What is the actual formula to construct the affinity kernel matrix?

[caodudu]

Hi. In the paper, The adaptive Gaussian kernel was given by $$M(x_i.x_j)=\frac{1}{\sqrt{2\pi(\sigma_i+\sigma_j)}}e^{\left(-\frac{1}{2}\frac{(x_i-x_j)^T(x_i-x_j)}{\sigma_i+\sigma_j}\right)}$$. However, in the script (~/SEACells/build_graph.py), I noted the method constructing affinity kernel matrix is more similar to $$M(x_i​,x_j​)=e^{(−\frac{|x_i​−x_j​|^2}{​σ_i​\cdotσ_j}​)}$$? So did I misunderstand something about it?

[caodudu]

And when I applid the nominal formula $$M(x_i.x_j)=\frac{1}{\sqrt{2\pi(\sigma_i+\sigma_j)}}e^{\left(-\frac{1}{2}\frac{(x_i-x_j)^T(x_i-x_j)}{\sigma_i+\sigma_j}\right)}$$ and assumpted $$x_i = x_j$$, We could find that $$M(x_i,x_j) \leq \frac{1}{\sqrt{2\pi}}\approx 0.4 $$. The upper bound seems weird.