  The Shapley value (SV) is a prevalent approach of allocating credit to
machine learning (ML) entities to understand black box ML models. Enriching
such interpretations with higher-order interactions is inevitable for complex
systems, where the Shapley Interaction Index (SII) is a direct axiomatic
extension of the SV. While it is well-known that the SV yields an optimal
approximation of any game via a weighted least square (WLS) objective, an
extension of this result to SII has been a long-standing open problem, which
even led to the proposal of an alternative index. In this work, we characterize
higher-order SII as a solution to a WLS problem, which constructs an optimal
approximation via SII and $k$-Shapley values ($k$-SII). We prove this
representation for the SV and pairwise SII and give empirically validated
conjectures for higher orders. As a result, we propose KernelSHAP-IQ, a direct
extension of KernelSHAP for SII, and demonstrate state-of-the-art performance
for feature interactions.
