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A Finslerian Approach for Embedding Directed Data

Gwendal Debaussart-Joniec, Théau Blanchard, Argyris Kalogeratos

arXiv:2609.37649v1stat.MLcs.LG

TL;DR

Directed-data embeddings often symmetrize relationships or map them into Euclidean space, where directionality is lost. This paper models data with a Finsler geometry and separates geometry from direction through symmetric and antisymmetric operators. Their limits recover a weighted Laplacian and a directional transport operator, and the paper reports recovery of geometry and direction in synthetic examples.

  • Problem

    Many directed-data methods symmetrize relationships or return Euclidean embeddings that cannot represent directionality.

    Source: §1 · pp. 1–2

  • Method

    The paper models data as samples from a Finsler manifold and uses symmetric and antisymmetric parts of a kernel operator to recover geometry and direction.

    Source: §1 · p. 2

  • Results

    Experiments on Randers and Matsumoto manifolds and directed stochastic block models show recovery of both geometry and directionality.

    Source: §1 · p. 2, §7 · p. 9

  • Takeaways & Limitations

    For Randers metrics, the explicit directional vector field supports an embedding that recovers manifold structure from the symmetric spectrum and underlying drift.

    Source: §1 · p. 2, §7 · p. 9

    The analysis assumes i.i.d. samples on a smooth, compact manifold without boundary, with a smooth density bounded away from zero and degrees bounded below.

    Source: §2.1 · p. 4, §2.2 · p. 4

Abstract

from arXiv · show

Many datasets carry an intrinsic directionality: citations point backward in time, cells differentiate along lineages, and traffic follows preferred routes. Spectral embedding methods, including most of their extensions to directed graphs, discard this information: they symmetrize the data and map it into a Euclidean space where asymmetry cannot be represented. We instead model directed data as sampled from a Finsler manifold, whose distance depends on the direction of travel, and study the kernel operator built from this asymmetric distance. Through a moment expansion of this operator, we show that its symmetric and antisymmetric parts separate geometry from direction. As the bandwidth of the kernel vanishes, the symmetric part converges to a weighted Laplacian, recovering diffusion maps in the Riemannian case, while the antisymmetric part converges to a first-order transport operator that encodes the directionality. We prove that the corresponding graph operators, built from finitely many samples, converge uniformly and almost surely to these limits. For Randers metrics, this vector field is explicit and yields an embedding algorithm recovering both the manifold structure, from the spectrum of the symmetric part, and the underlying drift. We illustrate the approach on synthetic directed graphs and point-clouds.

1 INTRODUCTION

The paper models directed data as samples from a Finsler manifold, so asymmetry arises from an underlying geometry rather than a hand-crafted kernel. Its operators separate geometry from direction, enabling a directed embedding and recovery of both structures.

1 INTRODUCTION

  • The paper addresses a gap in symmetric graph-Laplacian theory, whose kernel-based formulation cannot represent asymmetric relationships.Directed-data methods often restore symmetry or return Euclidean embeddings where directionality is lost.Source: §1 · pp. 1–2
  • The approach models directed data using an underlying Finsler geometry rather than hand-crafted asymmetry, then recovers a Randers approximation in the embedding space.The paper notes that many directed-graph methods restore symmetry, while existing Finslerian representations usually prescribe the metric form.Source: §1 · p. 2
  • The symmetric operator converges to a weighted Laplacian of the Binet–Legendre metric, while the antisymmetric operator converges to a derivative along the unit-ball centroid field.The limits recover diffusion maps in the Riemannian case; the centroid field encodes directionality.Source: §1 · p. 2
  • The corresponding finite-sample graph operators converge uniformly and almost surely to the continuum limits, with an explicit rate.Source: §1 · p. 2
  • The moment-Randers metric shares the limiting operators of the Finsler metric and can be estimated from the two graph operators to produce an embedding and a Randers metric on it.Source: §1 · p. 2
  • Experiments illustrate the approach on Randers and Matsumoto point-clouds and directed stochastic block models.Source: §1 · p. 2

2 BACKGROUND AND SETTING

The background defines Finsler metrics as direction-dependent generalizations of Riemannian metrics and introduces moments of their unit balls to represent geometry and direction. The setting uses sampled data, a Finsler-distance kernel, and forward and backward transport operators.

2.1 BASIC CONCEPTS

  • A Finsler metric is a positively homogeneous, sub-additive, strongly convex function on tangent vectors that defines curve lengths and an asymmetric distance; Riemannian metrics are its inner-product special case.Positive homogeneity need not hold for negative scalars, so reversing travel can change its cost.Source: §2.1 · p. 3
  • The Binet–Legendre metric is the second moment of the Finsler unit ball, while its first moment, the centroid field, measures asymmetry and carries direction.For Randers metrics, the centroid field is an explicit function of the metric components.Source: §2.1 · pp. 3–4
  • The sampling model assumes N i.i.d. observations from a probability measure with density relative to Busemann–Hausdorff measure.This measure is coordinate-independent and agrees with Riemannian volume in the Riemannian case.Source: §2.1 · p. 4

2.2 OUR SETTING

  • The setting assumes a smooth, connected, compact manifold without boundary, a smooth Finsler metric and density bounded away from zero, and a kernel with a sub-exponential tail.The degree functions are also assumed bounded below.Source: §2.2 · p. 4
  • The Finsler-distance kernel defines forward and backward transport operators; their difference motivates splitting the operator into symmetric and antisymmetric parts.A θ-normalization is used to reduce the influence of the unknown sampling density.Source: §2.2 · p. 4

3 CONTINUUM LIMITS: SEPARATING GEOMETRY FROM DIRECTION

A moment expansion separates the continuum operator’s geometry-bearing symmetric part from its direction-bearing antisymmetric part. Their limits are a weighted Laplacian and a derivative along the centroid field, respectively, with diffusion maps recovered in the Riemannian case.

3 CONTINUUM LIMITS: SEPARATING GEOMETRY FROM DIRECTION

  • The moment expansion of the Finsler transport operator provides the basis for deriving its continuum limits.It is the Finslerian counterpart of expansions used to derive continuum limits of graph Laplacians.Source: §3 · p. 5
  • As ε → 0, the symmetric and antisymmetric operators converge after rescaling by ε^-2 and ε^-1, respectively.The symmetric limit is elliptic and self-adjoint on a weighted L2 space.Source: §3 · p. 5
  • The symmetric limit is a weighted Laplacian with Binet–Legendre metric and reference measure ρ2(1−θ)mBH, while the antisymmetric limit differentiates along the centroid field.The unit ball’s first moment carries direction, while its second moment carries geometry.Source: §3 · p. 5
  • At θ = 1, the symmetric limit is independent of sampling density; θ = 0 retains the full density-induced drift, and θ = 1/2 halves it.The finite-ε symmetric operator is always diagonalizable with real eigenvalues.Source: §3 · p. 5
  • For a Riemannian metric, gBL = A and La = 0, so Equation (7) is the θ-normalized diffusion generator.For Berwald metrics, the Busemann–Hausdorff and Binet–Legendre volume measures coincide up to a constant factor.Source: §3 · p. 5

4 DISCRETE OPERATORS AND CONVERGENCE GUARANTEES

The discrete construction uses a kernel matrix on sampled points to define graph operators. With bandwidth shrinking under a stated sample-size condition, these operators converge uniformly and almost surely to their continuous counterparts.

4 DISCRETE OPERATORS AND CONVERGENCE GUARANTEES

  • The discrete construction starts from the empirical kernel matrix on i.i.d. sampled manifold points and defines out- and in-degrees.It then forms a θ-normalized matrix and its symmetric and antisymmetric parts.Source: §4 · p. 6
  • If ε(N) → 0 and Nε(N)^(m+2)/log N → ∞, the rescaled discrete operators converge uniformly and almost surely to the continuous operators.The theorem assumes f ∈ C3(M).Source: §4 · p. 6

5 MOMENT-RANDERS METRIC AND DIRECTED EMBEDDING

The moment-Randers metric approximates a Finsler metric by matching its Binet–Legendre metric and centroid field, quantities estimated from the symmetric and antisymmetric operators. This yields an embedding procedure that returns embedded samples, drifts, and an estimated Randers metric.

5.1 MOMENT-RANDERS APPROXIMATION

  • The moment-Randers approximation matches the Finsler metric’s Binet–Legendre metric and centroid field, though its Busemann–Hausdorff measure can differ.For a Randers input, the approximation equals the original metric.Source: §5.1 · p. 6

5.2 ESTIMATION THROUGH AN EMBEDDING

  • The carré du champ of the symmetric operator estimates the Binet–Legendre metric up to the embedding, while the antisymmetric operator estimates the pushed-forward centroid field.Together, these estimates define the embedded moment-Randers metric when its admissibility condition holds.Source: §5.2 · p. 7
  • The embedded moment-Randers metric is well defined when the estimated centroid’s squared norm under the estimated metric is below (m+3)^−1.This criterion is computable from the symmetric and antisymmetric operators.Source: §5.2 · p. 7

5.3 EMBEDDING ALGORITHM

  • The algorithm takes directed distances or a weighted directed graph, then returns embedded samples, drifts, and Randers metrics on the embedded points.The embedding may use a classical method such as Isomap or leading eigenvectors of the symmetric operator.Source: §5.3 · pp. 7–8

6 EXPERIMENTS

Experiments test recovery on a Randers Swiss roll, a non-Randers Matsumoto manifold, and directed stochastic block-model graphs. The reported examples recover geometry and directionality, including drift aligned with the Matsumoto height gradient.

Swiss roll dataset

  • Drift cosine is 1.00 in every Swiss-roll configuration, while relative error in the asymmetry norm is 0.07–0.17 and decreases with sample size.The experiment uses N = 2000 points for the embedding example; the table varies drift strength and sample size.Source: §6 · p. 8

Approximating Matsumoto metrics

  • For the non-Randers Matsumoto metric, the recovered drift aligns with the height-map gradient, its defining directional feature.This suggests the method captures the main directional structure of this non-Randers metric.Source: §6 · pp. 8–9, Figure 2 · p. 9

Embedding directed graphs

  • In the directed stochastic block model, the symmetric-operator eigenvectors cluster nodes by community, while the drift follows the dominant backward cyclic flow.Figure 2 presents the inputs in the top row and recovered embeddings and drifts in the bottom row.Source: §6 · p. 9, Figure 2 · p. 9

7 CONCLUSION AND PERSPECTIVES

The paper concludes that Finsler geometry provides a framework for directed graphs analogous to Riemannian geometry for undirected graphs. Experiments report recovery of both geometry and directionality; the authors identify operator spectra and extensions to standard directed graph operators as future work.

7 CONCLUSION AND PERSPECTIVES

  • The symmetric and antisymmetric graph operators recover, respectively, the Binet–Legendre metric and the unit-ball centroid field, supporting directed embeddings through a moment-Randers approximation.Experiments cover Randers and Matsumoto manifolds and directed stochastic block models.Source: §7 · p. 9
  • The authors propose future study of operator spectral properties, extensions to flow-based Laplacians, and Finsler-based methods for directed graph analysis.The paper also states that its datasets are synthetic and can be regenerated with publicly released code.Source: §7 · p. 9, REPRODUCIBILITY STATEMENT · p. 10

A TABLE OF NOTATIONS

B A PRIMER OF RIEMANNIAN AND FINSLER-RANDERS GEOMETRY

The primer contrasts Riemannian geometry’s symmetric, quadratic local metric with Finsler geometry, which permits nonquadratic and direction-dependent costs. It introduces Randers metrics and geometric quantities used to describe the metric, measure, and asymmetry.

B A PRIMER OF RIEMANNIAN AND FINSLER-RANDERS GEOMETRY

  • A Riemannian manifold has a smoothly varying tangent-space inner product that defines distances and angles, with distance measured along shortest paths.Its local cost is governed by the metric.Source: B A PRIMER OF RIEMANNIAN AND… · pp. 14–15
  • Finsler geometry relaxes quadraticity and symmetry, allowing local travel costs to depend on direction while retaining a formulation close to Riemannian geometry.Its metric is positively homogeneous and strongly convex.Source: B A PRIMER OF RIEMANNIAN AND… · p. 16
  • Unlike Riemannian geodesics, Finsler forward and backward distances can differ, and the fundamental tensor provides a local quadratic approximation.Shortest paths still minimize curve energy, allowing Riemannian shortest-path algorithms to be used at marginal additional cost.Source: B A PRIMER OF RIEMANNIAN AND… · p. 17, Figure 3 · p. 19
  • A Randers metric combines a Riemannian metric with a one-form, whose vector field shifts the metric’s ellipsoidal indicatrix and makes direction-dependent distances explicit.Its structure also provides explicit forms for the fundamental tensor and its inverse.Source: B A PRIMER OF RIEMANNIAN AND… · p. 17
  • The Busemann–Hausdorff measure makes each unit tangent ball have Euclidean-unit-ball volume, so the zeroth kernel moment is constant and degree normalization behaves as in the Riemannian case.The symmetric operator sees the Binet–Legendre metric, while the antisymmetric operator sees the unit ball’s centroid.Source: B A PRIMER OF RIEMANNIAN AND… · p. 18

C ADDITIONAL EXPERIMENTAL RESULTS

  • The additional experiments construct Randers metrics and drift fields for the Swiss Roll, approximate a Matsumoto metric, and study a directed stochastic block model.Illustrations display the recovered samplewise drift as an interpolated vector field on an embedding-space grid.Source: C ADDITIONAL EXPERIMENTAL RESULTS · p. 19

C.1 SWISS ROLL DATASET

On the Swiss Roll, the recovered drift direction aligns with its theoretical limit at every tested drift strength and sample size. The experiment uses a radius graph to avoid truncating strong, directionally extended kernels unevenly.

C.1 SWISS ROLL DATASET

  • The Swiss Roll experiment uses 2,000 noisy points, a tangential drift normalized to A^-1 norm 0.9, and an Isomap embedding in R2.The directed k-NN graph uses k = 10, with geodesic distances computed by GEORCE.Source: §C.1 · p. 20
  • A radius graph retains pairs within 3ε because strong drift extends the kernel farther along the drift than across it, which could bias moments in a k-NN graph.The bandwidth is the median Finsler distance to the 10th nearest neighbor; edge weights use a Gaussian kernel.Source: §C.1 · p. 20
  • The drift direction is recovered at every tested strength and sample size, with Table 1 reporting cosine similarity between the estimated drift and its limit.The embedding Jacobian is needed only for drift direction; the estimated drift norm does not depend on the embedding.Source: §C.1 · p. 20

C.2 MATSUMOTO METRIC APPROXIMATION

The appendix describes experiments on a Matsumoto metric and a directed stochastic block model, then records how forward and backward Finsler transport operators relate. In the block model, recovered drift tracks the known flow direction as community connectivity becomes asymmetric.

C.2 MATSUMOTO METRIC APPROXIMATION

  • The Matsumoto experiment samples 3,000 points from a height-map-induced metric, builds a k-NN graph with GEORCE geodesic weights, and estimates a Randers metric and drift after Isomap embedding.The graph uses k = 10 neighbors and the Gaussian-kernel bandwidth is the standard deviation of node geodesic distances.Source: §C.2 · pp. 20–21

C.3 DIRECTED STOCHASTIC BLOCK MODEL

  • The DiSBM assigns connection probabilities p, q, and r to the next, previous, and same communities, respectively, with the communities arranged in a cycle.Its adjacency entries are sampled independently using the corresponding community-pair probabilities.Source: §C.3 · p. 21, Figure 4 · p. 22
  • The DiSBM drift follows the known flow direction whenever p ≠ q: median signed cosine is ±0.89 at |p−q| = 0.02 and above 0.99 from |p−q| = 0.1.At p = q there is no preferred drift direction; estimated asymmetry strength is symmetric in p−q and grows linearly with |p−q|.Source: §C.3 · p. 22, Figure 5 · p. 22

D.1 FORWARD AND BACKWARD TRANSPORT OPERATORS

  • The backward transport operator is the L2(M,mBH) adjoint of the forward operator and equals transport under the reversed Finsler metric, whose geodesics reverse direction.The reversed metric’s distance satisfies dst←F(x,y) = dstF(y,x).Source: §D.1 · p. 23

D.2 PROOF OF THEOREM 1

The proof expands the kernel integral locally using the Finsler exponential map, function derivatives, kernel moments, and the Jacobian. The resulting moment expansion highlights terms arising from non-reversibility and S-curvature that vanish in Riemannian settings.

D.2 PROOF OF THEOREM 1

  • Non-reversibility generally makes the first kernel moment m1 nonzero, while the second-order expansion includes additional metric and curvature terms.The curvature term Υ multiplies f and reduces to an average of Ricci curvature in the Riemannian setting.Source: §D.2 · p. 25
  • S-curvature contributes first-order and curvature terms that vanish for Riemannian and, more generally, Berwald metrics.The proof obtains the antisymmetric expansion from the forward integral-operator expansion and then takes symmetric and antisymmetric parts.Source: §D.2 · pp. 24–25
  • The proof localizes the kernel integral near each point, expands the function and exponential-map Jacobian, then collects terms through order ε^2 to obtain the moment expansion.Sub-exponential kernel decay controls the far region, and the change of variables is valid within the injectivity radius.Source: §D.2 · p. 24, MAIN LEMMAS · p. 26, PROOF OF THE MOMENT EXPANSION OF THE… · pp. 29–30

D.3 PROOF OF THEOREM 2

This section proves Theorem 2 by deriving reverse-metric moment expansions and characterizing the normalized symmetric and antisymmetric transport limits, including their adjointness, divergence, and ellipticity properties. It also connects normalized kernel moments to unit-ball geometry, the centroid, and the Binet–Legendre metric.

D.3 PROOF OF THEOREM 2

  • Reversal preserves the geodesic spray and Ricci curvature after velocity reversal, while S-curvature and kernel moments obey corresponding parity identities.In particular, reverse moments satisfy ←m_n = (−1)^n m_n, ←Λ_k = Λ_k, ←s_n = (−1)^(n+1) s_n, and ←Υ = Υ.Source: LEMMAS ON THE REVERSE FINSLER METRIC… · pp. 30–33
  • Under metric reversal, odd-order terms in the transport-operator expansion change sign, while even-order terms remain unchanged.Consequently, the symmetric part retains even-order terms and the antisymmetric part retains odd-order terms through order ε^2.Source: LEMMAS ON THE REVERSE FINSLER METRIC… · p. 34
  • Centering removes the curvature-only term from the symmetric expansion and the S-curvature-only term from the antisymmetric expansion.The symmetric and antisymmetric normalized operators are centered around their degrees and vanish on constant functions.Source: θ-NORMALIZED TRANSPORT OPERATOR · pp. 36–37
  • The normalized antisymmetric operator is independent of sampling density and θ, whereas the symmetric limit depends on θ and loses its density dependence when θ = 1.For θ = 1, the sampling density disappears from the limit, as in the Riemannian setting.Source: θ-NORMALIZED TRANSPORT OPERATOR · p. 38
  • The moment coefficients make A1 skew-adjoint and A2 self-adjoint, yielding a divergence-form symmetric limit that is self-adjoint on L2(M, ωmBH).The manifold is assumed compact and without boundary.Source: θ-NORMALIZED TRANSPORT OPERATOR · pp. 38–39
  • The Binet–Legendre tensor is positive definite, completing the proof of Theorem 2.Source: θ-NORMALIZED TRANSPORT OPERATOR · p. 39

D.4 LINKS BETWEEN THE MOMENTS OF THE KERNEL AND THE UNIT BALL

  • The zeroth moment is spatially constant under Busemann–Hausdorff measure, so θ-normalization acts as in the Riemannian setting and θ = 1 removes sampling density from Ls.Here m0 = mµ0VolEuc(Bm), and qε = m0ρ + O(ε2).Source: IMPLICATIONS FOR THE LIMIT OPERATORS · p. 41
  • The normalized first moment is proportional to the unit-ball centroid: it vanishes for reversible metrics, while for non-reversible metrics the centroid field determines La as its directional derivative up to a kernel-dependent constant.A zero centroid does not imply reversibility; for Randers metrics, the centroid is explicit in terms of b.Source: IMPLICATIONS FOR THE LIMIT OPERATORS · pp. 41–42
  • The normalized second moment is proportional to the unit ball’s raw second moment and, up to the stated kernel constant, coincides with the Binet–Legendre metric.This Riemannian metric captures local Finsler geometry and is independent of bandwidth; normalization cancels shared volume factors, leaving centroid and second-moment averages with kernel dependence only through µn/µ0.Source: IMPLICATIONS FOR THE LIMIT OPERATORS · p. 42

D.5 PROOF OF COROLLARY 3

The symmetric limit is a Binet–Legendre Laplacian with a drift determined by the unit-ball volume density. For Berwald metrics that density is constant, so the operator simplifies; in the Riemannian case it reduces to the θ-normalized diffusion generator.

General Finsler metrics

  • For any Finsler metric, the difference from a weighted Binet–Legendre Laplace–Beltrami operator is a drift along the gradient of log w.Here w measures variation of the unit-ball volume across the manifold.Source: §D.5 · p. 42

Berwald metrics

  • For Berwald metrics, parallel transport preserves the unit balls and induces Binet–Legendre isometries, making their volumes constant and reducing Equation (24) to Equation (7).The Jacobian of parallel transport cancels against normalization by ball volume.Source: §D.5 · p. 42

Riemannian case

  • In the Riemannian case, vanishing S-curvature and reversibility recover the θ-normalized diffusion generator, with the first moment m1 equal to zero.The cited diffusion generator is the one of Coifman & Lafon (2006).Source: §D.5 · p. 43

D.6 PROOF OF THEOREM 4

This section analyzes convergence by separating deterministic bias from sampling fluctuation and specifying bandwidth conditions for uniform almost-sure convergence. It also derives closed-form Randers unit-ball moments and uses them to obtain the kernel moments.

D.6 PROOF OF THEOREM 4

  • The proof splits error into deterministic bias, which decreases with ε, and stochastic fluctuation, which grows as ε shrinks; concentration and a δ-net yield a uniform bound.The stochastic bound assumes Nε^m/logN → ∞ and holds almost surely for sufficiently large N.Source: FROM FINITE SAMPLES TO THE LIMIT… · p. 44, BOUNDING THE FLUCTUATION · p. 44
  • Under f ∈ C3(M), ε → 0, and Nε^m/logN → ∞, both operators converge uniformly and almost surely when the stated bandwidth conditions hold.The symmetric operator requires the stronger condition; the antisymmetric operator needs only Nε^m/logN → ∞.Source: COMBINING THE BIAS AND THE FLUCTUATION · p. 50
  • The bandwidth balances bias against fluctuation: larger ε blurs geometry, while smaller ε increases noise.The section identifies a balancing choice and corresponding error order, but their formulas are not present in the supplied passages.Source: COMBINING THE BIAS AND THE FLUCTUATION · p. 50

D.7 COMPUTATION OF THE MOMENTS IN THE RANDERS SETTING

  • For Randers metrics F(x,v) = √(vᵀA(x)v) + b(x)ᵀv, the unit-ball moments can be computed in closed form.The metric requires A(x) positive definite and ∥b(x)∥_{A⁻¹} < 1.Source: §D.7 · p. 50
  • The Randers unit ball is an ellipsoid centered at v0, and symmetry about that center makes the first moment shift with v0.The second moment also includes a term proportional to the centroid’s outer product; as b approaches zero, the Riemannian moment expressions are recovered.Source: §D.7 · pp. 51–52
  • A change of variables to the ellipsoid’s centered coordinates and the determinant lemma yield the unit-ball volume expression.The supplied passages describe this computation but do not provide the resulting formula.Source: §D.7 · p. 51
  • Proposition 45 gives the Randers kernel moments m0, ˜m1, and ˜m2 by applying Propositions 43 and 35.The supplied passages state this derivation but do not include the moment formulas.Source: §D.7 · p. 52

D.8 THE MOMENT-RANDERS APPROXIMATION

The moment-Randers approximation preserves a Finsler metric’s first two unit-ball moments, enabling its geometry and direction to be recovered from operators on an embedding. Its empirical version converges at the samples when the graph operators do.

D.8 THE MOMENT-RANDERS APPROXIMATION

  • The moment-Randers metric uniquely matches F’s centroid and raw second moment, and its unit ball is an ellipsoid centered at the centroid.If F is already Randers, the approximation equals F.Source: WELL-DEFINEDNESS OF THE MOMENT-RANDERS… · pp. 53–54
  • The antisymmetric operator recovers the centroid field from an embedding, while the symmetric operator’s carré du champ recovers the Binet–Legendre metric.The recovered centroid’s direction and Binet–Legendre magnitude do not depend on sampling density or normalization exponent.Source: RECOVERING THE CENTROID AND THE… · pp. 55–56
  • With a full-column-rank embedding and the admissibility condition, the estimated Randers metric is valid on the embedded tangent space, not all of R^ℓ.The admissibility condition is equivalent to ∥c∥_H < 1.Source: THE EMPIRICAL MOMENT-RANDERS METRIC · pp. 57–58
  • The empirical metric quantities converge at each sample when the discrete operators converge uniformly and almost surely to their continuous counterparts.The discrete quantities are computed from the graph operators and embedded samples.Source: THE EMPIRICAL MOMENT-RANDERS METRIC · p. 58

D.9 COMPUTATION OF THE KERNEL MOMENTS

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