^{1}

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As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the

Several years ago, motivated by considering heat flow on a manifold, R. Coifman proposed a diffusion distance—both for the case of a manifold and a discrete analog for a set of data points in

We see a drawback in Coifman's diffusion distance in that it finds the

Our main motivation for this paper is to propose an alternate diffusion metric, which finds the

In the case of heat flow on

We next give two proofs that the mean first passage cost—defined later in this paper as the cost to visit a particular point for the first time after leaving a specified point—satisfies the triangle inequality. (See Theorem

We calculate explicitly the normalized limit of the mean first passage time for the unit circle

The paper is organized as follows. After a section on notation, we discuss R. Coifman's diffusion distance for both the continuous and discrete cases in Section

In this paper, we will present derivations for both the continuous and discrete cases.

In the continuous situation, we assume there is an underlying Riemannian manifold

We will often specialize to the case when

For the discrete situation, the analog of

For

Finally,

Several years ago, R. Coifman proposed a novel diffusion distance based on the ideas of heat flow on a manifold or a discrete analog of heat flow on a set of data points (see, e.g, [

Referring to Section

Note that we thus have

Although Coifman's original definition used a kernel symmetric with respect to the space variable,

If, in addition to

Referring again to Section

An important benefit of introducing a diffusion distance as above can be illustrated by considering (

Analogous considerations hold in the discrete situation for

See [

We would now like to point out what we see some drawbacks of Coifman's distance, which led us to propose an alternative distance in Section

Let us consider (

The second point of concern is more general in nature. In the continuous case, Coifman's distance involves the

So the diffusion distance proposed by Coifman finds the

Note that

In the next section, we propose a variant of the diffusion distance discussed in this section. Our version will find the

In this section, we propose a new diffusion distance. Let us first define our alternate diffusion distance for the continuous case. Refer to Section

For any

For

As is clear from the defining equality in (

If

As an example, again consider the case where

For any, say, compact Riemannian manifold

In the discrete situation, where we start with a set of data points

If

In this section, we consider a slightly different topic: the mean first passage cost (defined below) between two states as a measure of separation in the discrete situation. We give two explicit proofs showing that the mean first passage cost satisfies the triangle inequality (in [

In [

In this section, as in the rest of the paper unless stated otherwise, we are

A model example we are thinking about is the following. Suppose we have a map grid and are tracking some localized storm which is currently at some particular location on the grid. We suppose that the storm behaves like a random walk and has a certain (constant in time) probability to move from one grid location to another at each “tick of the clock" (time step). We can thus model the movements of the storm by a Markov matrix

In the first part of this section, we give two proofs that the mean first passage cost/time, associated with a not-necessarily-symmetric Markov matrix

We conclude the section by exhibiting a connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem and verify our result numerically for the simple case of the unit circle.

In this section,

Let

We will give two proofs that the expected cost of going from one state to another satisfies the triangle inequality.

We again note that this proposition, for the case all costs are

(2) Our second proof is via explicit matrix computations. Let us define the following two quantities:

We will finish our second proof by showing that

We would like to point out that the decomposition of

We conclude this section by considering certain (suitably normalized) limiting values of the expected cost of going from state

Now, let us digress a little to describe a stochastic approach to solving certain boundary value problems. The description below follows very closely parts of Chapter

Let

Now we define

Let us transfer the above discussion to, say, a compact manifold

We thus see a connection between the (normalized) mean first passage time and the solution to the Dirichlet-Poisson problem discussed above.

Let us illustrate the preceding discussion by a simple example:

To numerically confirm (

The authors have presented a diffusion distance which uses

We thank Raphy Coifman for his continuous generosity in sharing his enthusiasm for mathematics and his ideas about diffusion and other topics. We would also like to thank the anonymous reviewer for his/her thorough critique of this paper and many helpful suggestions.