Some Remarks on Diffusion Distances

Abstract

As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the 𝐿² distance between two 𝐿²-normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds the 𝐿² distance between certain 𝐿¹ unit vectors. In the second part of the paper, we give two proofs that an extension of mean first passage time to mean first passage cost satisfies the triangle inequality; we do not assume that the underlying Markov matrix is diagonalizable. We conclude by exhibiting an interesting 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.

1. Introduction

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 ℝ^𝑛. In the continuous case, his distance can be written as the 𝐿² norm of the difference of two specified vectors, each of which has unit 𝐿¹ norm. (An analogous situation holds in the discrete case.) Coifman's distance can be successfully used in various applications, including data organization, approximately isometric embedding of data in low-dimensional Euclidean space, and so forth. See, for example, [1–3]. For a unified discussion of diffusion maps and their usefulness in spectral clustering and dimensionality reduction, see [4].

We see a drawback in Coifman's diffusion distance in that it finds the 𝐿² norm of the distance between two 𝐿¹ unit vectors, rather than 𝐿² unit vectors. As shown by a simple example later in this paper, two vectors (representing two diffusions), which we may want to consider to be far apart, are actually close to each other in 𝐿², even though the angle between them is large, because they have small 𝐿² norm, while still having unit 𝐿¹ norm. Additionally, applying Coifman's distance to heat flow in ℝ^𝑛, a factor of a power of time 𝑡 remains, with the exponent depending on the dimension 𝑛. It would be desirable not to have such a factor.

Our main motivation for this paper is to propose an alternate diffusion metric, which finds the 𝐿² distance between two 𝐿² unit vectors (with analogous statements for the discrete case). Our distance is thus the length of the chord joining the tips, on the unit hypersphere, of two 𝐿² normalized diffusion vectors, and is therefore based on cosine similarity (see (4.4) below). Cosine similarity (affinity) is popular in kernel methods in machine learning; see for example, [5, 6] (in particular, Section 3.5.1—Document Clustering Basics) and for a review of kernel methods in machine learning, [7].

In the case of heat flow on ℝ^𝑛, our proposed distance has the property that no dimensionally dependent factor is left. Furthermore, for a general manifold, our diffusion distance gives, approximately, a scaled geodesic distance between two points 𝑥 and 𝑦, when 𝑥 and 𝑦 are closer than √𝑡, and maximum separation when the geodesic distance between 𝑥 and 𝑦, scaled by √𝑡, goes to infinity.

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 4.2 in [8] in which the author states that the triangle inequality holds for the mean first passage time.) We give two proofs that do not assume that the underlying Markov matrix is diagonalizable; our proofs do not rely on spectral theory.

We calculate explicitly the normalized limit of the mean first passage time for the unit circle 𝑆¹ by identifying the limit as the solution of a specific Dirichlet-Poisson problem on 𝑆¹. We also provide numerical verification of our calculation.

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 3. In Section 4, we define and discuss our alternate diffusion distance. In Section 5, we give two proofs of the triangle inequality for mean first passage cost. We conclude the section by exhibiting an interesting 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 𝑆¹.

2. Notation and Setup

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 𝑀 with measure 𝑑𝑥;𝑥,𝑦,𝑢,𝑧,… will denote points in 𝑀. For 𝑡≥0, 𝜌_𝑡(𝑥,𝑦) will denote a kernel on 𝑀×𝑀, with 𝜌_𝑡(𝑥,𝑦)≥0 for all 𝑥,𝑦∈𝑀, and satisfying the following semigroup property:_𝑀𝜌_𝑡(𝑥,𝑢)𝜌_𝑠(𝑢,𝑦)𝑑𝑢=𝜌_𝑡+𝑠(𝑥,𝑦), (2.1) for all 𝑥,𝑦∈𝑀, and 𝑠,𝑡≥0. In addition, we assume the following property:_𝑀𝜌_𝑡(𝑥,𝑦)𝑑𝑥=1, (2.2) for all 𝑦∈𝑀 and all 𝑡≥0. The latter convention gives the mass preservation property_𝑀𝑇_𝑡𝑓(𝑥)𝑑𝑥=_𝑀𝑓(𝑦)𝑑𝑦, (2.3) where𝑇_𝑡𝑓(𝑥)≡_𝑀𝜌_𝑡(𝑥,𝑦)𝑓(𝑦)𝑑𝑦. (2.4)

We will often specialize to the case when 𝜌_𝑡(𝑥,𝑦)=𝜌_𝑡(𝑦,𝑥) for all 𝑥,𝑦∈𝑀 and 𝑡≥0, as in the case of heat flow. Note that when 𝜌_𝑡(𝑥,𝑦) is the fundamental solution for heat flow, we have 𝜌₀(𝑥,𝑢)=𝛿_𝑥(𝑢), where 𝛿_𝑥(𝑢) denotes the Dirac delta function centered at 𝑥. We will sometimes assume (as in the case of heat flow on a compact manifold) that there exist 0≤𝜆₁≤𝜆₂≤⋯, with each 𝜆_𝑗 corresponding to a finite dimensional eigenspace, and a complete orthonormal family of 𝐿² functions 𝜙₁,𝜙₂,…, such that𝜌_𝑡(𝑥,𝑦)=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡}𝜙_𝑗(𝑥)𝜙_𝑗(𝑦), (2.5) for 𝑡>0. We will also frequently use the following fact: if 𝜌_𝑡 is symmetric in the space variables, then for any 𝑥,𝑦∈𝑀,_𝑀𝜌_𝑡(𝑢,𝑥)𝜌_𝑡(𝑢,𝑦)𝑑𝑢=_𝑀𝜌_𝑡(𝑥,𝑢)𝜌_𝑡(𝑢,𝑦)𝑑𝑢=𝜌_2𝑡(𝑥,𝑦), (2.6) where we have used the symmetry of 𝜌_𝑡 and its semigroup property.

For the discrete situation, the analog of 𝜌_𝑡(𝑥,𝑦) is an 𝑁×𝑁 matrix 𝐴, with 𝐴=⟨𝑎_𝑖𝑗⟩^𝑁_𝑖,𝑗=1, every 𝑎_𝑖𝑗≥0. In keeping with the usual convention that 𝐴 is Markov if each row sum equals 1, that is, ∑_𝑗𝑎_𝑖𝑗=1 for all 𝑖, the analog of 𝑇_𝑡𝑓(𝑥)=∫_𝑀𝜌_𝑡(𝑥,𝑦)𝑓(𝑦)𝑑𝑦 is 𝐴^𝑇𝑣, where 𝐴^𝑇 is the transpose of 𝐴, and 𝑣 is an 𝑁×1 column vector. So the index 𝑖 corresponds to the second space variable in 𝜌_𝑡, the index 𝑗 corresponds to the first space variable in 𝜌_𝑡, and 𝑡=𝑛, 𝑛=1,2,…, corresponds to the 𝑛th power of 𝐴. The obvious analog of 𝜌_𝑡 symmetric in its space variables is a symmetric Markov matrix 𝐴, that is, 𝐴=𝐴^𝑇.

For 𝐴 as above, not necessarily symmetric, we think of 𝑎_𝑖𝑗 as the probability of transitioning from state 𝑠_𝑖 to state 𝑠_𝑗 in 𝑡=1 tick of the clock; 𝑆={𝑠₁,𝑠₂,…,𝑠_𝑁} is the underlying set of states. For 𝑋 a subset of the set of states 𝑆, the 𝑁×𝑁 matrix 𝑃_𝑋 will denote the following projection: all entries of 𝑃_𝑋 are 0 except for diagonal entries (𝑘,𝑘), when 𝑠_𝑘∈𝑋; the latter entries are equal to 1.

Finally, 𝟏 will denote the 𝑁×1 column vector where each entry is 1; 𝑒_𝑖 will denote the 𝑁×1 column vector with the 𝑖th component 1, and all others 0, and, for a set of states 𝑋, 𝑋 will denote the complement of 𝑋 with respect to 𝑆.

3. A Diffusion Distance Proposed by R. Coifman

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, [1, 2] for a thorough discussion). In this section, we will describe Coifman's distance using our notation, and consider some of its good points, and what we see as some of its drawbacks.

Referring to Section 2, for the continuous case, the unweighted version of Coifman's distance between 𝑥,𝑦∈𝑀, which we will denote by 𝑑_𝐶,𝑡(𝑥,𝑦), can be defined as follows:[𝑑_𝐶,𝑡(𝑥,𝑦)]²≡⟨𝑇_𝑡(𝛿_𝑥−𝛿_𝑦),𝑇_𝑡(𝛿_𝑥−𝛿_𝑦)⟩=⟨𝑇_𝑡𝛿_𝑥,𝑇_𝑡𝛿_𝑥⟩+⟨𝑇_𝑡𝛿_𝑦,𝑇_𝑡𝛿_𝑦⟩−2⟨𝑇_𝑡𝛿_𝑥,𝑇_𝑡𝛿_𝑦⟩. (3.1) Here,𝑇_𝑡𝛿_𝑧(𝑣)=_𝑀𝜌_𝑡(𝑣,𝑢)𝛿_𝑧(𝑢)𝑑𝑢=𝜌_𝑡(𝑣,𝑧), (3.2) for 𝑧∈𝑀. The ⟨⋯,⋯⟩ is the usual inner product on 𝐿²(𝑀). (In [1], the authors consider a weighted version of (3.1) which naturally arises when the underlying kernel does not integrate to 1 (in each variable). In terms of data analysis, this corresponds to cases where the data are sampled nonuniformly over the region of interest. For simplicity, we are just using Coifman's unweighted distance.)

Note that we thus have[𝑑_𝐶,𝑡(𝑥,𝑦)]²=_𝑀𝜌_𝑡(𝑣,𝑥)𝜌_𝑡(𝑣,𝑥)𝑑𝑣+_𝑀𝜌_𝑡(𝑣,𝑦)𝜌_𝑡(𝑣,𝑦)𝑑𝑣−2_𝑀𝜌_𝑡(𝑣,𝑥)𝜌_𝑡(𝑣,𝑦)𝑑𝑣=‖𝜌_𝑡(⋅,𝑥)‖²₂+‖𝜌_𝑡(⋅,𝑦)‖²₂−2⟨𝜌_𝑡(⋅,𝑥),𝜌_𝑡(⋅,𝑦)⟩. (3.3)

Although Coifman's original definition used a kernel symmetric with respect to the space variable, 𝑑_𝐶,𝑡(𝑥,𝑦) as given above need not be based on a symmetric 𝜌_𝑡. Note that, by the defining (3.1), 𝑑_𝐶,𝑡(𝑥,𝑦) is symmetric in 𝑥 and 𝑦 (even if 𝜌_𝑡 is not), and satisfies the triangle inequality. If 𝜌_𝑡 is symmetric in the space variables, from (2.6) we see that:[𝑑_𝐶,𝑡(𝑥,𝑦)]²=𝜌_2𝑡(𝑥,𝑥)+𝜌_2𝑡(𝑦,𝑦)−2𝜌_2𝑡(𝑥,𝑦), (3.4) a form matching one of Coifman's formulations for the continuous case.

If, in addition to 𝜌_𝑡 being symmetric in the space variables, we have that (2.5) holds, as in the case of heat flow, we easily see that:[𝑑_𝐶,𝑡(𝑥,𝑦)]²=^∞_𝑗=1𝑒^{−2𝜆𝑗𝑡}(𝜙_𝑗(𝑥)−𝜙_𝑗(𝑦))², (3.5) the original form proposed by Coifman. Note that the latter expression again explicitly shows that 𝑑_𝐶,𝑡(𝑥,𝑦) is symmetric in 𝑥 and 𝑦 and satisfies the triangle inequality (by considering, for example, the right-hand side as the square of a weighted distance in 𝑙²).

Referring again to Section 2, for the discrete situation, where we start with a set of data points 𝑆={𝑠₁,𝑠₂,…,𝑠_𝑁}, and 𝐴is a Markov matrix specifying the transition probabilities between the “states" of 𝑆, the distance between two data points 𝑠_𝑖 and 𝑠_𝑗 is given by[𝑑_𝐶,1(𝑠_𝑖,𝑠_𝑗)]²≡⟨𝐴^𝑇(𝑒_𝑖−𝑒_𝑗),𝐴^𝑇(𝑒_𝑖−𝑒_𝑗)⟩=⟨𝐴^𝑇𝑒_𝑖,𝐴^𝑇𝑒_𝑖⟩+⟨𝐴^𝑇𝑒_𝑗,𝐴^𝑇𝑒_𝑗⟩−2⟨𝐴^𝑇𝑒_𝑖,𝐴^𝑇𝑒_𝑗⟩=⟨𝐴𝐴^𝑇𝑒_𝑖,𝑒_𝑖⟩+⟨𝐴𝐴^𝑇𝑒_𝑗,𝑒_𝑗⟩−2⟨𝐴𝐴^𝑇𝑒_𝑖,𝑒_𝑗⟩=(𝐴𝐴^𝑇)_𝑖𝑖+(𝐴𝐴^𝑇)_𝑗𝑗−2(𝐴𝐴^𝑇)_𝑖𝑗, (3.6) where ⟨⋯,⋯⟩ is the usual inner product in ℝ^𝑁, and for a matrix 𝐵, (𝐵)_𝑖𝑗 denotes the 𝑖,𝑗 entry of 𝐵. Again, symmetry and the triangle inequality are easily verified. If 𝐴 is symmetric,[𝑑_𝐶,1(𝑠_𝑖,𝑠_𝑗)]²=(𝐴²)_𝑖𝑖+(𝐴²)_𝑗𝑗−2(𝐴²)_𝑖𝑗. (3.7) The “1" appearing in the subscript of 𝑑_𝐶,1(𝑠_𝑖,𝑠_𝑗) refers to the fact that 𝐴¹=𝐴 is used, corresponding to 𝑡=1 in the continuous case. As the diffusion along data points flows, after 𝑛 ticks of the clock, we can successively consider[𝑑_𝐶,𝑛(𝑠_𝑖,𝑠_𝑗)]²=(𝐴^𝑛(𝐴^𝑇)^𝑛)_𝑖𝑖+(𝐴^𝑛(𝐴^𝑇)^𝑛)_𝑗𝑗−2(𝐴^𝑛(𝐴^𝑇)^𝑛)_𝑖𝑗, (3.8) which, for a symmetric 𝐴, equals(𝐴^2𝑛)_𝑖𝑖+(𝐴^2𝑛)_𝑗𝑗−2(𝐴^2𝑛)_𝑖𝑗. (3.9)

An important benefit of introducing a diffusion distance as above can be illustrated by considering (3.5). If 𝜌_𝑡 is such that (3.5) holds for a complete orthonormal family {𝜙_𝑗}, we see that as 𝑡 increases, we are achieving an (approximate) isometric embedding of 𝑀 into successively lower-dimensional vector spaces (with a weighted norm). More specifically, for 𝜆_𝑗>0, if 𝑡 is large, the terms 𝑒^{−2𝜆𝑗𝑡}(𝜙_𝑗(𝑥)−𝜙_𝑗(𝑦))² are nearly 0. So, as 𝑡 increases, we see that the “heat smeared" manifold 𝑀 is parametrized by only a few leading 𝜙_𝑗's. Thus, “stepping" through higher and higher times, we are obtaining a natural near-parametrization of more and more smeared versions of 𝑀, giving rise to a natural ladder of approximations to 𝑀.

Analogous considerations hold in the discrete situation for 𝐴 symmetric, when we easily see that the eigenvalues of 𝐴² are between 0 and 1 and decrease exponentially for 𝐴^2𝑛, as 𝑛 increases (the “heat smeared" data points are now parametrized by a few leading eigenvectors of 𝐴, associated to the largest eigenvalues).

See [1–3] for more discussion and examples of the natural embedding discussed above, along with illustrations of its power to organize unordered data, as well as its insensitivity to noise.

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 4.

Let us consider (3.4) for the case where𝜌_𝑡(𝑥,𝑦)=(4𝜋𝑡)^−𝑛/2𝑒^{−|𝑥−𝑦|2/4𝑡}, (3.10) the fundamental solution to the heat equation in ℝ^𝑛. Then,[𝑑_𝐶,𝑡(𝑥,𝑦)]²=2−2𝑒^{−|𝑥−𝑦|2/8𝑡}(8𝜋𝑡)^𝑛/2. (3.11) If |𝑥−𝑦|²/8𝑡 is small, then to the leading order in |𝑥−𝑦|²/4𝑡,[𝑑_𝐶,𝑡(𝑥,𝑦)]²=1(8𝜋𝑡)^𝑛/2(|𝑥−𝑦|²4𝑡+𝒪((|𝑥−𝑦|²)4𝑡²)). (3.12) Thus, if |𝑥−𝑦|≪√𝑡, we do recover the geodesic distance between 𝑥 and 𝑦 but, due to the 1/𝑡^𝑛/2 term in front, normalized by a power of 𝑡 which depends on the dimension 𝑛. As pointed out by the reviewer, for ℝ^𝑛 itself, the normalization does depend on 𝑛, but is simply a global change of scale, for each 𝑡, and thus basically immaterial. Suppose, however, that the data we are considering come in two “clumps", one of dimension 𝑛 and the other of dimension 𝑚, with 𝑛≠𝑚. Let us also suppose these clumps are somehow joined together and, far away from the joining region, each clump is basically a flat Euclidean space of the corresponding dimension. Then, far away from the joint, heat diffusion in a particular clump would behave as if it were in ℝ^𝑛, respectively ℝ^𝑚 (until the time that the flowing heat “hits" the joint region). Thus, in the part of each clump that is far from the joint, the diffusion distance would be normalized differently, one normalization depending on 𝑛 and the other on 𝑚. An overall change of scale would not remove this difference, thus we would not recover the usual Euclidean distance in the two clumps simultaneously, as we would like.

The second point of concern is more general in nature. In the continuous case, Coifman's distance involves the 𝐿² distance between 𝑇_𝑡𝛿_𝑧, when 𝑧=𝑥, and 𝑇_𝑡𝛿_𝑧 when 𝑧=𝑦; see (3.1). The 𝐿¹ norm of 𝑇_𝑡𝛿_𝑧 is 1, since_𝑀𝑇_𝑡𝛿_𝑧(𝑣)𝑑𝑣=_𝑀(_𝑀𝜌_𝑡(𝑣,𝑢)𝛿_𝑧(𝑢)𝑑𝑢)𝑑𝑣=1, (3.13) using the mass preservation assumption of (2.2). For the discrete case, 𝟏^𝑇𝐴^𝑇𝑒_𝑖=1, where 𝟏^𝑇 is the 1×𝑁 vector of 1's.

So the diffusion distance proposed by Coifman finds the 𝐿² (resp., 𝑙²) distance between 𝐿¹ (resp., 𝑙¹) normalized vectors. Let us illustrate by an example for the discrete situation, with 𝑁=10,000, in which this may lead to undesired results. Without specifying the matrix 𝐴, suppose that after some time has passed, we have the following two 1×10,000 vectors giving two different results of diffusion:𝑣₁1=(1100,…,100,0,…,0), (3.14) where the first one hundred entries are each 1/100, and the rest 9,900 entries are 0, and𝑣₂1=(,110,000110,000,…,10,000), (3.15) where each entry is 1/10,000.

Note that 𝑣₁ and 𝑣₂ both have 𝑙¹ norm 1. Now, considering two canonical basis vectors 𝑒^𝑇_𝑖 and 𝑒^𝑇_𝑗, 𝑖≠𝑗, each of which has 𝑙¹ norm 1, we see that ⟨𝑒^𝑇_𝑖−𝑒^𝑇_𝑗,𝑒^𝑇_𝑖−𝑒^𝑇_𝑗⟩=2. So, a distance of √2 gives the (in fact, maximum) separation between two completely different (𝑙¹ unit) diffusion vectors. Return to 𝑣₁ and 𝑣₂, note that 𝑣₂ corresponds to total diffusion, while 𝑣₁ has only diffused over 1% of the entries. We would thus hope that 𝑣₁ and 𝑣₂ would be nearly as much separated as 𝑒^𝑇_𝑖 and 𝑒^𝑇_𝑗, that is, have diffusion distance not much smaller than √2. But a trivial calculation shows that⟨𝑣₁−𝑣₂,𝑣₁−𝑣₂⟩<.1, (3.16) which seems much smaller than what we would like. The problem is that ⟨𝑣₁−𝑣₂,𝑣₁−𝑣₂⟩ is small since the 𝑙² norm of each of 𝑣₁ and 𝑣₂ is small, even though the 𝑙¹ norm of each is 1.

In the next section, we propose a variant of the diffusion distance discussed in this section. Our version will find the 𝐿² (resp., 𝑙²) distance between vectors which are normalized to have 𝐿² (resp., 𝑙²) norm to be 1, rather than 𝐿¹ (or 𝑙¹) norm 1.

4. An Alternate Diffusion Distance

In this section, we propose a new diffusion distance. Let us first define our alternate diffusion distance for the continuous case. Refer to Section 2 for the definitions of functions and operators used below.

For any 𝑧∈𝑀, let𝜓_𝑧𝛿(𝑢)≡^𝑧(𝑢)‖𝑇_𝑡𝛿_𝑧‖₂=𝛿_𝑧(𝑢)∫_𝑀𝜌_𝑡(𝑣,𝑧)𝜌_𝑡(𝑣,𝑧)𝑑𝑣. (4.1) Then,𝑇_𝑡𝜓_𝑧(𝑢)=_𝑀𝜌_𝑡𝛿(𝑢,𝑤)^𝑧(𝑤)‖𝑇_𝑡𝛿_𝑧‖₂𝜌𝑑𝑤=^𝑡(𝑢,𝑧)‖𝑇_𝑡𝛿_𝑧‖₂. (4.2) Note that 𝑇_𝑡𝜓_𝑧(⋅) has 𝐿² norm 1:_𝑀[𝑇_𝑡𝜓_𝑧(𝑢)]²𝑑𝑢=_𝑀𝜌²_𝑡(𝑢,𝑧)∫_𝑀𝜌²_𝑡(𝑣,𝑧)𝑑𝑣𝑑𝑢=1. (4.3)

For 𝑥,𝑦∈𝑀, we define our diffusion distance, 𝑑_2,𝑡(𝑥,𝑦), as follows:[𝑑_2,𝑡(𝑥,𝑦)]²≡⟨𝑇_𝑡(𝜓_𝑥−𝜓_𝑦),𝑇_𝑡(𝜓_𝑥−𝜓_𝑦∫)⟩=2−2^𝑀𝜌_𝑡(𝑢,𝑥)𝜌_𝑡(𝑢,𝑦)𝑑𝑢‖𝜌_𝑡(⋅,𝑥)‖₂‖𝜌_𝑡(⋅,𝑦)‖₂=2−2⟨𝜌_𝑡(⋅,𝑥),𝜌_𝑡(⋅,𝑦)⟩‖𝜌_𝑡(⋅,𝑥)‖₂‖𝜌_𝑡(⋅,𝑦)‖₂, (4.4) where we have used (4.3). Here again, ⟨⋯,⋯⟩ is the usual inner product on 𝐿²(𝑀). Note the analogy to (3.3).

As is clear from the defining equality in (4.4), 𝑑_2,𝑡(𝑥,𝑦) is symmetric in 𝑥 and 𝑦 and satisfies the triangle inequality:𝑑_2,𝑡(𝑥,𝑧)≤𝑑_2,𝑡(𝑥,𝑦)+𝑑_2,𝑡(𝑦,𝑧), (4.5) for all 𝑥,𝑦,𝑧∈𝑀. Geometrically, 𝑑_2,𝑡(𝑥,𝑦) is the length of the chord joining the tips of the unit vectors 𝑇_𝑡𝜓_𝑥 and 𝑇_𝑡𝜓_𝑦. We have that0≤𝑑_2,𝑡(𝑥,𝑦)≤√2, (4.6) for all 𝑥,𝑦∈𝑀 and 𝑡≥0.

If 𝜌_𝑡 is symmetric in the space variables, by (2.6), we have that[𝑑_2,𝑡(𝑥,𝑦)]²𝜌=2−2^2𝑡(𝑥,𝑦)𝜌_2𝑡(𝑥,𝑥)𝜌_2𝑡(𝑦,𝑦). (4.7)

As an example, again consider the case where 𝜌_𝑡(𝑥,𝑦)=(4𝜋𝑡)^−𝑛/2𝑒^{−|𝑥−𝑦|2/4𝑡}, the fundamental solution to the heat equation in ℝ^𝑛. Then,[𝑑_2,𝑡(𝑥,𝑦)]²=2−2𝑒^{−|𝑥−𝑦|2/8𝑡}. (4.8) Note that if |𝑥−𝑦|≪√𝑡, then[𝑑_2,𝑡(𝑥,𝑦)]²≈2−2(1−|𝑥−𝑦|²8𝑡)=|𝑥−𝑦|²4𝑡, (4.9) so 𝑑_2,𝑡(𝑥,𝑦) gives (approximately) the geodesic distance in ℝ^𝑛, in the “near regime" where |𝑥−𝑦|≪√𝑡, and with scale √𝑡. Note that unlike (3.12), no 𝑡^𝑛/2 term appears. (Also see the discussion following (3.12).) Also note that if |𝑥−𝑦|²/8𝑡 is large, 𝑑_2,𝑡(𝑥,𝑦)≈√2 (the greatest possible distance, see (4.6)), so for such 𝑡 the points 𝑥 and 𝑦 are (nearly) maximally separated. Hence, 𝑑_2,𝑡(𝑥,𝑦), for the case of heat flow in ℝ^𝑛, gives a scaled geodesic distance when 𝑥 is close to 𝑦, with 2√𝑡 as the unit of length and near maximum separation when 𝑥 is far from 𝑦 at the scale √𝑡.

For any, say, compact Riemannian manifold 𝑀, if 𝜌_𝑡(𝑥,𝑦) is the fundamental solution to the heat equation on 𝑀, we have that𝜌_𝑡(𝑥,𝑦)=(4𝜋𝑡)^−𝑛/2𝑒^{−𝑑2(𝑥,𝑦)/4𝑡}(1+𝒪(𝑡)),as𝑡→0⁺, (4.10) where 𝑑(𝑥,𝑦) is the geodesic distance on 𝑀 (see [9]). Hence, repeating the expansion in (4.9) for a compact manifold 𝑀, with 𝑡 small, and 𝑑(𝑥,𝑦)≪√𝑡, we have that 𝑑_2,𝑡(𝑥,𝑦)≈𝑑(𝑥,𝑦)/2√𝑡, again recovering (scaled) geodesic distance. (The discussion following (3.12) gives an example for which it would be preferable not to have presented a normalization factor which depends on the dimension.) Exponentially decaying bounds on the fundamental solution of the heat equation for a manifold 𝑀 (see [9, Chapter XII, Section 12]), suggest that 𝑥 and 𝑦 become nearly maximally separated, as given by 𝑑_2,𝑡(𝑥,𝑦), when 𝑑(𝑥,𝑦) (scaled by √𝑡) is large, just as in the Euclidean case.

In the discrete situation, where we start with a set of data points 𝑆={𝑠₁,𝑠₂,…,𝑠_𝑁}, and 𝐴is a Markov matrix specifying the transition probabilities between the “states" of 𝑆, for 𝑛=1,2,…, we let𝑣_𝑖,𝑛=𝑒_𝑖‖(𝐴^𝑇)^𝑛𝑒_𝑖‖, (4.11) where 𝑒_𝑖 is the 𝑖th canonical basis vector (see Section 2), and ‖⋯,⋯‖ is the 𝑙² vector norm. For 𝑠_𝑖,𝑠_𝑗∈𝑆 and 𝑛=1,2,…, we define 𝑑_2,𝑛(𝑠_𝑖,𝑠_𝑗) by[𝑑_2,𝑛(𝑠_𝑖,𝑠_𝑗)]²≡⟨(𝐴^𝑇)^𝑛(𝑣_𝑖,𝑛−𝑣_𝑗,𝑛),(𝐴^𝑇)^𝑛(𝑣_𝑖,𝑛−𝑣_𝑗,𝑛)⟩=2−2⟨(𝐴^𝑇)^𝑛𝑒_𝑖,(𝐴^𝑇)^𝑛𝑒_𝑗⟩‖(𝐴^𝑇)^𝑛𝑒_𝑖‖‖(𝐴^𝑇)^𝑛𝑒_𝑗‖=2−2(𝐴^𝑛(𝐴^𝑇)^𝑛)_𝑖𝑗(𝐴^𝑛(𝐴^𝑇)^𝑛)_𝑖𝑖(𝐴^𝑛(𝐴^𝑇)^𝑛)_𝑗𝑗, (4.12) where ⟨⋯,⋯⟩ and ‖⋯,⋯‖ are, respectively, the usual inner product and norm in ℝ^𝑁, and, for a matrix 𝐵, (𝐵)_𝑖𝑗 denotes the 𝑖,𝑗 entry of 𝐵.

If 𝐴 is symmetric,[𝑑_2,𝑛(𝑠_𝑖,𝑠_𝑗)]²=2−2(𝐴^2𝑛)_𝑖𝑗(𝐴^2𝑛)_𝑖𝑖(𝐴^2𝑛)_𝑗𝑗. (4.13) As before, 𝑛 represents the 𝑛th tick of the clock.

5. The Mean First Passage Cost Satisfies the Triangle Inequality: An Example of Its Normalized Limit

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 [8], the author states this result when all costs are equal to 1 as Theorem 4.2, but the proof is not very explicit in our opinion).

In [10–12], as well as some of the references listed therein, it is shown that the symmetrized mean first passage time and cost are metrics (for mean first passage cost see, in particular, [10]; also, in the above sources the symmetrized mean first passage time is called the commute time). “Symmetrized" refers to the sum of the first cost (time) to reach a specified state from a starting state and to return back to the starting state. This symmetrization is necessary to ensure a quantity symmetric in the starting and destination states. In the sources cited above, the fundamental underlying operator is the graph Laplacian 𝐿, which, using the notation of [12], is defined as 𝐿=𝐷−𝑊. Here, 𝑊=⟨𝑤_𝑖𝑗⟩ is the adjacency matrix of a graph, and 𝐷 is the diagonal degree matrix, with the 𝑖th entry on the diagonal equaling ∑_𝑗𝑤_𝑖𝑗. In addition to assuming the nonnegativity of the 𝑤_𝑖𝑗's, the authors in the above works assume that 𝑊 is symmetric. The resulting symmetry (and positive semi-definiteness of 𝐿) implies the existence of a full set of nonnegative eigenvalues of 𝐿, and the diagonalizability of 𝐿 is used heavily in the proofs that the commute time/cost is a distance. In the random walk interpretation, see, for example, [12], the following normalized Laplacian is relevant: 𝐿_𝑟𝑤=𝐼−𝐷⁻¹𝑊. To make a connection with the notation in the present paper, 𝐷⁻¹𝑊=𝐴, a Markov matrix giving the transition probabilities of the random walk. Although 𝐷⁻¹𝑊 is not necessarily symmetric, it is easy to see that 𝐷⁻¹𝑊=𝐷^−1/2{𝐷^−1/2𝑊𝐷^−1/2}𝐷^1/2 (see the discussion in [12]). Hence 𝐷⁻¹𝑊, while not itself symmetric in general, is conjugate to the symmetric matrix 𝐷^−1/2𝑊𝐷^−1/2, and thus too has a full complement of eigenvalues.

In this section, as in the rest of the paper unless stated otherwise, we are not assuming that the Markov matrix 𝐴 is symmetric or conjugate to a symmetric matrix; hence 𝐴 may not be diagonalizable (i.e., 𝐴 may have Jordan blocks of dimension greater than 1). We thus do not have spectral theory available to us. Furthermore, we do not wish to necessarily symmetrize the mean first passage time/cost to obtain a symmetric quantity; we are not actually going to get a distance, but will try to obtain the “most important" property of being a distance, namely, the triangle inequality.

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 𝐴, with the 𝑛th power of 𝐴 giving the transition probabilities after 𝑛 ticks of the clock. If there is no overall wind, the matrix 𝐴 could reasonably be assumed to be symmetric, and we could use spectral theory. But suppose there is an overall wind in some fixed direction, which is making it more probable for the storm to move north, say, rather than south. Then the matrix 𝐴 is not symmetric; there is a preferred direction of the storm to move in, from one tick of the clock to the next; spectral theory cannot, in general, be used. Furthermore, it may not be reasonable in this situation to consider the commute time—the symmetrized mean first passage time—since we may rather want to know the expected time to reach a certain population center from the current location of the storm, and may not care about the storm's return to the original location. Thus the mean first passage time would be the quantity of interest.

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 𝐴, does indeed satisfy the triangle inequality; our proofs do not rely on spectral theory. We think that satisfying the triangle inequality, while in general failing to be symmetric, is still a very useful property for a bilinear form to have.

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, 𝑆={𝑠₁,𝑠₂,…,𝑠_𝑁} is a finite set of states and 𝐴 is a Markov matrix giving the transition probabilities between states in one tick of the clock (see Section 2). 𝐶 will denote an 𝑁×𝑁 matrix with non-negative entries, 𝐶=⟨𝑐_𝑖𝑗⟩^𝑁_𝑖,𝑗=1. We will think of each 𝑐_𝑖,𝑗 as the “cost" associated with the transition from state 𝑠_𝑖 to state 𝑠_𝑗. By a slight abuse of notation, for 1≤𝑚,𝑛≤𝑁, 𝑃_𝑛 will be the 𝑁×𝑁 matrix in which all entries are 0, except the (𝑛,𝑛) entry which is 1 (this corresponds to 𝑋={𝑠_𝑛} in Section 2). Also, 𝑃_𝑚𝑛 will be the 𝑁×𝑁 matrix in which all entries are 0, except the (𝑚,𝑚) and (𝑛,𝑛) entries each of which is 1 (this corresponds to 𝑋={𝑠_𝑚,𝑠_𝑛} in Section 2).

Let 𝑌_𝑚𝑛 be the random variable which gives the cost accumulated by a particle starting at state 𝑠_𝑚 until its first visit to state 𝑠_𝑛 after leaving 𝑠_𝑚. In other words, if a particular path of the particle is given by the states 𝑠_𝑚,𝑠_𝑗1,𝑠_𝑗2,…,𝑠_𝑗𝑝,𝑠_𝑛, the value of 𝑌_𝑚𝑛 is 𝑐_𝑚𝑗1+𝑐_𝑗1𝑗2+⋯+𝑐_𝑗𝑝𝑛. We suppose 𝐴 has the property that for every 𝑖,𝑗, there exists an 𝑛 such that (𝐴^𝑛)_𝑖𝑗>0, that is, every state 𝑗 is eventually reachable from every state 𝑖. Then, as is shown in [13] (using slightly different notation), we have the following formula for 𝔼(𝑌_𝑚𝑛), which is the expected cost of going from state 𝑠_𝑚 to state 𝑠_𝑛:𝔼(𝑌_𝑚𝑛)=𝑒^𝑇_𝑚𝐼𝐼−𝐴(𝐼−𝑃_𝑛)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏, (5.1) where ⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩ is the 𝑁×𝑁 matrix with 𝑝,𝑞 entry equal to 𝑐_𝑝𝑞𝑎_𝑝𝑞. (In particular, it is shown in [13] that 𝐼−𝐴(𝐼−𝑃_𝑛) is invertible and ‖(𝐴(𝐼−𝑃_𝑛))^𝑘‖→0, as 𝑘→∞.) See [14, 15] for discussion of related expected values, and [8, 10–12, 16–18] for discussion of mean first passage times and related concepts.

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

Proposition 5.1. 𝔼(𝑌_𝑖𝑗)≤𝔼(𝑌_𝑖𝑘)+𝔼(𝑌_𝑘𝑗).

We again note that this proposition, for the case all costs are 1, is stated as Theorem 4.2 in [8], but we feel the proof is not very explicit. (In our proofs below, we assume 𝑗≠𝑘; if 𝑗=𝑘, the inequality in Proposition 5.1 is immediate.)

Proof. (1) Our first proof is probabilistic. Let a random walker start at state 𝑠_𝑖 and accumulate costs given by the matrix 𝐶 as he moves from state to state. As soon as the walker reaches state 𝑠_𝑗, we obtain a sample value of 𝑌_𝑖𝑗. Now, at this point of the walk, there are two possibilities. Either the walker has passed through state 𝑠_𝑘 before his first visit to 𝑠_𝑗 after leaving 𝑠_𝑖, or he has not. In the first instance, we have obtained sample values of 𝑌_𝑖𝑘 and 𝑌_𝑘𝑗 along the way, and 𝑌_𝑖𝑗=𝑌_𝑖𝑘+𝑌_𝑘𝑗 for this simulation. In the second case, we let the walker continue until he first reaches 𝑠_𝑘, to obtain a sample value of 𝑌_𝑖𝑘, and walk still more until he reaches 𝑠_𝑗 for the first time since leaving 𝑠_𝑘, thus giving a sample value of 𝑌_𝑘𝑗 (note that by the memoryless property, this sample value of 𝑌_𝑘𝑗 is independent of the walker's prior history). In the second case, we thus clearly have that 𝑌_𝑖𝑗<𝑌_𝑖𝑘+𝑌_𝑘𝑗. Combining the two cases, we have 𝑌_𝑖𝑗≤𝑌_𝑖𝑘+𝑌_𝑘𝑗. Repeating the simulation, averaging, and taking the limit as the number of simulations goes to infinity, we obtain that 𝔼(𝑌_𝑖𝑗)≤𝔼(𝑌_𝑖𝑘)+𝔼(𝑌_𝑘𝑗).
(2) Our second proof is via explicit matrix computations. Let us define the following two quantities:𝑄₁≡𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞𝑄⟩𝟏,₂≡{𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)𝐴𝑃_𝑘𝟏}𝑒^𝑇_𝑘𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏. (5.2) (See Section 2 and the paragraphs before the statement of Proposition 5.1.) Now, we have 𝑄₁≡𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝑒^𝑇_𝑖^∞_𝑚=0[𝐴(𝐼−𝑃_𝑗𝑘)]^𝑚⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏≤𝑒^𝑇_𝑖^∞_𝑚=0[𝐴(𝐼−𝑃_𝑘)]^𝑚⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑘)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝔼(𝑌_𝑖𝑘); (5.3) see (5.1). Also, 𝑄₂≡{𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)𝐴𝑃_𝑘𝟏}𝑒^𝑇_𝑘𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏={𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)𝐴𝑃_𝑘𝟏}𝔼(𝑌_𝑘𝑗). (5.4) But 𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)𝐴𝑃_𝑘𝟏≤𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑘)𝐴𝑃_𝑘𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑘)(𝐼−𝐴(𝐼−𝑃_𝑘))𝟏=1, (5.5) where we have used a series expansion to show the first inequality (all entries are non-negative), and the fact that 𝐴𝑃_𝑘𝟏=(𝐼−𝐴(𝐼−𝑃_𝑘))𝟏, since 𝐴𝟏=𝟏. Thus, 𝑄₂≤𝔼(𝑌_𝑘𝑗).
We will finish our second proof by showing that 𝑄₁+𝑄₂=𝔼(𝑌_𝑖𝑗). First note that𝑄₂≡{𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)𝐴𝑃_𝑘𝟏}𝑒^𝑇_𝑘𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘)𝐴𝑃_𝑘𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏, (5.6) using that 𝑃_𝑘=𝑒_𝑘𝑒^𝑇_𝑘=𝑃_𝑘𝟏𝑒^𝑇_𝑘. Thus, 𝑄₁+𝑄₂=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘){𝐼−𝐴(𝐼−𝑃_𝑗)+𝐴𝑃_𝑘}𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘){𝐼−𝐴(𝐼−𝑃_𝑗−𝑃_𝑘𝐼)}𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗𝑘){𝐼−𝐴(𝐼−𝑃_𝑗𝑘𝐼)}𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏=𝔼(𝑌_𝑖𝑗). (5.7) Here we have used the fact that 𝑃_𝑗+𝑃_𝑘=𝑃_𝑗𝑘 (as mentioned earlier, we are assuming 𝑗≠𝑘; the triangle inequality we are proving holds trivially for the case 𝑗=𝑘).

We would like to point out that the decomposition of 𝔼(𝑌_𝑖𝑗)=𝑄₁+𝑄₂ in the second proof above is not a “miraculous" guess. We arrived at this decomposition by writing 𝔼(𝑌_𝑖𝑗) as the derivative (evaluated at 0) of the characteristic function (Fourier transform) of 𝑌_𝑖𝑗 (see [13]), and breaking up the expression to be differentiated into a sum of terms: one term corresponding to the random walk going from 𝑠_𝑖 to 𝑠_𝑗 without visiting 𝑠_𝑘 first, and one term corresponding to visiting 𝑠_𝑘 before reaching 𝑠_𝑗. After differentiation, the resulting six pieces, when suitably combined into two terms, yielded 𝑄₁ and 𝑄₂.

We conclude this section by considering certain (suitably normalized) limiting values of the expected cost of going from state 𝑠_𝑖 to state 𝑠_𝑗, for the first time after leaving 𝑠_𝑖, given by (5.1). For this discussion, we will take all the costs to be identically 1, that is, 𝑐_𝑝𝑞=1 for all 𝑝,𝑞. Then, we see from (5.1) that𝔼(𝑌_𝑖𝑗)=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝐴𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝐴(𝐼−𝑃_𝑗)𝟏+𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝐴𝑃_𝑗𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝐴(𝐼−𝑃_𝑗)𝟏+𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)(𝐼−𝐴(𝐼−𝑃_𝑗))𝟏=𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝐴(𝐼−𝑃_𝑗)𝟏+1, (5.8) where we have used 𝐴𝑃_𝑗𝟏=(𝐼−𝐴(𝐼−𝑃_𝑗))𝟏, since 𝐴𝟏=𝟏.

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 9 in [19]. Some statements are excerpted verbatim from that work, with minor changes in some labels. The background results below are well known and are often referred to as Dynkin's formula (see, e.g, [20]). We are presenting them for the reader's convenience and will use them to exhibit an interesting connection between the mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem; see (5.16).

Let 𝐷 be a domain in ℝ^𝑛, and let 𝐿 denote a partial differential operator on 𝐶²(ℝ^𝑛) of the form:𝐿=^𝑛_𝑖=1𝑏_𝑖𝜕(𝑥)𝜕𝑥_𝑖+^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜕(𝑥)²𝜕𝑥_𝑖𝜕𝑥_𝑗, (5.9) where 𝑎_𝑖𝑗(𝑥)=𝑎_𝑗𝑖(𝑥). We assume each 𝑎_𝑖𝑗(𝑥)∈𝐶²(𝐷) is bounded and has bounded first and second partial derivatives; also, each 𝑏_𝑖(𝑥) is Lipschitz. Suppose 𝐿 is uniformly elliptic in 𝐷 (i.e., all the eigenvalues of the symmetric matrix ⟨𝑎_𝑖𝑗(𝑥)⟩ are positive and stay uniformly away from 0 in 𝐷). Then, for 𝑔∈𝐶^𝛼(𝐷), some 𝛼>0, and 𝑔 bounded, and for 𝜙∈𝐶(𝜕𝐷), the function 𝑤 defined below solves the following Dirichlet-Poisson problem:𝐿𝑤=−𝑔in𝐷,lim_{𝑥→𝑦,𝑥∈𝐷}𝑤(𝑥)=𝜙(𝑦),forallregularpoints𝑦∈𝜕𝐷. (5.10) (Regular points in this context are defined in [19] and turn out to be the same as the regular points in the classical sense, i.e., the points 𝑦 on 𝜕𝐷 where the limit of the generalized Perron-Wiener-Brelot solution coincides with 𝜙(𝑦), for all 𝜙∈𝐶(𝜕𝐷).)

Now we define 𝑤. We choose a square root 𝜎(𝑥)∈ℝ^𝑛×𝑛 of the matrix 2⟨𝑎_𝑖𝑗(𝑥)⟩, that is,12𝜎(𝑥)𝜎^𝑇(𝑥)=⟨𝑎_𝑖𝑗(𝑥)⟩. (5.11) Next, for 𝑏=⟨𝑏_𝑖(𝑥)⟩, let 𝑋_𝑡 be an Itô diffusion solving𝑑𝑋_𝑡=𝑏(𝑋_𝑡)𝑑𝑡+𝜎(𝑋_𝑡)𝑑𝐵_𝑡, (5.12) where 𝐵_𝑡 is n-dimensional Brownian motion. Then,𝑤(𝑥)=𝔼_𝑥[𝜙(𝑋_𝜏)]+𝔼_𝑥[^𝜏₀𝑔(𝑋_𝑢)𝑑𝑢],for𝑥∈𝐷, (5.13) is a solution of (5.10). Here, the expected values are over paths starting from 𝑥∈𝐷, and 𝜏 is the first exit time from 𝐷.

Let us transfer the above discussion to, say, a compact manifold 𝑀, rather than ℝ^𝑛. We sample 𝑀 and let the “states" 𝑠_𝑘 be the sample points. We construct a transition matrix 𝐴 to give a discretized version of (5.12). Let 𝜖>0 be the approximate separation between the sample points. Fix a sample point 𝑠_𝑗, and let 𝐷=𝑀⧵𝐵_𝜖(𝑠_𝑗) be the domain in 𝑀 consisting of the complement of the closure of the ball of radius 𝜖 in 𝑀, center 𝑠_𝑗. Let 𝑠_𝑖 be a sample point in 𝐷. For this situation, in (5.10), let 𝜙 be the 0 function, and 𝑔 be the constant 1 function. Then (5.13) becomes:𝑤(𝑥)=𝔼_𝑠𝑖[^𝜏₀𝑑𝑢], (5.14)𝜏 is the first exit time from 𝐷, that is, first visit time to the 𝜖 neighborhood of 𝑠_𝑗. (Compare with Proposition 8B in [21] which discusses the case of the Dirichlet-Poisson problem (5.10) with 𝜙=0 and 𝑔=1 for a manifold.) As shown in [13] (with slightly different notation), a discrete version of (5.14) is(𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩𝟏)Δ𝑡, (5.15) where 𝑐_𝑝𝑗=0, all 𝑝, and 𝑐_𝑝𝑞=1, for all 𝑞 not equal to 𝑗, and all 𝑝. Thus, ⟨𝑐_𝑝𝑞𝑎_𝑝𝑞⟩=𝐴(𝐼−𝑃_𝑗). Combining (5.15) and (5.8), we see that:𝔼(𝑌_𝑖𝑗)Δ𝑡−Δ𝑡=(𝑒^𝑇_𝑖𝐼𝐼−𝐴(𝐼−𝑃_𝑗)𝐴(𝐼−𝑃_𝑗)𝟏)Δ𝑡≈𝑤(𝑠_𝑗), (5.16) for Δ𝑡 small.

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: 𝑀=𝑆¹, the unit circle. We will consider 𝑑²/𝑑𝜃², the Laplacian on 𝑆¹, and sample 𝑆¹ uniformly. We will let the transition matrix 𝐴 take the walker from the current state to each of the two immediate neighbor states, with probability 1/2 for each. The variance is then (Δ𝜃)². Since 𝑑𝑋_𝑡=√2𝑑𝐵_𝑡, see (5.12), we must have (Δ𝜃)²=2Δ𝑡, and we should use (Δ𝜃)²/2 as our value of Δ𝑡 in (5.16). Using symmetry, we can take 𝑠_𝑗=0, the 0 angle on 𝑆¹, without loss of generality. Let𝑤_𝜖1(𝜃)=2(𝜃−𝜖)((2𝜋−𝜃)−𝜖),𝜖≤𝜃≤2𝜋−𝜖. (5.17) Note that𝑑²𝑑𝜃²𝑤_𝜖𝑤(𝜃)=−1,𝜖<𝜃<2𝜋−𝜖,_𝜖(𝜖)=𝑤_𝜖(2𝜋−𝜖)=0. (5.18) So 𝑤_𝜖 is the unique solution satisfying (5.10) for our example, on the domain 𝑆¹⧵[−𝜖,𝜖].

To numerically confirm (5.16), we ran numerical experiments in which we discretized 𝑆¹ into 𝑁 equispaced points, with the transition matrix 𝐴 taking a state to each of its 2 immediate neighbors with probability 1/2, and used (Δ𝜃)²/2 as the value of Δ𝑡 in (5.16) to calculate 𝔼(𝑌_𝑖𝑗)Δ𝑡−Δ𝑡. We took 𝑠_𝑗 to be the angle 0, and 𝑠_𝑖 to be the closest sample point to the angle with radian measure 1, for example. Letting 𝜖→0 in (5.17), we compared the value of 𝔼(𝑌_𝑖𝑗)Δ𝑡−Δ𝑡 with 𝑤(𝜃)=(1/2𝜃)(2𝜋−𝜃). For instance, with 𝑁=1000, 𝑠_𝑗=0, and 𝑠_𝑖 the nearest sample point to the angle with radian measure 1, the relative error is less than 0.08%. Note that 𝑤(𝜃) is, for 𝜃 close to 0, essentially a scaled geodesic distance on 𝑆¹ (from our base angle 0).

6. Conclusions

The authors have presented a diffusion distance which uses 𝐿² unit vectors, and which is based on the well-known cosine similarity. They have discussed why the normalization may make diffusion distances more meaningful. We also gave two explicit proofs of the triangle inequality for mean first passage cost, and exhibited a connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem.

Acknowledgments

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.