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I recently came across Varadhan's formula (see e.g. [1], [2], [3], [4], [5]): $$ {d_{\text{g}}(x,y)^2}{} = -\lim_{t \rightarrow 0} 4 t \log K_t(x,y) $$ where $d_\text{g}$ is the geodesic distance and $K_t$ is the heat kernel (i.e. the fundamental solution to the manifold heat equation $\Delta_g u = \partial_t u$). Essentially, it gives a way to compute the geodesic distance between two points, using the heat kernel.

Completely separately, there is a notion of a diffusion map (see e.g. [1], [2], [3], [4]). The idea is to use a "similarity" function $\kappa(x,y)$ to construct the transition matrix of a Markov process representing a random walk on a set of points (from a manifold): $$ p(x,y) = \dfrac{\kappa(x,y)}{\sqrt{\left(\sum_u\kappa(x,u)\right)\left(\sum_v\kappa(y,v)\right)}} $$ which is the chance of the walk moving from $x$ to $y$ in one step. The chance of moving from $x$ to $y$ in $n$ steps is given by the matrix power $p_n(x,y)=p^{(n)}(x,y)$. The diffusion distance is then: $$ d_m(x,y)^2 = \sum_{z} || p_m(x,z) - p_m(y,z) ||^2_d $$ where $m$ is fixed and $||\cdot||_d$ is some distance metric. Informally, the diffusion distance is related to the length of all paths between the two points. This is quite distinct from the geodesic distance.

Of course, the quantity $K_t(x,y)$ is akin to a similarity function (since it is the probability of getting from $x$ to $y$ in time $t$ for a random walk), and could also be used to construct a notion of distance directly, I suppose.


Questions

  • Is there a clear mathematical connection between these different ways to measure distances between points on a manifold using the heat kernel?

  • What other connections are there between the heat kernel, diffusion, and paths/distances between points on a manifold?

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