I read the paper about SHAP.
I think this paper is very interesting ! I would like to understand the algorithm, but I cannot follow the below fact.
\begin{equation} X^T WX=\dfrac{1}{M-1}I+cJ \end{equation}
where $W$ is a diagonal matrix with Shapley kernel weights for each arrow of $X$, $I$ is the identity matrix and $J$ is the matrix of all ones.
And I cannot follow the below fact. \begin{equation} (X^T WX)^{-1}=I+\dfrac{1}{M-1}(I-J) \ \ (c\rightarrow \infty) \end{equation}
Could you mind telling me how to prove this fact.
I think there is typographical error in the NIPS supplemental paper.
\begin{equation} X^TWX = \dfrac{1}{M-1}I+cJ \ \ (\mathrm{NIPS}) \end{equation}
\begin{equation} X^TWX = \dfrac{M-1}{M}I+cJ \ \ (\mathrm{Update}) \tag{1} \end{equation}
If we suppose equation (1), using sherman-morrison formula, we can get equation (2).
\begin{eqnarray} (X^T WX)^{-1}&=&\dfrac{M}{M-1}I-\dfrac{\dfrac{M}{M-1}\dfrac{M}{M-1}cJ}{1+\dfrac{M}{M-1}cM}\\ &\rightarrow&\dfrac{M}{M-1}I-\dfrac{1}{M-1}J \ \ (c\rightarrow \infty)\\ &=&I+\dfrac{1}{M-1}(I-J) \ \ (c\rightarrow \infty) \tag{2} \end{eqnarray}