrained and efficiency-constrained optimizations, the paper unifies many AFA definitions under a single framework and explores theoretical connections to cooperative game theory. The results provide a toolkit for designing kernel-based explanations with tailored weighting schemes, while highlighting avenues for empirical validation and potential axiomatization of the newly proposed AFAs.

Abstract

This study first derives a general and analytical expression of AFA (Additive Feature Attribution) in terms of the kernel in LIME (Local Interpretable Model-agnostic Explanations). Then, we propose some new AFAs that have appropriate properties of kernels or that coincide with the LS prenucleolus in cooperative game theory. We also revisit existing AFAs such as SHAP (SHapley Additive exPlanations) and re-examine the properties of their kernels.

Alternative Methods to SHAP Derived from Properties of Kernels: A Note on Theoretical Analysis

TL;DR

This work establishes a general analytical linkage between Additive Feature Attribution (AFA) and the kernel used in LIME, deriving expressions that map any AFA to its kernel and enabling construction of new AFAs with desirable kernel properties. It shows SHAP arises from a specific kernel and demonstrates how alternative kernels yield variants such as LN Kernel (LnK), ExK, Cncav, and others aligned with LS prenucleolus or LIME-like weightings. By presenting explicit closed-form solutions for unconstrained and efficiency-constrained optimizations, the paper unifies many AFA definitions under a single framework and explores theoretical connections to cooperative game theory. The results provide a toolkit for designing kernel-based explanations with tailored weighting schemes, while highlighting avenues for empirical validation and potential axiomatization of the newly proposed AFAs.

Abstract

This study first derives a general and analytical expression of AFA (Additive Feature Attribution) in terms of the kernel in LIME (Local Interpretable Model-agnostic Explanations). Then, we propose some new AFAs that have appropriate properties of kernels or that coincide with the LS prenucleolus in cooperative game theory. We also revisit existing AFAs such as SHAP (SHapley Additive exPlanations) and re-examine the properties of their kernels.
Paper Structure (17 sections, 1 theorem, 45 equations)

This paper contains 17 sections, 1 theorem, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A brief review on LIME and kernel
  4. Results
  5. Symmetric Condition on Kernel
  6. Analytical solution to the optimization problem with no constraint
  7. Analytical solution to the optimization problem with the efficiency constraint
  8. SHAP
  9. ES and FESP in Condevaux2023
  10. AFA based on LS preucleolus
  11. AFA with a reasonable kernel (I)
  12. AFA with a reasonable kernel (II)
  13. AFA with a reasonable kernel (III)
  14. Conclusion
  15. Coincidence of AFAs in additive prediction models
  16. ...and 2 more sections

Key Result

Lemma A.1

If the prediction model $f$ is additive with respect to $X_j$, then the characteristic function form game $(N, v_{\tau})$ defined in (vtau) is additive.

Theorems & Definitions (2)

  • Definition A.1
  • Lemma A.1