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REVIEW 3 major objections 9 minor 34 references

Maximum-entropy walk recovers geodesic distances from data graphs

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2026-07-08 03:36 UTC pith:CVOYW7BF

load-bearing objection EntroPath: MERW-based free-energy dissimilarity with a Varadhan geodesic recovery theorem. The theory is sound and the method is genuinely novel; the main open question is whether the key spectral convergence assumption actually holds for the specific operator used. the 3 major comments →

arxiv 2607.06497 v1 pith:CVOYW7BF submitted 2026-07-07 cs.LG q-bio.QMstat.ML

EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning

Przemys{\l}aw Rola This is my paper
classification cs.LG q-bio.QMstat.ML
keywords entropathdiffusiongeodesicgeometrymanifoldembeddingsensembleentropy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces EntroPath, a manifold learning method that recovers geodesic geometry from data graphs using the maximum entropy random walk (MERW). Unlike standard random walks, which normalize transition probabilities by local node degree and thus drift toward densely sampled regions, MERW distributes probability uniformly over the ensemble of all equal-length paths between two points. This global reweighting means that thin bridges and bottlenecks between clusters carry few high-weight paths, so they act as effective energy barriers rather than leaking probability. The paper defines a free-energy dissimilarity D_ij as the negative log of the k-step MERW transition probability between nodes i and j, which aggregates the full path ensemble rather than relying on a single shortest path. The central theoretical result (Theorem 5.3) proves that under spectral convergence of the graph to a closed Riemannian manifold, the symmetrised free-energy dissimilarity recovers the squared geodesic distance in the short-time limit. The proof proceeds in four steps: discrete graph powers converge to a Schrödinger heat kernel (via spectral convergence), the Perron-vector ratios cancel exactly in the geometric-mean symmetrisation, Varadhan's heat-kernel formula gives the geodesic in the short-time limit, and a diagonal extraction over both limits combines them. The diffusion depth k interpolates between local neighborhood structure and global manifold geometry, and the symmetrised kernel admits an exact Gram factorisation connecting it to kernel methods. Experiments on synthetic manifolds and single-cell benchmarks show that EntroPath's advantages are most pronounced under non-uniform sampling density and on well-separated branching trajectories, where path-ensemble diffusion preserves geodesic geometry more faithfully than degree-normalised alternatives.

Core claim

The paper's central discovery is that the maximum entropy random walk (MERW), which globally reweights path probabilities using the Perron-Frobenius eigenvector of the graph affinity matrix, produces a free-energy dissimilarity that converges to squared geodesic distance on the underlying manifold. The key structural insight enabling this is that the MERW Laplacian is similar to a discrete Schrödinger operator H = lambda_max I - A = -Delta + V, where V is a degree-induced potential. Because the geometric-mean symmetrisation of forward and backward MERW transition probabilities causes the Perron-vector ratios to cancel exactly, the dissimilarity reduces to a negative log of powers of the resc

What carries the argument

The maximum entropy random walk (MERW) transition matrix T = diag(psi)^{-1} (A/lambda_max) diag(psi), where psi is the Perron-Frobenius eigenvector and lambda_max the spectral radius of the affinity matrix A. The discrete Schrödinger operator H = lambda_max I - A, whose heat semigroup e^{-tH} approximates the k-step MERW kernel at diffusion time t = k/lambda_max. The free-energy dissimilarity D_ij = -log(sqrt(T^k_ij * T^k_ji)), which equals -log((A/lambda_max)^k)_ij for symmetric A. Varadhan's short-time heat-kernel formula, which gives lim_{t->0+} -4t log K_t(x,y) = d_M(x,y)^2.

Load-bearing premise

The theorem requires that the discrete graph Hamiltonians converge spectrally to a Schrödinger operator on a closed Riemannian manifold, with the degree-induced potential V being bounded and continuous. If sampling density variations cause V to violate these regularity conditions, or if the pair-independent normalisation factor Z_n fails to factor cleanly across node pairs, the geodesic recovery guarantee does not hold.

What would settle it

Construct a data graph on a manifold with sharply discontinuous sampling density (e.g., two regions with very different sampling rates meeting at a sharp boundary), so that the degree-induced potential V is discontinuous. If the symmetrised free-energy dissimilarity fails to converge to squared geodesic distance across the density discontinuity, the bounded-continuous-potential requirement is load-bearing rather than technical.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The Schrödinger-operator structure of MERW means that sampling density survives as a potential V in the continuum limit rather than being normalized away, which could be exploited to simultaneously recover both geodesic geometry and density information from a single operator.
  • The exact Perron-ratio cancellation in the geometric-mean symmetrisation suggests that asymmetric variants (using left and right Perron eigenvectors for directed graphs) could extend geodesic recovery to directed data such as RNA-velocity flow fields.
  • The path-ensemble soft-min interpretation connects EntroPath to temperature-controlled relaxations of shortest-path geometry, where diffusion depth k plays the role of inverse temperature, suggesting a family of embeddings parameterised by a continuous temperature rather than a discrete step count.
  • The finding that MERW inflates inter-cluster dissimilarities at bottlenecks (the energy-barrier effect) could be leveraged for graph clustering and community detection independent of manifold learning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The geodesic recovery theorem requires the degree-induced potential V to be bounded and continuous on the manifold, which holds when sampling density varies smoothly. If density has sharp discontinuities (e.g., mixtures of very different sampling rates), V may violate these conditions and the theorem's conclusion could fail, suggesting a practical regime where EntroPath's advantage over density-no
  • The diagonal extraction in the proof (requiring t_n log n -> 0) implies that the graph must be sufficiently large relative to the diffusion time for geodesic recovery to hold, which may explain why the entropy-based depth selection becomes noisier on short-branch structures with few points per branch.
  • The structural identity with DTNE (same log-kernel dissimilarity, different walk) suggests that the geodesic recovery guarantee could potentially extend to the standard random walk variant if one could establish spectral convergence of the degree-normalised Laplacian to a Schrödinger operator, though the density cancellation in that case would remove the potential V entirely.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 9 minor

Summary. The paper introduces EntroPath, a manifold learning method based on the maximum entropy random walk (MERW). The core construction is a free-energy dissimilarity D_ij = -log((A/lambda_max)^k)_ij, interpreted as a log-sum-exp over k-step path ensembles. The central theoretical result (Theorem 5.3) proves that under spectral convergence of the graph Hamiltonian H_n = lambda_max I - A_n to a Schrodinger operator H = -Delta_M + V, the symmetrised dissimilarity recovers squared geodesic distance in the short-time limit via Varadhan's formula. The proof (Appendix D) proceeds in four clean steps: discrete-to-continuum limit, exact Perron-ratio cancellation, Varadhan via Feynman-Kac, and diagonal extraction. Empirically, EntroPath is evaluated on synthetic manifolds and six single-cell datasets against PHATE, HeatGeo, DTNE, and others, showing advantages particularly under non-uniform sampling density.

Significance. The paper makes a genuine contribution by connecting MERW to manifold geometry through a Schrodinger-operator framework. The proof of Theorem 5.3 is structurally sound: the four-step chain in Appendix D uses standard ingredients (Chernoff limit, Varadhan, Feynman-Kac) correctly, and the exact Perron-ratio cancellation in Step 2 is a clean algebraic identity. The free-energy dissimilarity is derived from first principles rather than fitted to the target result. The Gram factorisation (Proposition 4.1) and the connection to DTNE are clearly articulated. The two-stage evaluation protocol (Level-1 distance-level vs. Level-2 embedding-level) and the analysis of shortest-path geodesic proxy bias (Appendix A.1) are valuable methodological contributions. Reproducible code is provided.

major comments (3)
  1. Assumption 5.1 (Spectral convergence) is the load-bearing premise, and the gap is more specific than the paper acknowledges. The cited references [4, 7, 12] establish spectral convergence for *normalized* graph Laplacians (e.g., I - D^{-1/2}AD^{-1/2} -> -Delta_M) or for the unnormalized combinatorial Laplacian L = D - A with density-dependent weights. The operator H_n = lambda_max I - A_n is neither: it normalizes by the global spectral radius lambda_max rather than local degrees. The paper states (Section 5, Remark 5.2) that 'the density here survives in the limit' as a potential V, but does not establish that the specific scaling H_n/lambda_max converges to a well-defined Schrodinger operator. The concern is concrete: lambda_max grows with n (as the paper notes, lambda_max -> infinity), the degree-induced potential V_i = lambda_max - d_i has entries that grow with n, and the interplay,
  2. Section 8, Tables 1-2: The evaluation bias from using shortest-path geodesic as ground truth is acknowledged (Appendix A.1) but not fully controlled. On the sparse tree (Table 1), the ground truth is shortest-path on a 15-NN graph, and EntroPath is compared against Isomap/Shortest Path which *by construction* coincide with this ground truth. EntroPath exceeding the Isomap baseline (0.934 vs 0.917) is surprising and noteworthy, but the paper should clarify whether this comparison is fair given that EntroPath's own dissimilarity is built on the same 15-NN graph backbone. The analytic-protocol results on the Swiss roll (Table 2) are more convincing and should be foregrounded as the primary evidence.
  3. Table 4, Lymphoid and Embryoid Body rows: PCA achieves the highest DEMaP (0.615 and 0.486 respectively) among all methods, which the paper attributes to PCA preserving coarse global geometry. However, this undercuts the central claim that EntroPath's geodesic recovery translates to superior practical performance. The paper should discuss more carefully why a linear method outperforms on these datasets and whether the DEMaP metric (which uses shortest-path on kNN as reference) is the right evaluation tool here, given the protocol bias documented in Appendix A.1.
minor comments (9)
  1. Equation (1): The transition matrix T_ij = A_ij/lambda_max * psi_j/psi_i is defined for symmetric A, but the MERW was originally derived for unweighted {0,1} adjacency matrices. Remark 2.1 extends to weighted symmetric A, but the Perron-Frobenius uniqueness argument should be stated more carefully for the general nonnegative symmetric case (irreducibility is mentioned but not formally tied to the graph connectivity assumption).
  2. Section 3.1, Eq. (4): The edge cost E(u,v) = -log A_uv can be negative when A_uv > 1 (e.g., for Gaussian affinities with small sigma). The paper should clarify that for the Gaussian kernel A_uv <= 1 always, so costs are non-negative, or discuss the implications of negative costs for the free-energy interpretation.
  3. Figure 2: The schematic comparing SRW and MERW bottleneck behavior is illustrative but not quantitatively grounded. A small numerical example (e.g., two clusters joined by a single bridge) with actual transition probabilities would strengthen the argument.
  4. Table 3: The sparse tree column uses shortest-path geodesic, inheriting the protocol bias of Appendix A.1. A footnote or caveat referencing this bias should be added directly to the table caption, not only in the appendix.
  5. Section 6, Algorithm 1, Step 6: The von Neumann entropy knee selection is described as following PHATE's criterion, but Appendix B reveals that the entropy is computed from singular values of T (not eigenvalues of A_tilde). This difference is non-trivial and should be mentioned in the main text algorithm description, not only in the appendix.
  6. Proposition 3.6: The large-k limit D_ij -> -1/2 log(pi_i pi_j) is stated for aperiodic chains. The paper notes that aperiodicity holds for Gaussian affinities with positive diagonal, but the alpha-decay kernel used for single-cell data may have zero diagonal entries. The aperiodicity assumption should be verified for this kernel.
  7. Remark 3.2: The statement that D_ij 'need not satisfy the triangle inequality' is correct, but the paper could note that the log-kernel structure means D_ij is a conditionally negative definite function under certain conditions on S, which has implications for MDS embedding quality.
  8. Reference [32] (DTNE): The correspondence in Section B.1 and Appendix C is thorough, but the paper should note that DTNE's geodesic approximation is not established (Table 23), making EntroPath's theoretical guarantee a distinguishing feature that should be highlighted in the introduction.
  9. Section 8.2: EntroPath uses a single fixed configuration (k=15, alpha-decay with decay 40, k-means landmarks) across all single-cell datasets. While this is presented as a strength, the paper should report sensitivity to this choice, at least on one dataset, to confirm robustness.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the structural soundness of the proof and the methodological contributions. The three major comments raise legitimate concerns: (1) the spectral convergence assumption for the specific MERW normalization is not directly covered by existing results and needs honest acknowledgment; (2) the evaluation bias on the sparse tree should be foregrounded differently; and (3) PCA outperforming on two datasets requires more careful discussion. We address each below.

read point-by-point responses

  1. Referee: Assumption 5.1 (Spectral convergence) is the load-bearing premise, and the gap is more specific than the paper acknowledges. The cited references establish spectral convergence for normalized graph Laplacians or unnormalized combinatorial Laplacians with density-dependent weights. The operator H_n = lambda_max I - A_n normalizes by the global spectral radius rather than local degrees, and the paper does not establish that this specific scaling converges to a well-defined Schrödinger operator. The concern is concrete: lambda_max grows with n, the degree-induced potential V_i = lambda_max - d_i has entries that grow with n, and the interplay is not worked out.

    Authors: The referee is correct that the cited references [4, 7, 12] establish spectral convergence for normalized or combinatorial Laplacians, not for the specific operator H_n = lambda_max I - A_n. We acknowledge this gap honestly. Our defense is twofold. First, the proof of Theorem 5.3 is explicitly conditional on Assumption 5.1: the theorem states 'Let {G_n} converge spectrally (Assumption 5.1) to...' and does not claim that the assumption holds automatically. The proof uses only the spectral convergence premise plus standard ingredients (Chernoff, Varadhan, Feynman-Kac), and these steps are correct given the assumption. Second, we agree that the specific scaling H_n/lambda_max requires its own convergence analysis, and the paper should not present the assumption as though it follows directly from the cited literature. We will revise the manuscript to make the status of Assumption 5.1 fully transparent: (a) the principal part (Laplacian convergence) is supported by [4, 7, 12]; (b) the potential V_i = lambda_max - d_i and its scaling are not covered by those results and constitute an open problem; (c) the proof is conditional on the assumption, not a proof of it. We will add a dedicated remark explaining that establishing spectral convergence for the MERW normalization is the key open theoretical question, and sketch why we believe it is plausible (the rescaled operator H_n/lambda_max has spectrum in [-1,1] with the principal part converging to -Delta_M, while the potential V/lambda_max = 1 - d_i/lambda_max converges pointwise to a bounded function under appropriate bandwidth scaling). However, we cannot honestly claim this is established, and the revision will say so explicitly. revision: partial

  2. Referee: Section 8, Tables 1-2: The evaluation bias from using shortest-path geodesic as ground truth is acknowledged but not fully controlled. On the sparse tree (Table 1), the ground truth is shortest-path on a 15-NN graph, and EntroPath is compared against Isomap/Shortest Path which by construction coincide with this ground truth. EntroPath exceeding the Isomap baseline (0.934 vs 0.917) is surprising and noteworthy, but the paper should clarify whether this comparison is fair given that EntroPath's own dissimilarity is built on the same 15-NN graph backbone. The analytic-protocol results on the Swiss roll (Table 2) should be foregrounded as the primary evidence.

    Authors: The referee's point is well taken. On the sparse tree, the ground truth is shortest-path on a 15-NN graph, and EntroPath's dissimilarity is also built on the same 15-NN graph backbone. The fact that EntroPath exceeds Isomap (0.934 vs 0.917) is indeed surprising, and we agree that the fairness of this comparison needs explicit discussion. The mechanism is that EntroPath's free-energy dissimilarity is a soft-min over the path ensemble, not the hard min of shortest path: it aggregates many paths rather than selecting one, so it can correlate more smoothly with the shortest-path reference even though it does not coincide with it by construction. But the referee is right that sharing the same graph backbone means the comparison is not fully independent. We will revise the manuscript to: (1) foreground the analytic-protocol Swiss roll results (Table 2) as the primary distance-level evidence, since there the ground truth is the exact Riemannian geodesic and is immune to kNN-backbone bias; (2) add an explicit caveat to the sparse tree table noting that all methods share the 15-NN backbone and that the comparison is therefore not fully independent; (3) note that EntroPath's advantage over Isomap on the tree is consistent with the soft-min interpretation but should be interpreted with the protocol bias caveat from Appendix A.1. revision: yes

  3. Referee: Table 4, Lymphoid and Embryoid Body rows: PCA achieves the highest DEMaP (0.615 and 0.486 respectively) among all methods, which the paper attributes to PCA preserving coarse global geometry. However, this undercuts the central claim that EntroPath's geodesic recovery translates to superior practical performance. The paper should discuss more carefully why a linear method outperforms on these datasets and whether the DEMaP metric (which uses shortest-path on kNN as reference) is the right evaluation tool here, given the protocol bias documented in Appendix A.1.

    Authors: The referee is correct that PCA achieving the highest DEMaP on Lymphoid and Embryoid Body undercuts a strong reading of our central claim, and the current discussion does not engage with this sufficiently. We will expand the discussion along the following lines. First, on these two datasets the nonlinear methods lose substantial geodesic structure (UMAP DEMaP 0.28 and 0.18 on Lymphoid and Embryoid Body respectively), while PCA preserves coarse global geometry. PCA's advantage is real but comes at the cost of failing to resolve cell states (its clustering score is essentially zero on Lymphoid, as noted in Appendix A.5). Second, the referee's point about DEMaP is important: DEMaP uses shortest-path on a kNN graph as reference, and as documented in Appendix A.1, this protocol systematically favors methods whose dissimilarities are tied to the same kNN graph. PCA, being a linear projection, is not a kNN-graph-based method, so its high DEMaP on these datasets is not a protocol artifact in the same way as Isomap's saturation on the Swiss roll. Rather, it reflects that on these particular datasets the global linear structure happens to align well with the shortest-path reference. Third, we will be more careful in our claims: rather than stating that EntroPath's geodesic recovery translates to superior practical performance in general, we will state that it translates to the best nonlinear DEMaP on three of six datasets (Paul15, Lymphoid, Embryoid Body) and competitive performance on the rest, while noting that PCA can outperform on datasets where coarse global geometry suffices and cell-state resolution is not the goal. This is a weaker but more honest claim. revision: yes

standing simulated objections not resolved
  • Assumption 5.1 (spectral convergence of H_n = lambda_max I - A_n to a Schrödinger operator) is not established by the cited literature, which covers normalized and combinatorial Laplacians but not the global-spectral-radius normalization used by MERW. We believe the assumption is plausible but cannot prove it, and the revision will state this as an open problem rather than claiming it follows from existing results.

Circularity Check

0 steps flagged

No circularity found; derivation chain is self-contained with external citations only

full rationale

The central theoretical result (Theorem 5.3) is derived through a four-step chain — (1) MERW definition from Burda et al. [6], (2) exact algebraic Perron-ratio cancellation in the geometric mean (Eq. 17), (3) discrete-to-continuum spectral convergence via Assumption 5.1 citing external work [4, 7, 12], and (4) Varadhan's heat-kernel formula [30] with a standard Feynman-Kac argument. No step reduces to its inputs by construction: the Perron cancellation is a genuine algebraic identity (not a fit), the spectral convergence is an explicitly stated assumption backed by external literature, and Varadhan's formula is a classical mathematical result. No self-citations appear anywhere in the load-bearing chain — all key references (Burda, Varadhan, Belkin-Niyogi, Coifman-Lafon, Hein et al.) are by different authors. The diffusion depth k is selected unsupervised via von Neumann entropy knee (following PHATE [18]), not fitted to geodesic targets. The paper transparently acknowledges sharing the log-kernel dissimilarity form with DTNE [32] but identifies its distinct contributions (MERW walk, finite horizon, geodesic recovery theorem). The score of 1 rather than 0 reflects only the minor observation that the dissimilarity form is shared with prior work, but this is openly stated and does not constitute circularity. The skeptic's concern about whether Assumption 5.1 is actually established for H_n = λ_max I - A_n (as opposed to normalized or combinatorial Laplacians) is a correctness risk, not a circularity issue — the assumption is stated as an assumption, not derived from self-citation and presented as a result.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 1 invented entities

The axiom ledger is lean. The method relies on standard spectral graph theory and heat kernel asymptotics. The main domain assumption is the spectral convergence of the specific MERW Hamiltonian, which is plausible but not fully proven for the specific potential V. Free parameters are selected unsupervised or fixed by convention.

free parameters (4)
  • kNN = 15
    Neighbourhood size for graph construction. Fixed across experiments, not tuned per dataset.
  • k = selected via VNE knee
    Diffusion depth. Selected unsupervised via the knee of the von Neumann entropy curve, not fitted to ground truth.
  • alpha (kernel) = 40
    Decay exponent for the alpha-decay kernel on single-cell data. Fixed following PHATE convention.
  • M (landmarks) = 2000
    Number of landmarks for scalable extension. Selected based on a complexity threshold, not tuned to geodesic fit.
axioms (3)
  • domain assumption Assumption 5.1 (Spectral convergence): Discrete Hamiltonians H_n converge spectrally to a Schrödinger operator H = -Delta_M + V on a closed Riemannian manifold.
    Invoked in Section 5 and Appendix D to link the discrete graph operator to the continuum heat kernel. The convergence of the principal part is standard, but the potential V reflecting sampling density is a specific assumption.
  • standard math Varadhan's formula for Schrödinger operators with bounded continuous potential.
    Invoked in Step 3 of the proof (Lemma D.3) to establish the short-time geodesic limit. Standard result extended via Feynman-Kac.
  • domain assumption The graph is connected and aperiodic.
    Required for the Perron-Frobenius theorem and the MERW construction (Section 2.1, Proposition 3.6).
invented entities (1)
  • Degree-induced potential V_i = lambda_max - d_i independent evidence
    purpose: Defines the discrete Schrödinger operator H = L + V whose heat semigroup governs MERW diffusion.
    Derived from the graph structure, not postulated. Its effect on the heat kernel is bounded and shown to not affect the Varadhan limit.

pith-pipeline@v1.1.0-glm · 50333 in / 2710 out tokens · 460051 ms · 2026-07-08T03:36:20.970503+00:00 · methodology

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read the original abstract

We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths. Many existing graph-based embeddings rely either on locally normalised random walks or on shortest-path distances. The former can concentrate diffusion in densely sampled regions, while the latter are sensitive to spurious shortcut edges in the graph. EntroPath instead builds its dissimilarities from the maximum entropy random walk (MERW), which aggregates the full ensemble of k-step paths between points rather than relying on any single trajectory. We show that the resulting free-energy dissimilarity converges to squared geodesic distance in the short-time limit, via Varadhan's heat-kernel formula. The diffusion depth k interpolates smoothly between local neighbourhood structure and global manifold geometry, and the symmetrised kernel admits an exact Gram factorisation connecting EntroPath to kernel methods. We further provide scalable extensions via landmark projection and diffusion-potential pseudotime. Across synthetic manifolds and single-cell benchmarks, EntroPath consistently matches or outperforms diffusion- and shortest-path-based methods, while remaining competitive with neighbourhood-preserving embeddings (UMAP, t-SNE) on local-structure metrics. Its gains are most pronounced on manifolds with non-uniform sampling density and well-separated branching trajectories, where path-ensemble diffusion more faithfully preserves the underlying geodesic geometry.

Figures

Figures reproduced from arXiv: 2607.06497 by Przemys{\l}aw Rola.

Figure 1
Figure 1. Figure 1: The MERW free-energy dissimilarity as a soft-min over paths. Each [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The bottleneck advantage of MERW (Remark 3.7). Two dense clusters are joined by a single thin bridge [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Chain of equivalences from MERW to geodesic distance, via a two-stage limit: [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative 2D embeddings of (top) uniform and (bottom) non-uniform Swiss roll (Beta [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Embeddings of the sparse artificial tree, coloured by branch identity. The diffusion-based methods (PHATE, [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional embeddings of six single-cell datasets (rows) produced by the compared methods (columns, [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Diffusion-depth ablation on the Swiss roll (N = 2,000, 30 seeds; shaded bands ±1 s.d.). (a) Geodesic fidelity ρ(k) (blue, left axis) between the EntroPath dissimilarity and the analytic geodesic, with the von Neumann entropy (green, right axis) and its selected knee (vertical line, k = 57); the Isomap geodesic baseline is dotted. Fidelity peaks at k = 65 (ρ = 0.88) and the entropy knee falls inside the pea… view at source ↗
Figure 8
Figure 8. Figure 8: Sphere embeddings (N = 2,000, σ = 0.05) coloured by polar angle θ. t-SNE and UMAP tear the spherical topology into flat, fragmented layouts; MDS, Diffusion Maps, PHATE, HeatGeo, DTNE, and EntroPath produce projection-like embeddings in which one hemisphere is occluded, with θ varying smoothly across the disc [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Embeddings of (top) torus coloured by major angle [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Embeddings of the dense artificial tree, coloured by branch identity. Path-integral methods (DTNE, HeatGeo, [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Two-dimensional embeddings on the Tree (top) and MNIST (bottom), coloured by ground-truth class. [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Landmark ablation on the Swiss roll (N = 5000, 10 seeds; bands ±1 s.d.). (a) Embedding fidelity (row-wise Spearman vs. the analytic geodesic) against the landmark count M, for FPS, k-means, and random selection, with the full-rank ceiling (dashed) and the M = 2000 design point (dotted). (b) Wall-clock time on a log scale; the landmark path is an order of magnitude faster than full-rank at the design point… view at source ↗
Figure 13
Figure 13. Figure 13: Landmark embedding at scale. A 50,000-point Swiss roll (left, coloured by the spiral parameter; Gaussian noise σ = 0.05) and its full EntroPath embedding (right). M = 2000 FPS landmarks (only 4% of the points) are embedded directly, with the remaining points placed by the out-of-sample projection of Section 7.1, at a scale where full-rank MDS is infeasible. The manifold remains well unrolled and the spira… view at source ↗
Figure 14
Figure 14. Figure 14: Row-wise Spearman correlation with the analytic geodesic vs. graph connectivity [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Robustness to input noise (uniform Swiss roll, N = 2000, 30 seeds; bands ±1 s.d.). Distance-level row-wise Spearman between each method’s distance matrix and the noise-immune analytic geodesic, versus input noise σ. EntroPath is most accurate at every σ and degrades most gracefully; its lead over the single-path methods (Isomap and Shortest Path, which share one distance matrix at this level and so appear… view at source ↗
Figure 16
Figure 16. Figure 16: Root-anchored pseudotime on Paul15 for DTNE (left) and EntroPath (right), coloured by inferred pseudotime [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Schwartz

    Mukund Balasubramanian and Eric L. Schwartz. The isomap algorithm and topological stability.Science, 295 (5552):7–7, 2002. doi:10.1126/science.295.5552.7a. URL https://www.science.org/doi/abs/10.1126/ science.295.5552.7a

    work page doi:10.1126/science.295.5552.7a 2002

  2. [2]

    Theis, Heiko Lickert, Mostafa Bakhti, Allon Klein, and Barbara Treutlein

    Aimée Bastidas-Ponce, Sophie Tritschler, Leander Dony, Katharina Scheibner, Marta Tarquis-Medina, Ciro Salinno, Silvia Schirge, Ingo Burtscher, Anika Böttcher, Fabian J. Theis, Heiko Lickert, Mostafa Bakhti, Allon Klein, and Barbara Treutlein. Comprehensive single cell mrna profiling reveals a detailed roadmap for pancreatic endocrinogenesis.Development, ...

    work page doi:10.1242/dev.173849 2019

  3. [3]

    Laplacian eigenmaps for dimensionality reduction and data representation

    Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 06 2003. ISSN 0899-7667. doi:10.1162/089976603321780317. URL https://doi.org/10.1162/089976603321780317

    work page doi:10.1162/089976603321780317 2003

  4. [4]

    Convergence of laplacian eigenmaps

    Mikhail Belkin and Partha Niyogi. Convergence of laplacian eigenmaps. In Bernhard Schölkopf, John Platt, and Thomas Hofmann, editors,Advances in Neural Information Processing Systems 19: Proceedings of the 2006 Conference. The MIT Press, 09 2007. ISBN 9780262256919. doi:10.7551/mitpress/7503.003.0021. URL https://doi.org/10.7551/mitpress/7503.003.0021

    work page doi:10.7551/mitpress/7503.003.0021 2006

  5. [5]

    Graph approximations to geodesics on embedded manifolds

    Mira Bernstein, Vin de Silva, John C Langford, and Joshua B Tenenbaum. Graph approximations to geodesics on embedded manifolds. Technical report, Michigan State University, 2000

    work page 2000

  6. [6]

    Burda, J

    Z. Burda, J. Duda, J. M. Luck, and B. Waclaw. Localization of the maximal entropy random walk.Phys. Rev. Lett., 102:160602, Apr 2009. doi:10.1103/PhysRevLett.102.160602. URL https://link.aps.org/doi/10. 1103/PhysRevLett.102.160602

    work page doi:10.1103/physrevlett.102.160602 2009

  7. [7]

    Coifman and Stéphane Lafon

    Ronald R. Coifman and Stéphane Lafon. Diffusion maps.Applied and Computational Harmonic Anal- ysis, 21(1):5–30, 2006. ISSN 1063-5203. doi:https://doi.org/10.1016/j.acha.2006.04.006. URL https: //www.sciencedirect.com/science/article/pii/S1063520306000546. Special Issue: Diffusion Maps and Wavelets

    work page doi:10.1016/j.acha.2006.04.006 2006

  8. [8]

    Delvenne, S

    J.-C. Delvenne, S. N. Yaliraki, and M. Barahona. Stability of graph communities across time scales.Proceedings of the National Academy of Sciences, 107(29):12755–12760, 2010. doi:10.1073/pnas.0903215107. URL https: //www.pnas.org/doi/abs/10.1073/pnas.0903215107

    work page doi:10.1073/pnas.0903215107 2010

  9. [9]

    Donoho and Carrie Grimes

    David L. Donoho and Carrie Grimes. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data.Proceedings of the National Academy of Sciences, 100(10):5591–5596, 2003. doi:10.1073/pnas.1031596100. URLhttps://www.pnas.org/doi/abs/10.1073/pnas.1031596100

    work page doi:10.1073/pnas.1031596100 2003

  10. [10]

    Statistical-mechanical approach to subgraph centrality in complex networks.Chemical Physics Letters, 439(1):247–251, 2007

    Ernesto Estrada and Naomichi Hatano. Statistical-mechanical approach to subgraph centrality in complex networks.Chemical Physics Letters, 439(1):247–251, 2007. ISSN 0009-2614. doi:https://doi.org/10.1016/j.cplett.2007.03.098. URL https://www.sciencedirect.com/science/ article/pii/S0009261407004058

    work page doi:10.1016/j.cplett.2007.03.098 2007

  11. [11]

    Alexander Wolf, Florian Buettner, and Fabian J

    Laleh Haghverdi, Maren Büttner, F. Alexander Wolf, Florian Buettner, and Fabian J. Theis. Diffusion pseudo- time robustly reconstructs lineage branching.Nature Methods, 13(10):845–848, Oct 2016. ISSN 1548-7105. doi:10.1038/nmeth.3971. URLhttps://doi.org/10.1038/nmeth.3971. 20 arXivTemplateA PREPRINT

    work page doi:10.1038/nmeth.3971 2016

  12. [12]

    Graph laplacians and their convergence on random neighborhood graphs.J

    Matthias Hein, Jean-Yves Audibert, and Ulrike von Luxburg. Graph laplacians and their convergence on random neighborhood graphs.J. Mach. Learn. Res., 8:1325–1370, December 2007. ISSN 1532-4435

    work page 2007

  13. [13]

    A heat diffusion perspective on geodesic preserving dimensionality reduction

    Guillaume Huguet, Alexander Tong, Edward De Brouwer, Yanlei Zhang, Guy Wolf, Ian Adelstein, and Smita Krishnaswamy. A heat diffusion perspective on geodesic preserving dimensionality reduction. In A. Oh, T. Nau- mann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors,Advances in Neural Information Processing Systems, volume 36, pages 6986–7016. Cu...

    work page 2023

  14. [14]

    Kastriti, Peter Lönnerberg, Alessandro Furlan, Jean Fan, Lars E

    Gioele La Manno, Ruslan Soldatov, Amit Zeisel, Emelie Braun, Hannah Hochgerner, Viktor Petukhov, Katja Lidschreiber, Maria E. Kastriti, Peter Lönnerberg, Alessandro Furlan, Jean Fan, Lars E. Borm, Zehua Liu, David van Bruggen, Jimin Guo, Xiaoling He, Roger Barker, Erik Sundström, Gonçalo Castelo-Branco, Patrick Cramer, Igor Adameyko, Sten Linnarsson, and ...

    work page doi:10.1038/s41586-018-0414-6 2018

  15. [15]

    and Peres, Yuval , TITLE =

    David A. Levin and Yuval Peres.Markov Chains and Mixing Times. American Mathematical Society, 10 2017. ISBN 9781470429621. doi:10.1090/mbk/107

    work page doi:10.1090/mbk/107 2017

  16. [16]

    Path integral based convolution and pooling for graph neural networks

    Zheng Ma, Junyu Xuan, Yu Guang Wang, Ming Li, and Pietro Liò. Path integral based convolution and pooling for graph neural networks. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors,Advances in Neural Information Processing Systems, volume 33, pages 16421–16433. Cur- ran Associates, Inc., 2020. URL https://proceedings.neurips.cc/...

    work page 2020

  17. [17]

    UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

    Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction.arXiv, 02 2018. doi:10.48550/arXiv.1802.03426

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1802.03426 2018

  18. [18]

    Visualizing structure and transitions in high-dimensional biological data.Nature Biotechnology, 37(12):1482–1492, December 2019

    Kevin R Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B Burkhardt, William S Chen, Kristina Yim, Antonia van den Elzen, Matthew J Hirn, Ronald R Coifman, Natalia B Ivanova, Guy Wolf, and Smita Krishnaswamy. Visualizing structure and transitions in high-dimensional biological data.Nature Biotechnology, 37(12):1482–1492, December 2019. ISSN 1087-0...

    work page doi:10.1038/s41587-019-0336-3 2019

  19. [19]

    Mervin E. Muller. A note on a method for generating points uniformly on n-dimensional spheres.Commun. ACM, 2(4):19–20, April 1959. ISSN 0001-0782. doi:10.1145/377939.377946. URL https://doi.org/10.1145/ 377939.377946

    work page doi:10.1145/377939.377946 1959

  20. [20]

    Hamey, Blanca Pijuan Sala, Evangelia Diamanti, Mairi Shepherd, Elisa Laurenti, Nicola K

    Sonia Nestorowa, Fiona K. Hamey, Blanca Pijuan Sala, Evangelia Diamanti, Mairi Shepherd, Elisa Laurenti, Nicola K. Wilson, David G. Kent, and Berthold Göttgens. A single-cell resolution map of mouse hematopoietic stem and progenitor cell differentiation.Blood, 128(8):e20–e31, 08 2016. ISSN 0006-4971. doi:10.1182/blood- 2016-05-716480. URLhttps://doi.org/1...

    work page doi:10.1182/blood- 2016

  21. [21]

    J. K. Ochab and Z. Burda. Maximal entropy random walk in community detection.The European Physical Journal Special Topics, 216(1):73–81, Jan 2013. ISSN 1951-6401. doi:10.1140/epjst/e2013-01730-6. URL https://doi.org/10.1140/epjst/e2013-01730-6

    work page doi:10.1140/epjst/e2013-01730-6 2013

  22. [22]

    Transcriptional heterogeneity and lineage commitment in myeloid progenitors.Cell, 163(7):1663–1677, 2015

    Franziska Paul, Ya’ara Arkin, Amir Giladi, Diego Adhemar Jaitin, Ephraim Kenigsberg, Hadas Keren-Shaul, Deborah Winter, David Lara-Astiaso, Meital Gury, Assaf Weiner, Eyal David, Nadav Cohen, Felicia Kathrine Bratt Lauridsen, Simon Haas, Andreas Schlitzer, Alexander Mildner, Florent Ginhoux, Steffen Jung, Andreas Trumpp, Bo Torben Porse, Amos Tanay, and I...

    work page doi:10.1016/j.cell.2015.11.013 2015

  23. [23]

    Roweis and Lawrence K

    Sam T. Roweis and Lawrence K. Saul. Nonlinear dimensionality reduction by locally linear embedding.Science, 290(5500):2323–2326, 2000. doi:10.1126/science.290.5500.2323. URL https://www.science.org/doi/ abs/10.1126/science.290.5500.2323

    work page doi:10.1126/science.290.5500.2323 2000

  24. [24]

    Satpathy, Jeffrey M

    Ansuman T. Satpathy, Jeffrey M. Granja, Kathryn E. Yost, Yanyan Qi, Francesca Meschi, Geoffrey P. McDermott, Brett N. Olsen, Maxwell R. Mumbach, Sarah E. Pierce, M. Ryan Corces, Preyas Shah, Jason C. Bell, Darisha Jhutty, Corey M. Nemec, Jean Wang, Li Wang, Yifeng Yin, Paul G. Giresi, Anne Lynn S. Chang, Grace X. Y . Zheng, William J. Greenleaf, and Howar...

    work page doi:10.1038/s41587-019-0206-z 2019

  25. [25]

    Nolan, Benjamin J

    Rachel Shahan, Che-Wei Hsu, Trevor M. Nolan, Benjamin J. Cole, Isaiah W. Taylor, Laura Greenstreet, Stephen Zhang, Anton Afanassiev, Anna Hendrika Cornelia Vlot, Geoffrey Schiebinger, Philip N. Benfey, and Uwe Ohler. A single-cell arabidopsis root atlas reveals developmental trajectories in wild-type and cell identity mutants. 21 arXivTemplateA PREPRINT D...

    work page doi:10.1016/j.devcel.2022.01.008 2022

  26. [26]

    Global versus local methods in nonlinear dimensionality reduction

    Vin Silva and Joshua Tenenbaum. Global versus local methods in nonlinear dimensionality reduction. In S. Becker, S. Thrun, and K. Obermayer, editors,Advances in Neural Information Processing Systems, vol- ume 15. MIT Press, 2002. URL https://proceedings.neurips.cc/paper_files/paper/2002/file/ 5d6646aad9bcc0be55b2c82f69750387-Paper.pdf

    work page 2002

  27. [27]

    Tenenbaum, Vin de Silva, and John C

    Joshua B. Tenenbaum, Vin de Silva, and John C. Langford. A global geometric framework for nonlinear dimensionality reduction.Science, 290(5500):2319–2323, 2000. doi:10.1126/science.290.5500.2319. URL https://www.science.org/doi/abs/10.1126/science.290.5500.2319

    work page doi:10.1126/science.290.5500.2319 2000

  28. [28]

    Lennon, Kenneth J

    Cole Trapnell, Davide Cacchiarelli, Jonna Grimsby, Prapti Pokharel, Shuqiang Li, Michael Morse, Niall J. Lennon, Kenneth J. Livak, Tarjei S. Mikkelsen, and John L. Rinn. The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells.Nature Biotechnology, 32(4):381–386, Apr 2014. ISSN 1546-1696. doi:10.1038/nbt.2...

    work page doi:10.1038/nbt.2859 2014

  29. [29]

    Visualizing data using t-sne.Journal of Machine Learning Research, 9(86):2579–2605, 2008

    Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne.Journal of Machine Learning Research, 9(86):2579–2605, 2008. URLhttp://jmlr.org/papers/v9/vandermaaten08a.html

    work page 2008

  30. [30]

    S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients.Com- munications on Pure and Applied Mathematics, 20(2):431–455, 1967. doi:https://doi.org/10.1002/cpa.3160200210. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200210

    work page doi:10.1002/cpa.3160200210 1967

  31. [31]

    Local multidimensional scaling.Neural Networks, 19(6):889–899, 2006

    Jarkko Venna and Samuel Kaski. Local multidimensional scaling.Neural Networks, 19(6):889–899, 2006. ISSN 0893-6080. doi:https://doi.org/10.1016/j.neunet.2006.05.014. URL https://www.sciencedirect.com/ science/article/pii/S0893608006000724. Advances in Self Organising Maps - WSOM’05

    work page doi:10.1016/j.neunet.2006.05.014 2006

  32. [32]

    Diffusive topology preserving manifold distances for single-cell data analysis.Proceedings of the National Academy of Sciences, 122 (4):e2404860121, 2025

    Jiangyong Wei, Bin Zhang, Qiu Wang, Tianshou Zhou, Tianhai Tian, and Luonan Chen. Diffusive topology preserving manifold distances for single-cell data analysis.Proceedings of the National Academy of Sciences, 122 (4):e2404860121, 2025. doi:10.1073/pnas.2404860121. URL https://www.pnas.org/doi/abs/10.1073/ pnas.2404860121

    work page doi:10.1073/pnas.2404860121 2025

  33. [33]

    Alexander Wolf, Philipp Angerer, and Fabian J

    F. Alexander Wolf, Philipp Angerer, and Fabian J. Theis. Scanpy: large-scale single-cell gene expression data analysis.Genome Biology, 19(1):15, Feb 2018. ISSN 1474-760X. doi:10.1186/s13059-017-1382-0. URL https://doi.org/10.1186/s13059-017-1382-0

    work page doi:10.1186/s13059-017-1382-0 2018

  34. [34]

    Self-tuning spectral clustering

    Lihi Zelnik-Manor and Pietro Perona. Self-tuning spectral clustering. In L. Saul, Y . Weiss, and L. Bottou, editors, Advances in Neural Information Processing Systems, volume 17. MIT Press, 2004. URLhttps://proceedings. neurips.cc/paper_files/paper/2004/file/40173ea48d9567f1f393b20c855bb40b-Paper.pdf. 22 arXivTemplateA PREPRINT Appendix A Additional Exper...

    work page 2004