Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs
Authors: Raif M. Rustamov
Subhabrata Majumdar
Venue: NeurIPS 2022 Submitted (2022)

Abstract: Collections of probability distributions arise in a variety of statistical applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions are represented by histograms over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of histograms over such general domains. We propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the histogram collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures, one based on resampling and another on combining $p$-values from coordinate-wise tests. Our experiments in a variety of data settings show that the resulting tests are powerful and the $p$-values are well-calibrated. Example applications to user activity patterns and spatial data are provided.

Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs
Anonymous Author(s) Affiliation Address email
Abstract
Collections of probability distributions arise in a variety of statistical applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions are represented by histograms over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections histograms over such general domains. We propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances Hilbert embeddable, allowing to reduce the histogram collection comparison problem to more familiar mean testing problem a Hilbert space. We provide testing procedures, based on resampling and another on combining p-values from coordinate- tests. Our experiments in a variety of data settings show that the resulting tests are powerful and the p-values are well-calibrated. Example applications to user activity patterns and spatial data are provided.
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Introduction
15 Distributional data arise in a variety of statistical applications. In practice these are not limited to
16 distributions over real intervals, but are often defined over manifolds and graphs. For instance, even in the simplest case of analyzing 24-hour activity patterns by constructing histograms of activity counts by time, the resulting histograms are really supported real line. If in addition to the time of activity, the observations come with real number such as the intensity of the activity, then we end up with a histogram over a cylindrical domain. Spatial datasets recorded at some geographic region level are another example: one can build a distribution over the region adjacency graph by capturing the normalized counts of events each region. When analyzing distributions over such general domains it is desirable to rely methods that take into account the connectivity and geometry of the underlying domain, respect the distributional nature of the data, and lead to efficient practical algorithms.
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circle rather than an interval on the
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26 In this paper we consider the problem of comparing two collections of distributions, namely testing for meanoggneiiy-whethee all of the distributions come from the same mneaadisttibbtion While 28 conceptually similar two-sample testing, this higher order notion in the sense that our units
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29 of analysis are distribution//histograas Letting (X'' denote the set of Borel probability measures
30 on metric space X, consider the space P(P(X)). Let P,Q P(P(X)). and assume that we are
31 given two collections of probability measures { and [u,],, that are drawn from P and Q, M4 P and P4 Q in independent and identical manner. Our goal is to test whether P Q and, 33 morcover, to be able to conduct such tests for general X. Such test would be useful numerous practical situations. For instance, an online retailer may aggregate a customer's monthly activity
Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

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Pushiorward
est
osta
Prelfoowwrd
Figure I: Schematic of the proposed intrinsic slicing construction. Given two probability measures on the sphere (here the darkest blue corresponds zero mass), different aspects their dissimilarities become apparent after pushforward to the real line using the eigenfunctions of the Laplace-Beltrami Beltrami operator, { this case spherical harmonics.
particular example of our general construction, the (squared) intrinsic sliced 2- 2-Wasserstein distance (SW is the weighted sum of the dissimi- larities of the corresponding pushforwards of μ and v as measured by squared 2- 2-Wasserstein distance W2 (+,))
the real line.
35 into histogram over cylinder capturing the time of the day and amount of purchase for each transaction. By considering collections of histograms for various customer segments, one can conduct tests to determine if there are statistically significant differences between behavioral patterns of these segments.
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We attack this problem using insights from recent developments that utilize Hilbert embeddings for simplifying distributional data problems (see e.g. [18, 24] for particular examples). The simplification comes as result of linearity of Hilbert spaces, which allows adapting existing statistical approaches such functional data methodology distributional data. A crucial requirement on the embedding is that the distance in the embedding space should give meaningful distance between measures; it is this property that renders quantities computed in the embedding space such means and variances meaningful. Thus, embedding constructions should be driven by specifying appropriate distances on space of measures. Of course, not every distance can embedded and Hilbert embeddable distances are called Hilbertian; see [19] for an overview of this notion.
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The focus in this paper will on transportation based distances between distributionshiistogaaas [25]. Other approaches such as bin-wise treatment of histograms may result in increased variability
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when horizontal variation is present, leading eventually less powerful methods. Transportation
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based distances are more efficient at capturing this and other aspects of distributional data [3, 17,
52 19]. However, adopting the transportation theoretic approaches our problem immediately hits the real line is Hilbert 54 embeddable, fails be so on general domains [19]. The general Hilbert embedding framework of [18] is not tied to distance between probability distributions and so can be problematic for capturing
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roadblock beyond the real line case: while 2- Wasserstein distance
56 the location and variability aspects of distribution collections. addition, [18] has difficulties in higher dimensions and does not provide constructions suitable for manifolds or graphs.
Inspired by the sliced 2-sssserstein distances high dimensional spaces [11, 12], we introduce new slicing construction (Figure 1) that leverages the cigenvalues and eigenfunctions of the Kaplace-Beltrame operator on manifolds and graph Laplacians capture the intrinsic geometry and connectivity of the domain. We apply this slicing construction to obtain novel class of intrinsic sliced 62 2- Wasserstein distances on manifolds and graphs. The resulting distances are Hilbert embeddable.
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63 have a number of desirable properties, and can be truncated to obtain finite miie-dimenniioal embeddings.
64 Using the corresponding embedding allows us to reduce the histogram collection comparison problem
to the comparison of means in nigh-dimensional Euclidean space. We provide two approaches
86 for hypothesis testing and verify via extensive experiments on synthetic and real data examples in variety of data settings that these tests are powerful, and the paluuues are well-calibrated.

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Comparing with closely related work, while the Generalized Sliced Wasserstein (GSW) distance [11] sets up the idea of approximating Wasserstein distances using multiple nonlinear projections, is presented in extrinsic terms (i.e. Euclidean space) and can suffer from the curse of dimensionality when low dimensional data manifold lives in aigh--imensional space. Our choice of eigenfunctions for projection is very different from the one one-parameter function families in GSW. Moreover, the GSW construction does not directly apply to graphs. While the tree-sliced variant of GSW [14] can be applied in an intrinsic manner (the clustering variant), it relies on a different type of distance, in the limit related to the euclidean/geodesic distance. This can be seen by comparing our lower bound to
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theirs: our lower bound for ISW is in terms of the MMD using the spectral distance (Proposition 5). Finally, the robust sliced Wasserstein distance of [13] does make use of the geometric properties of the underlying manifold. However, their goal is compute a correspondence between two manifolds
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minimizing some version of Fuclidean slicing.
Preliminaries
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Given compact metric space X. let ((2) denote the set of Borel probability measures X. Our main interest is in the case where X' is a graph or a manifold with the shortest/geodesi distance as 84 the metrie, and thus the compactness restriction. The 2-Wasserstein distance can be defined on P(X) using the metric of X' as the ground distance [17, 19]. giving W P(x))= x P(X) + R20: due
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86 to the repeated use of the real line case we use the shorthand W2 = W Central to our study are
distributions on the space of probability measures P()(X)) = ((((()), where ((P(X))
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is the Borel o-algebra generated by the topology induced by WA [3]. To avoid confusion, we will
refer to the elements of ((P(X)) as meta pneta-distributions.
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Let P,Q& E P(P(X)), and assume that we are given two collections of probability measures {us ) and {2,), that are drawn from P and Q: 14; P and in ~ Q in independent-ande -ntt-anndddddialllyy distributed (hereafter i.i.d.) manner. Our goal is to use this sample to test the null hypothesis of whether P Q. While this is conceptually a two-sample test, note that our data points are distributions; in practice, the distributions μ4 or i4 are given by histograms.
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Remark 1. Let compare this with the usual two-sample testing. Consider P E P(P(2)) con- structed as follows. Let n= E P(X) be a fixed probability measure. Let Iι, 22 II. u and construct the histogram summarizing this sample: 1 Σm 6z,- Now, A Ed 6z. E P(2) is one sample drawn from P. In our testing scenario one gets the collection { where each histogram is obtained as above: 14; ~ P. Similarly, consider Q E P(P(X)) of the same type based on some i= the corresponding collection of histograms. Testing whether P = Q in the limit boils down to 2 = v², When compared to the usual two-sample testing this may seem rather inefficient, requiring A times more samples (resp. N1 and A2A samples from μ²' and
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other fixed v² E P(X), and let [
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V*) However, in our setup it is not assumed that the histograms in the collections come from meta- distributions of the above simple type (i.e. all μ4; are generated by drawing from the same underlying distribution μ²). I In fact, the target use-case for our approach is when these histograms are collected by observing different individuals who have their person-specifie behavior/dtistriuttion..
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107 Let D(-.)) P(X) x P(x) + Ez0 be a distance between probability distributions. D(-.-) is called Hilbertian (this is just naming convention; no implication that the map is Hilbert map) if there exist a Hilbert space and a map y) P(X)) - H such that D(u, v) = [[л(()) - 17(x)) ( For example, it is well-known that 2- 2-Wasserstein distance on X' = IR is Hilbertian [19] (also see Section 3.2) and Maximum Mean Discrepancy (MMD) on any X is Hilbertian [8]; however, the 2- Wasserstein distance W on general X' is not Hilbertian [19].
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Since the map 1) takes every measure on X' to a point in H. we see that process P P(P(X)) gives a rise to a measure on H. given by pushforward operation, n P Pon E P(H). In addition, if a finite dimensional approximation ni P(X)) - IR D of n is available, then #D is a measure
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on RR, This observation is enormously useful: problems about the elements of the rather abstract
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space P(P())) are reduced to problems about familiar measures on H or even RR For example,
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the usual notions of mean and variance can be applied to the measure t P to gain insights about the
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meta-distribution P. The validity of these insights hinges on the map coming from Hilbertian distance, as distances are central to the statistical quantities of interest.
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Testing for T#P n#Q can serve as a proxy for our original testing problem of = = Q. As typical with two-sample tests, various aspects of the equality n = can be tested, such as the mean or variance equality; unspecific tests of equality can be applied as well. We will concentrate on testing certain aspects of the equality so that one can easily drill down on the results. This is similar to the regular two-sample testing where checking for equality of, say, means is often preferrable as it gives immediately interpretable insights, whereas a general test that only says there are unspecified differences between the distributions is less useful for interpretation. To obtain succint and interpretable tests we concentrate on the mean of the resulting pushforward measure in H.
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Definition 1. For a meta-distribution / E (X)(X)). we define its Hilbert centroid with respect to the Hilbertian distance D as CR# = [[[[a] E H. assuming it exists.
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Our testing procedure is based checking the equality Cr#p C or more explicitly: En-P [n()]] = En- Eu>e( Intuitively, each "dimension" of the map probes some aspect of the two involved dett--iisributions and makes sure that they are in agreement in expectation. One of our testing approaches will use the statistic
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Q(P.Q) |Cm#p Ca#Q like
(2.1)
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to capture the deviations from equality; this quantity can be written directly in terms of pairwise distances.
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Proposition 1. For P.Q E ((P(X))) the following holds:
- E P n')l (
T(P.Q) Q) Eu (n, m
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Next we give an example of what Hilbert centroid equality means in an important special case.
Example 1. Let X [0. T] C R with DD being the 2- Wassersseei distance W/. Given a probability
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measure μ E ([,,T) let Fj. be its cumulative distribution function: (x) - u=(0,, x/) = -o du.
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The generalized inverse of cumulative distribution function (CDF) is defined by (s) := inf(ar e
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[0, T] : F2 (z:) s). The squared 2-Wasserstein distance has rather simple expression in terms of
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the inverse CDF [19]:
W2 = { (F)
(2.2)
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This formula immediately establishes the Hilbertianity of W2 through the map 1) : P((0, 7)) Le (10, TT) defined by T7(1) = F=-.. Note that 17 is invertible for increasing normalized functions in the embedding space. Using this insight, we see that the corresponding "average measure" of Pe E P(P(X)) can be introduced via Pav = - (En-pl(())) It is easy to prove that Pav satisfies the following: Pav = arg pi) (M2(,, ())]] which is the definition of the Fréchet mean, see for example [19]. In this setting, C Cn#QQ boils down to having the same Fréchet means,
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Pa Qav-
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We will later see that the Hilbert embedding corresponding to the intrinsic sliced 2- Wasserstein distance is assembled of embeddings like in Example applied after pushforwards (see Figure 1 for an intuition). This means that the resulting equality Cr#t Cy##Q becomes more stringent, making
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it a better proxy for detecting the deviations from P = Q without losing the interpretability aspect.
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3 Intrinsic Sliced 2- Wasssersteinn Distance
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We introduce a Hilbertian version of W2 on manifolds and graphs via a construction we call intrinsic slicing due to its use of the domain's 's intrinsic geometric properties. To focus our discussion concentrate on the manifold case, as the graph case is simpler and is obtained by replacing the Laplace-Beltran Beltrami operator by the graph Laplacian.
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Let de, Φz; 0, 1.... be the eigenvalues and eigenfunctions of the Laplace- Beltrami operator on X' with Neumann boundary conditions. The eigenfunctions are sorted by increasing eignevalue and assumed to be orthonormal with respect to some fixed (e.g. uniform) measure on X; also and the corresponding spectral distance on the manifold d(x, yy a) + k(y. y) 2k(2,, y) =
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on const and X O. One can define the spectral kemne k(r, =
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[a()))(o(()= ?, where a Rxo + R2o is a function that controls contribution from
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each spectral band. By setting ((x) = e- tA for some > O, we get the heat/diffusion kernel and the corresponding diffusion distance [4]. Another important case a(A) 1/A² if X 0 and
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(D) = 0, which gives the biharmonic kernel and distance [15]. In both of these constructions a(-) decreasing function, allowing the smoother low-frequency (i.e. smaller Ae) eigenfunctions to contribute more.
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3.1 Definition and properties
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A real-valued function Φ : X + R can be used to map the manifold X' onto the real line. Any probability measure P(X) can likewise be projected onto the real line using the pushforward of o, which we denote by odu = o 6-1 E P(R). While the pushforward notions used here and in previous sections are conceptually the same, for clarity use for measures and # for meta-distributions. We define intrinsic slicing as follows.
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Definition 2. Given a function a R>00 + R20 and a probability distance D(:s-) on P(R), we define the intrinsic sliced distance LSD(-,) -) on P(X) by
ISD = Σ ((Nx)DD dxtu).
The choice of the Laplacian cigenfunctions in the definition can be justified by number of their properties. Eigenfunctions are intrinsic quantities of manifold and are ordered by smoothness. Thus, they allow capturing the intrinsic connectivity of the underlying domain. Furthermore, due to the orthogonality of eigenfunctions, their pushforwards can capture complementary aspects.
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While the definition is general, our focus in this paper is on the case when D W2; we remind that we always use W2 to denote the 2-Wasserstein distance on P(R). We call the resulting distance Intrinsic Sliced Wasserstein Distance, and denote it by ISW2. First, we discuss the convergence of the infinite sum in Definition 2.
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Proposition 2. If X is smooth compact n-dimensional manifold and Ee ( 1)/2 1)/12////)) a(Ae) < co. then ISW2 is well-defined.
(n-1)/2
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Next, we prove number of properties of ISD.
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Proposition 3. Df is a Hilbertian probability distance such that ISD is well-defined, then (i) ISD Hilbertian, and (ii) ISD satisfies the following metric properties: non-negativity, symmetry, the triangle inequality, and ISD(u, p.) =0.
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Proof. By Hilbertian property of 'D, there exists Hilbert space H and a map 1/ P(R) + 7 such that P2) (nn) 17 (p2) ( for all Pi. E (R). Plugging this into Definition 2 we have ISD(u,,) = [[(()) - n(()]],.. where H H and the b-th component of 17())
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is (a(X))no(retu) H. The second part of Proposition 3 directly follows from the Hilbert property.
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Since W2 is Hilbertian on P(TR), the application of Proposition 3 yields that T.W is also Hilberitan, making it possible to 7SW22 for our hypothesis tests in Section 4.
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The following result shows that ISW2 inherits an important property of the Wasserstein distances, namely that the distance between two Dirac delta measures equals to specific ground distance between their locations.
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Proposition 4. When μ μ= 8-),v -5(() 6-(-) for two points x. E X, we have ISW2(H, = d(x,)) y). where d(.,.) -) is the spectral distance corresponding to the choice of al-).
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For a simple choice of distance D on P(R)), namely the absolute mean difference, the corresponding intrinsic sliced distance is the well-known MMD [8].
[En=pp [x] - Ey--pz for ISD is equivalent to the MMD with the spectral kernel k(.,)).
Proposition 5. Let D(p,, p2) =
for pl, E P(R), then the corresponding
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When k(z, 3) is the heat kernel, the sliced distance in Proposition 5 is very much like the MMD with the Gaussian kernel, with the parameter t in a a(A) e -tà controlling the kernel width. Indeed,
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the two kernels coincide on R and on general manifolds Varadhan's ' formula gives asymptotic equivalence for small [2].
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An interesting insight derived from the above result that TSW2 is in a sense a "stronger" distance than MMD that uses the corresponding spectral kemell. The TSW2 compares the quantiles of
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pushforward distributions (Eq. (2.2)), whereas MMD compares their expectations only. We formalize
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this notion next, also providing a theoretical reason for preferring 1SW2 for hypothesis testing.
Proposition 6. MM v) s 1SW2(,, when the same a(-) is used in both constructions.
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Proof. This follows directly from the fact that for PL, P2 E P(R) the inequality [Exppp [z]] By-pu oa will < W2(P1, P2) holds.
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We are now a position to prove that 1SW2 is a true metric.
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Theorem 1. (f) > 0 for all > 0, then ISW2 is a metric on ((()).
We remind that 2- Wassessstein distance can be defined directly P(X) using the geodesic distance
as the ground metrie; we denote this distance as W Lipschitz properties of the eigenfunctions imply the following:
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Proposition? 7. There exists a constant depending only on X C R" such for all p, E P(X)
the inequality ISW2(H,)) S eW2 (H, E. A (n+31/2 (<xe) holds.
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Our final result looks at the quantity T defined using ISW2 by Eq. (2.1). We will be using T computed on finite collections of measures as test statistic in the next section. We show that it enjoys robustness with respect to small perturbations of the measures the collection.
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Proposition 8. Let { and { be two collections of probability measures on P(X), such
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that vi, W (ni, 14:) S e, then T( {vr ) S C²e.. Here C' = C=cyE
E. X (m+3)/2 a(Ae) from
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previous proposition and is assumed to be finit.
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This bound implies that the distributions in a collection undergo horizontal shifts that are small as measured by the geodesic Wasserstein distance W then T is small well.
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3.2
Approximate Hilbert Embedding
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An important aspect of ISWa that its Hilbert map 17 P(X) H. can be approximated by embedding пnd PP(') + RDD such that TSW2(1, v) = [l (D()) This is useful for practical computation and for one of our hypothesis testing approaches.
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Using the formula for ISW2 on P(R) in terms of the quantile function, Eq. (2.2), the Hilbert map is defined by (n) F-1 We have W2(u, v) [1n7(()) (v) where the norm involves integration. We can discretize the integral using the Riemann sum for equidistant knots
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1-1 1,11, 1,... D', define the approximate embedding nb. P(R) + R2 as:
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Dr
VD :4
FF- (81), (sD:).
(3.1)
Now, (2, ~ Inn (v) with approximation quality depending on the embedding dimension D'
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To approximate the Hilbert map for ISW, we truncate the series defining ISW, and use a finite number of eigenfunctions for pushforward: de, ( 1, 1,..., I, where de is dropped since it is constant. By inspecting the proof of Proposition and using Eq. (3.1), we can define Ni : P(x') - RD with
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D LD' the concatenation of L maps:
(mp)e : a(he)
[F Sett (s1), F (o.).( (SDr)]
D'
Spectral decompositions of the Beltrami operators for general manifolds [4, 20] or graph Laplacians can be computed numerically. For applications that involve simple manifolds, the cigenvalues and eigenfunctions can be computed analytically (see Appendix).
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Hypothesis Testing
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Let ( and 1
No be two i.i.d collections of measures drawn from P.Q E P(P(X)) respectively. Our goal is use these samples to test the null hypothesis Hu CD#P C+#Q. where is the Hilbert embedding of the sliced distance ISW2 on P(X).
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4.1
Resampling Based Test
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We use the quantity T(;-) from Eq. (2.1) as the test statistic. Its sample version is computed by replacing the expectations by the empirical means, and excluding the diagonal terms to achieve unbiasedness
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(1))
E (SW3(ve,)) vi) 1.j 2N - 1)
ISW N1
T
Σ ド:
+
2N
Note that ET = T(P.Q) In practice, the 15W2 values are computed from the approximate embedding: ISW2(PI, P2) x [| - (p2)))l We denote the resulting statistic by TT,D.
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The difference between T TL.D' and the population version (i.e. T - -.. can be decomposed as TT T)- + (T- (T TI,"',, where the summands inside the terms TL and TL correspond to partial sums that approximate ISW3 (:--) by E (( ( and (oi Φit)) by I ( T) ( respectively. We show in Appendix that a) summands in the second and third terms in the sum can be made infinitesmally small by choosing large enough L and D', respectively; b) an asymptotic result for the first difference can be obtained by extending the tools from [8. 23]. These results are based on several assumptions detailed in the Appendix. Combining the two results, we establish asymptotic distributions of TL.D<
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Theorem 2. Assume relevant conditions (see Appendix) hold. Define N N₁ + N2, and suppose
that N1. N2 co, have N1/N - Pi,N N2/N P2 = Pt. Pirfor some fixed O P1 < 1.With
272 L 22 LN, D' 2 Dx chosen in an appropriate way (see Appendix), under Ho Cn#p = C we
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have
NT
Im 7m(A2 (A - 1), Σ m=
where Am N NO, 1) for m 1,2,.., ., and 7/m are the eigenvalues of certain operator that depends on P and Q. Further, under H, Oupp + Ca#Q /N (TI.D- T) is asymptotically Gaussian with mean 0 and finite variance.
-
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contiguous alternatives H {(P, Q): Cu#p Cu#Q + ŠN, 1, .).. where the deviation from null is quantified collectively by pushforward differences deN E ÓN Del @e(varden) ) that are made to approach 0 as N - oo. The following theorem establishes consistency of our testing procedure against a family of such local alternatives.
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Theorem 3. Assume conditions (i)-tiii) hold, and let L, D' be chosen as in Theorem Then for the sequence of contiguous alternatives HAN such that N N11ōx li - co, the test based on TT LT is consistent for any a E (0, 1), that is as N -> oc the asymptotic power approaches I.
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Testing Procedure In practice, to obtain the p-value for the -ttttiiiiiii we use a bootstrap procedure. Remember that TT.D is computed via the approximate embedding yn with D = LD'. The collection {p, [ is mapped to the collection {X, (1,) N,,, of vectors in I drawn in an i.d manner from 1 P P nn E P(RP). Similarly, for the other collection we have a sample ( ())()) drawn from di #Q Now, the null Ho Cu#p = Cn#qQ implies that the means
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of the distributions # P and d # coincide in R.D.
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The bootstrap null distribution for TI.D' can be obtained as follows. Let X and Y be the sample means; construct the combined sample (X,-X+ X+ X+Y
2 1.4 U(Y-UU----+ X+y Nr This centers both 2
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samples
X+Y, Now, from the combined sample select with replacement N1 (resp. N₂) samples
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294 to make bootstrap sample { J==1 (resp. (Y)))) Repeat this process B times (we take B = 1000
in our experiments), and collect the null test statistic values T L, TT. ( for
296 b .... B. The approximate p-value is then given by: p B+1 (11b: Th.0 2 +
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4.2
Testing via p-value Combination
The bootstrap test above incurs a high computational cost and the granularity of the p-values is determined by the number of resamples, which can be too coarse in massive multiple comparison settings often seen in industrial applications. Thus, propose approach that avoids resampling.
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As explained above, testing Ho Cr#P = CT#Q can be interpreted as testing whether the means of the distributions TD#PP and nD# coincide in RPP To this end, we adopt the approach pro- posed by [22] in spatial statistics context. First, we apply the Sthrens-FFiserr Welch t-test (with- out assuming equality of variances) to each coordinate of the samples {X, = (D()) and
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(Y (D)) [== to obtain the p-values Pł, k = 1, ..., D. Second, overall p-value is com- puted via the harmonic mean p-value combination method which is robust to dependencies [6, 26]:
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where the function H has a known form described in [26].
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Another approach for combining p- p-values is the Cauchy combination test [16], but in our numerical experiments we found that the Cauchy combination approach encounters problems when any of the
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p-values is very close to 1, which can happen in our setting due to the form of the embedding yp.
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Therefore, in contrast to [22], for us the harmonic combination is the only appropriate choice.
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To guarantee size control, we establish a version of Theorem 1 from [16] for the harmonic mean p-value. Assume that test statistic Z RDD has null distribution with zero mean and every pair of coordinates of Z follows bivariate Gaussian distribution. Compute the coordinate-wis two-sided
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p-values Pk 2(1 ())))) where & is the standard Gaussian CDF.
Theorem 4. Lel pk: ..., D be the null p-values as above and pd computed via harmonic mean 1,...
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approach, then
Prob(p" S Sa) 1
lim o
a
318 In the Appendix we show that this theorem applies in our setting, so the proposed procedure asymptotically controls the size of the test for small a. Our experimental results show that the control is already achieved for moderate sample sizes and the commonly used a = 0.05.
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5 Experiments
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Synthetic Experiments We compare the performance of our tests on distributions over finite interval, circle, and cylinder with existing methods, and settings of the embedding parameters L, D'. For evaluation, we use empirical power at different degrees of departure from the null hypothesis (captured by 8); further details can be found in the Appendix. The summary results presented in Figure show that all methods maintain nominal size (power at 5 = O is close to a). On the inite interval: 1) from Figure (a) our combination test (ISD comb) outperformed all the other tests, but the bootstrap test (ISD boot) performs worse than others except Fmaxb; 2) from Figure 2 (b) combination test with sliced I,S/2 improves over the unsliced version-withh more eigenfunctions, the power first improves considerably, then become similar to the unsliced version. For circular domain, Figure (c) shows that our tests maintain considerably higher power than existing methods for all 8. Figure (d) shows that our combination test maintains nominal size on cylindrical domain.
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NHANES data on physical activity monitoring This data [10] contains physical activity pattern readings for 6839 individuals. Data for each individual corresponds to activity monitor intensity values for 7 days. Since the time dimension periodic, we get person-specife probability distributions
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over the cylinder S (T:) x T2). We check if activity patterns vary across age groups The
337
p- value combination test results are shown below the diagonal in Table 1. Our method detects
statistically significant differences between all pairs of groups, except the 36-45 and 46-55 groups.
339
As expected, the control p-values-obaann
by mixing samples between two age groups and splitting arbitrarily-do do not concentrate near zero. More details are in the Appendix.
8

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Meboc
Mered
Neteed
Nibed
1.40
(a)
(b)
(c)
Figure 2: Performance on synthetic finite interval and manifold data. Finite interval: (a) comparison with existing methods a test based on basis function representation (FP) [7], a sum-type lz norm- based test (L2b) [27], and max-type test [28] that uses the maximum of coordinate- wise F statistic (Fmaxb); (b) unsliced vs. different settings of (L, D"). Manifold data: (c) circular data, comparing with Fréchet ANOVA [5], and the DISCO nonparametric test [21]; (d) harmonic combination tests on cylindrical data for I, Dotted lines indicates nominal size of tests (a = 0.05).
Crime Type
Tue vs Thu Tue Sat
Ages
6-15
16-25
26-35
36-45
46-55
56-65
66-75
76-85
Theft
0,428
4.2e-06
6-15
0.394
0.098
0.555
0.882
0.985
0.919
0.997
Decept Pract
0.313
0.001
16-25 1.2e-13
0.575
0.967
0.126
0.921
0.911
0.977
Battery
0.430
0.001
26-35 3.1e-21 2.7c-04
0.459
0.197
0.996
0.919
0.565
Robhery
0.119
0.003
36-45 6.1e-22 7.9e-08
0.042
0.864
0.637
0.849
0.991
Narcotics
0,854
0.004
46-55 8.2e-22 4.7e-05
0.011
0.343
0.841
0.165
0.554
Criminal Dam
0.855
0.02
65-65166 1.3e-25 0.001
0.001
5,6e-05
0.003
0.991
0.962
Other Offense
0.931
0.052
66-75 3.6e-35 7.8e-12 1.5e-11
4.6e-15
1.8e-13
0.001
0.989
Burglary
0.142
0.261
76-85 3.8e-46 1.4e-26 1.7e-30 8.4e-37 2.1e:35 1.3e-17
6.5e-09
Assault
0.997
0.38
Mot Veh Theft
0.858
0.416
Table I: Activity intensity comparison across age groups in the NHANES data. Below diagonal: p-values for the actual data comparisons. Above diagonal: null p-values obtained by combining and randomly splitting the two groups. Bold entries correspond to rejected hypotheses with the BH procedure at FDR level 0.1.
Table 2: Chicago Crime analy- sis p-values. Bold entries cor- respond to rejected hypotheses with the BH procedure at FDR level 0.1.
Chicago Crime We use the Chicago Crimes 2018 dataset [1] to demonstrate the use of our methodology on histograms over graphs. Each beat (geographic area subdivision used by police) corresponds to a vertex, and two vertices are connected by an edge if the corresponding beats share a geographic boundary. For each crime type and day, the normalized counts of that crime type for each beat gives daily probability distribution over the graph. Our goal is compare the collection of distributions of, say, theft occurring on Tuesday those of Thursday and Saturday. The Tuesday versus Thursday comparison is intended null case, we do not expect to any differences between them [22]. We detect statistically significant differences between Tuesday and Saturday patterns for six categories of crime, and expected, differences between Tuesday and Thursday patterns. See Appendix for more details and for another graph based application in the context of Brain Connectomics.
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Conclusion
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The construction of ISWa provides novel embedding of probability distributions into Hilbert space. This can be used to adapt many inferential methods to general spaces where the existence of Fréchet means or higher moments are not guaranteed. The ISW2 can also be useful for machine
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learning applications where prediction targets live in general domain. Given that rigorous Fréchet baand methodology for such problems has only been proposed recently [9], development of prediction models for manifold-valus data that are free of restrictive assumptions an attractive
358
future line of research.

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3-17, 2019.
Checklist
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For all authors...
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(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [N/A] This is fairly upstream methodological work.
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]

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(b) Did you include complete proofs of all theoretical results? [Yes]
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3. If you ran experiments..
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [Yes]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
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(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [No] Evaluation metrics are proportions, so they are already computed using multiple replications.
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minimal enough to not warrant this consideration, i.e.
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experiments finish within minutes to hours.
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12


Reviews:
[{"review_id": "zLkPrSbYwlB", "reviewer": "Reviewer_LPx3", "summary": "The authors propose the intrinsic sliced Wasserstein (ISW) distances for collections of probability measures. The ISW is a variant of sliced Wasserstein (SW) which exploits the closed-form solution of optimal transport on 1-dimensional space. The authors propose to use the eigenfunctions to project supports into 1-dimensional space and use corresponding eigenvalues as its weights.\n\nISW is Hilbert embeddable which allows to reformulate testing problems over collections of probability measures to traditional ones in Hilbert space. The authors demonstrate it for hypothesis testing via resampling based test and testing by p-value combination (from coordinate-wise tests).\n", "questions": "+ The proposed ISW is a variant of sliced-Wasserstein (SW) by leveraging the eigenfunctions and eigenvalues (closely related to SW, GSW, TSW as in line 68-80; and the recent Sobolev transport – a variant of optimal transport for measures supported on graphs). The authors should compare them in experiments.\n+ The motivation of using eigenfunctions (for slicing) is too vague (the authors only give a short discussion in line 179-182). I think the authors should describe its advantage rigorously in theory and/or experiments.\n\nSome other concerns are as follow:\n+ What is the relation between T in Equ. (2.1) and in Proposition 1?\n+ For Proposition 6, is it hold for any kernel used in MMD? Please specify it.\n+ In Proposition 8, why can one assume that C is finite? (it is not clear whether the bound is useful in Proposition 8 without this assumption)\n+ For line 266-269, how to choose L, D’ to obtain infinitesimally small error estimation (e.g., to obtain error less than epsilon)?\n+ For experiments, the authors should discuss the results, e.g., how to choose L, D’ (in Figure 2b) follow the results in Section 3.2 and Section 4?   \n\nMinor:\n+ In line 193-194, \\eta^0 maps to H^0, but D is a norm on H’?\n", "limitations": "+ The authors have not discussed the limitations of their works clearly in the text. (as mentioned in the checklist).\n+ The authors have not discussed the potential negative societal impacts of their work as in the checklist.\n", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n+ The authors propose a novel intrinsic sliced Wasserstein (ISW) for collections of probability measures by leveraging eigenfunctions for 1-dimensional projection and eigenvalues as weights.\n+ The paper is easy to follow.\n+ The authors derive hypothesis testing in the Hilbert for collections of probability measures problems under ISW.\n\nWeaknesses\n+ The ideas to use eigenfunctions and eigenvalues for sliced Wasserstein are interesting. However, the authors have not demonstrated the advantages of the proposed approach over other approaches (e.g., sliced-Wasserstein, general sliced-Wasserstein, tree-sliced-Wasserstein or the recent Sobolev transport – a variant of optimal transport on graphs. Note that all these approaches are Hilbert embeddable as the proposed method) in theory and/or experiments.\n+ The authors should discuss the complexity of the proposed intrinsic sliced Wasserstein (and report time consumption in experiments)\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "k0LU8-7EpHb", "reviewer": "Reviewer_tW4F", "summary": "Consider the set of probability measures on some manifold. The authors propose a new distance on this set, that behaves in some way like the sliced Wasserstein distance. This distance is defined as the weighted average of the Wasserstein distance between the pushforwards of the two measures by the eigenfunctions of the Laplace-Beltrami operator on the manifold. Some properties of this distance are given, and the distance is then used in the two-sample testing problem, where each sample is a sample of iid probability measures on a given manifold. Some numerical illustrations of the two-sample tests are then given.", "questions": "On the implementation:\nIn a lot of situations, the measures at stake will be atomic, of the form $\\mu_i= \\sum a_i \\delta_{x_i}$. Then, it looks like the most efficient way to compute the ISW distance is to compute $\\phi_k(x_i)$ for every eigenfunction $\\phi_k$, and then compute the 1D matching between two 1D atomic distributions (that is we do the same as for the classical sliced Wasserstein distance). Why is this approach not at all proposed in the paper? \n\nOn the numerical illustrations:\nThe authors test their procedures against other reasonable testing procedures, but not against the one that appears to be the most standard in their situation, that would be a two-sample test based on the Wasserstein distance. (that is we reject if $W_2(\\mu_n,\\nu_n)>\\alpha$, and we accept otherwise). CLTs for empirical Wasserstein distances could probably be used to obtain asymptotic confidence levels on such a test, or one could use bootstrap or other approaches. Is there a reason for which this was not implemented (at least for synthetic experiments where the number of observations is reasonably small, so that W2 should be easy to compute)?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Sliced Wasserstein (SW) distances have been originally proposed as computationally cheap versions of the Wasserstein distance on the Euclidean space $\\mathbb{R}^D$. The idea of the original SW is the following: instead of computing the Wasserstein distance between two measures in high dimension, let us look at a certain number of \"projections\" of the measures on lines in $\\mathbb{R}^D$; then one can compute efficiently a distance between the two measures by averaging the distance between the projections.  This paper proposes a very reasonable generalization of this idea, called the Intrinsic SW (ISW) when the support of the measure is a manifold or a graph: as lines are no longer defined, the distance is defined as an average of \"projections\" along the eigenvalues of the Laplace-Beltrami operator, that should describe the geometry of the manifold (and therefore replace the lines in $\\mathbb{R}^D$).\n\nThe paper focuses on the notion of ''Hilbert embeddability'' to justify the use of this distance. This means that their distance $ISW(\\mu,\\nu)$ is actually equal to $\\|\\eta(\\mu)-\\eta(\\nu)\\|_H $ for some feature map $\\eta$ and some Hilbert space $H$. However, the authors seemed to have miss a key link with kernel methods. Indeed, the feature map $\\eta$ defines a kernel on the space $P(X)$ of probability measures on $X$ (through $k(\\mu,\\nu) = <\\eta(\\mu),\\eta(\\nu)>_H$), and the method they propose for the two-sample tests is the kernel two-sample test method from Gretton & al. The authors seemed to have understand partially the connection, as their proofs are adaptations of the ones from Gretton & al, but the authors have missed that they could actually directly used the results from Gretton & al, without any adaptation. \n\nThere are other concepts defined in the paper that can be rephrased in the more standard language of kernels:\n- a Hilbert centroid (def. 1) is a kernel mean embedding. \n- T(P,Q) (eq 2.1) is actually the MMD associated with the kernel on P(X) given by the intrinsic sliced Wasserstein distance.\n- Most proofs corresponding to Section 4.1 can be removed: it suffices to check that the assumptions given in Gretton & al are satisfied for the kernel given by the intrinsic SW distance.\n\nOther comments:\n- The intrinsic SW distance depends on the choice of a weight function (describing the importance given to each eigenvector). Having to tune such an important hyperparameter is clearly a drawback of this distance compared to the Sliced Wasserstein distance.\n- Notation in Prop.9 (in the Appendix) are extra confusing: when defining the eigenvectors of some operator (that are defined on some Hilbert space H), the variable \"x\" is used, whereas x was used before to note an element of the underlying manifold. As we also consider eigenvectors on the manifold, this created a lot of confusion at first.\n\nOn my grade: I put 4 so far, as I believe the link with kernel methods is not made very clear in the paper, whereas it is central to the approach proposed. I could improve my grade if this link is made more explicit (e.g. say explicitly that the test IS a kernel two-sample test and that $T(P,Q)$ IS a MMD), and if proofs are significantly shortened by using Gretton & al.'s theorems directly. Please tell me if I am missing something and if this test is not a kernel two-sample test.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "WgLlzOUjFf", "reviewer": "Reviewer_ib9Y", "summary": "It is well-known that the Wasserstein distance between two 1D measures can be computed easily. Hence, a common approach in a higher dimensional Euclidean space $\\mathbb{R}^d$ consists in _slicing_ the space (formally, projecting on 1D subspaces), compute the distances on the slices, and integrating to retrieve a proper distance between the input measures. \n\nThis work proposes to adapt this idea in the context of measures supported on a (compact) manifold $\\mathcal{M}$. The key idea is to rely on the Laplacian-Beltrami operator on the manifold, which provides a collection of maps $\\phi_\\ell : \\mathcal{M} \\to \\mathbb{R},\\ \\ell \\in \\mathbb{N}$ (eigenvectors of the operator), which can be used to push-forward the measures we want to compare to $\\mathbb{R}$. Namely, they introduce \n$$ISW_2(\\mu,\\nu) = \\sum_\\ell \\alpha(\\lambda_\\ell) W_2(\\phi_\\ell \\mu, \\phi_\\ell \\nu), $$\nwhere $\\lambda_\\ell$ is the eigenvalue of the Laplacian operator, $\\alpha$ is some weighting function. \n\nAuthors then prove that $ISW_2$ defines a metric (under mild assumptions), the key aspect being that it induces a universal kernel which yields the separation property $ISW_2(\\mu,\\nu) = 0 \\Rightarrow \\mu = \\nu$ (other properties are mostly straightforward). \n\nOnce this distance is introduced, authors provide some approximation results for practical computations (e.g. we do not have access to all the $(\\phi_\\ell)_\\ell$ and similarly just a sampling of the (projected) measures. Afterward, as this metric is _Hilbertian_, meaning that the metric space $(P(X), ISW_2)$ isometrically embeds in a Hilbert space, they design a two-sample-test between distribution supported on $P(X)$ whenever $X$ is a compact manifold. \nThey showcase the relevance of their approach in different numerical experiments. \n\n(note : $\\phi \\mu$ denotes the pushforward for $\\mu$ by $\\phi$ here, hashtag not used due to rendering issues.)", "questions": "# Questions\n\n1. In the Chicago data experiment, why is the manifold/graph structure \"crucial\" as stated in the Caption of Figure 4 in the appendix ? i guess that we can reasonably assume that this graph is planar in which case we can consider the measures are supported on $\\mathbb{R}^2$. As far as I can tell, there is no comparison between $ISW_2$ and $W_2^{\\mathbb{R}^2}$ (or a standard sliced-$W_2$, so that one has a kernel embedding and can run a two-sample-test here).\n2. Related to the comparison between the $ISW_2$ and the usual Sliced-Wasserstein (assuming our manifold is embedded in $\\mathbb{R}^n$, what is the computational burden in terms of running time related to the computation of $ISW_2$ vs $SW_2$ (in situations where we do not have access to the $\\phi_\\ell$ and $\\lambda_\\ell$ in close form)? I think a proper comparison of the benefits of using $ISW_2$ instead of a naive $SW_2$ is worth of interest. \n\n# Minor remarks and suggestions\n\n- I think there are few typos in the Proof of Proposition 3 (some $\\mathcal{H}$ should be $\\mathcal{H}^0$, or $\\eta$ should be $\\eta^0$).\n- The proof of Theorem 1 in the appendix could be enhanced (with some more precise quote of the results used in the aforementioned references). E.g. what precise statement in [17] and [12] yield the implication $\\alpha(\\lambda_\\ell) > 0 \\Rightarrow$ the kernel is universal hence characteristic? Right now, it is cumbersome to properly check the proof. A similar comment holds for other proofs as well (e.g. the proof of Theorem 2 is quite expeditious).\n- In think the quantitative results mentioned in section 4.1 and detailed in the appendix should actually be mentioned in section 3.2 directly. Indeed, when reading section 3.2 at first, one may be worried by the lack of quantitative assertion there. \n- line 325 : inite-->finite (typo). ", "limitations": "Few limitations of the work should be discussed further. \n\nI did not identify ethical concern specific to this work. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Originality\n\nUsing the Laplace-Beltrami operator as a way to build a \"canonical\" way to slice measures supported on a manifold feels pretty satisfying and is a novel idea to the best of my knowledge. \nOn the other hand, one could argue that once the idea of using the Laplace-Beltrami operator to slice the measures has been introduced, most of the work (aside from the experimental section) is essentially an adaptation of standard techniques in two-sample-tests literature.\n\n# Clarity\n\nThe paper was overall well-written. It motivates the need to consider that our measures are supported on manifold in a convincing manner (e.g. on a circle for daily event records) and aside from few paragraphs (see below), it is fairly easy to read. \n\n# Significance\n\nI think this paper provides a new interesting approach that may motivate further practical developments of Optimal Transport for measures supported on non-flat domain. \n\n# Quality\n\nI think this is a competent paper which introduces a new idea that may be of interest for the Optimal Transport community. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Reject