Hypergraphs on high dimensional time series sets using signature transform

Paul Minchella; R\'emi Vaucher

REVIEW 4 major objections 5 minor 12 references

Hypergraphs on high dimensional time series sets using signature transform

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that averaging hyper-adjacency tensors over randomly subsampled time points builds a stable simplicial complex over a collection of multivariate time series, and that this is the first method to do so.

desk verdict Plausible incremental extension with an honest preliminary result, but the central linearity assumption is unproven and the higher-order case is not even dimensionally defined; the paper deserves referee time but needs serious revision. read the letter →

arxiv 2507.15802 v1 pith:T7DP4TAF submitted 2025-07-21 stat.ML cs.LGstat.CO

Rémi Vaucher , Paul Minchella This is my paper · ORCID

classification stat.MLcs.LGstat.CO MSC 55N3162M1060L10

keywords multivariatetimeserieshypergraphsimplicialcomplexsignaturetransformtopologicaldataanalysishyper-adjacencytensorrandomsubsamplingLasso

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that a hypergraph (a simplicial complex in the TDA language) can be built over a collection of multivariate time series, not just over a single one. The authors take the existing signature-based construction for one series, treat each series as a vertex, and add controlled randomness by repeatedly subsampling the time points and averaging the resulting hyper-adjacency tensors into probability tensors. They report that on synthetic nearest-neighbour ODE systems this averaged construction recovers the expected interaction edges with accuracy between 66% and 71%. If true, this would be the first method that puts higher-order structure on top of many multivariate series simultaneously, which matters for epidemiology, finance, and any field where the data is a set of related multi-channel recordings.

What carries the argument

The signature transform maps each bounded-variation path to a sequence of iterated integrals that encodes the order of the path's increments; it is used here to turn each multivariate series into a feature of the same type that can enter a linear regression. Algorithm 1 and its generalization Algorithm 2 express the signature of one vertex as a sparse linear combination of signatures of subsets of other vertices, using Lasso and an $R^2 > 0.67$ threshold to decide which subset-simplexes enter the link. The new ingredient is Algorithm 3: it picks $N_{\text{tries}}$ random subsets of the sampling times, builds the simplicial complex on each subset, stores the result as a hyper-adjacency tensor, and averages these tensors into a probability tensor. That average is the object claimed to be a stable estimate of each potential hyperedge's probability, which can then be thresholded into a fixed complex.

What would settle it

Run Algorithm 3 on a synthetic ensemble where the ground-truth interactions are known and strictly nonlinear (for example, a series coupled to its neighbours through a product or a threshold function), keeping everything else fixed; if the recovered hypergraph no longer matches the planted edges, the failure is evidence that the sparse-linear signature assumption, not just the threshold, is the load-bearing part. A cleaner disambiguation is to compare the recovered edges under the $R^2 > 0.67$ cutoff against the ranking of Lasso coefficients across many subsamples: if the same false edges keep appearing with high probability but low $R^2$, the threshold is doing the work; if true edges disappear when the linear approximation is poor, the model is doing the work.

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Extended reading notes

Core claim

The central claim is that Algorithm 3—subsample the observation times, run the signature regression on each subsample, and average the hyper-adjacency tensors—yields a probability tensor whose thresholding recovers the simplicial complex of genuine interactions among a collection of multivariate time series. The paper presents this as a generalization of the single-series algorithm of [3] to the setting where each vertex is itself a multidimensional path, and it states that to the authors' knowledge no prior method builds high-dimensional structure over this kind of data. Validation is limited to 1-dimensional complexes (ordinary edges) on synthetic ODE systems where each series is coupled only to its neighbours; the reported accuracy is 66–71% depending on the number of vertices, with precision falling from 0.56 to 0.45 as the network grows from 5 to 8 vertices. The authors explicitly label the results as preliminary and list as future work the refinement of metrics, hyperparameter optimization, and testing on 2-dimensional hyperedges.

Load-bearing premise

The whole construction rests on the unproven assumption that the signature of a vertex can be written as a sparse linear combination of signatures of subsets of the other vertices, and that the Lasso coefficients surviving the arbitrary $R^2 > 0.67$ cutoff correspond exactly to real interactions; if that linear model is wrong, the recovered hypergraph has no guaranteed meaning.

Editorial extensions

If this is right

If the averaging construction works, a collection of multivariate time series can be summarized by a single simplicial complex whose edges reflect probable higher-order interactions, enabling TDA filtrations on top of the collection.
The method yields a per-edge probability rather than a hard decision, so thresholds can be tuned to the application's precision/recall trade-off.
The random subsampling makes the construction stable under noise in the time series, since edges that only appear by chance in one subsample wash out in the average.
The approach extends naturally to higher-order simplexes (triangles, tetrahedra) once synthetic datasets with dimension-2 hyperedges are generated, as the authors state they plan to do.
The framework handles series with missing or heterogeneous dimensions through projection or augmentation, so it can be applied when channels differ across vertices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test is to run Algorithm 3 on coupled systems with known nonlinear or multiplicative interactions; the recovered hypergraph should fail in a characteristic way if the sparse-linear signature assumption is the wrong model.
The probability tensors themselves could be fed into a persistence filtration without thresholding, replacing the fixed-threshold choice with scale-space analysis of the hypergraph.
The random subsampling step is reminiscent of ensemble or bagging ideas, so the variance of the probability tensor across tries could be used to build a confidence interval per potential hyperedge, something the paper does not discuss.
Since the authors only test 5 to 8 vertices, a useful stress test is scaling to 50 or 100 vertices where the number of candidate subsets explodes; how the Lasso step scales is unclear.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is small but real: prior work built a simplicial complex from a single multivariate time series, and this paper adapts that construction to a collection of multivariate series, adding random time-subsampling with averaged tensors. That extension is coherent for the pairwise case, and the synthetic experiments do what they claim—66–71% accuracy against an independently generated ground-truth adjacency. The authors are also honest that this is preliminary, and they flag their own limitations (only pairwise edges, no hyperparameter optimization, only LASSO). Credit where due: the code is linked, the setup is simple enough to reproduce, and the writing is clear.

The soft spots are real and roughly match the reader's report. First, Equation (1) is the load-bearing assumption: vertex signatures are modeled as sparse linear combinations of subset signatures, and coefficients survive only if R^2 > 0.67. No derivation, consistency result, or ablation isolates that linearity from the threshold, the truncation order, or the subsampling parameters. The 66–71% accuracy does not establish that the recovered hypergraph has meaning outside this specific ODE family. Second, for k>1 the equation is dimensionally ambiguous as written: S_i lives in T(R^d) while S_{i1...ik} lives in T(R^{kd}), and no alignment or projection is specified. Since only k=1 edges are tested, the paper does not validate the hypergraph claim in the general setting. Third, the experiments report point estimates without error bars despite Algorithm 3 being randomized, and key hyperparameters (l, N_tries) are unreported if they were set. Finally, the “first method” claim is stated without a comparison; given the cited literature, it may be true, but it is not demonstrated.

None of this is fatal to the core idea. The paper is a reasonable workshop-level contribution that could grow into a solid journal paper with: error bars, a null-model test, at least one real-data case, and a careful treatment of the k>1 alignment issue. The stress-test note about Eq. (1) holding for k>1 is correct, and the authors should be pushed to address it directly.

Who is this for? Researchers working on TDA over time series who want a quick look at a plausible extension. It deserves a serious referee, but the referee should ask for the missing experiments and a fix to the higher-order definition before publication.

Referee Report

4 major / 5 minor

Summary. The paper proposes an extension of a signature-transform-based algorithm from Chrétien et al. to build hypergraphs (simplicial complexes) over a collection of multivariate time series. For each vertex, the algorithm computes the path signature of that vertex and regresses it, via Lasso, on signatures of other vertices or subsets of vertices; coefficients surviving a fixed R^2 threshold define hyperedges. To increase robustness, Algorithm 3 repeats this construction on randomly subsampled time points and averages the resulting hyper-adjacency tensors into "probability tensors." The method is validated only on synthetic ODE chain data with n = 5,...,8 vertices, restricted to 1-dimensional simplicial complexes (ordinary edges), with reported accuracy between 66% and 71%.

Significance. If the claimed extension is established, constructing higher-order structure over collections of multivariate time series would be a useful addition to the TDA/signature toolbox, and the paper's idea of averaging adjacency tensors over random time subsamples is a reasonable robustness heuristic. The paper deserves credit for providing a code link and for evaluating against an independently generated ground-truth adjacency derived from the ODE model, which avoids direct circularity. However, the current evidence is preliminary: only pairwise edges are tested, the central linear-signature assumption is not derived or isolated from thresholds, and the randomized algorithm is reported without variance information. The significance of the claimed 'first method' for high-dimensional structure over collections of multivariate time series is therefore not yet established by the presented experiments.

major comments (4)

[Algorithm 1, Eq. (1)] For k > 1, Eq. (1) is dimensionally ill-defined as written: the signature S_i of a single multivariate vertex lies in T(R^d), whereas the signature S_{i1...ik} of the k-dimensional path formed by k vertices lies in T(R^{kd}); no embedding, concatenation order, or alignment is specified. Since Section 3 explicitly restricts experiments to k = 1, the reported 66-71% accuracy does not validate the hypergraph (higher-order edge) claim. The authors should either specify a dimensionally consistent definition for k > 1 and test k = 2, or limit the central claims to graph construction.
[Section 3, Table 1] All metrics in Table 1 concern 1-dimensional links only; no true hyperedges of dimension 2 or higher are generated or evaluated. The conclusion that this is 'the first method to build high dimensional structure over this type of data' is stronger than the evidence presented, because the only tested structure is an ordinary graph. Higher-order experiments with [v_i v_j v_k] hyperedges, as the authors acknowledge as future work, are necessary before a hypergraph claim can be sustained.
[Eq. (1) and Section 3] The recovery procedure is built on the unproven modeling assumption that a vertex signature is a sparse linear combination of subset signatures, and the threshold R^2 > 0.67 is arbitrary. No theorem, consistency result, null-model test, or ablation separates the linearity assumption from the truncation order, Lasso penalty, R^2 threshold, or subsampling parameters. A permutation/independence test or a synthetic setup where the interaction structure is known but the linear-signature relation is deliberately violated would clarify whether the 66-71% accuracy reflects the model assumption or the method's actual signal.
[Section 2.2 and Table 1] Algorithm 3 is randomized through uniform subsampling of l time points, but Table 1 reports point estimates with no error bars, and the paper does not state the values of l and N_tries used nor the random seeds. The claim that averaging over random subsamples 'enhances robustness' is therefore not quantitatively supported. At minimum, the experiments should report means and standard deviations over repeated runs of Algorithm 3 and specify all hyperparameter values.

minor comments (5)

[Section 1.4] Definition 3 is not the standard definition of a simplicial complex; the condition 'for every sigma_i in C, there exists j with sigma_i ∩ sigma_j a sub-simplex of both' does not express closure under faces. This should be corrected or replaced by the standard face-closure condition, or the paper should consistently use hypergraph terminology.
[Section 3] The synthetic ODE parameters c, h, sigma_epsilon, and sigma_startId are not reported, and the signature truncation order is said to be fixed 'at values that seemed correct with respect to literature' without giving the actual value. These omissions prevent replication of the experiments.
[Algorithm 3] The text says probability tensors 'can be used to' create a fixed complex by thresholding or by Metropolis Hastings, but no thresholding rule or MH procedure is specified; a short description of the thresholding used in Section 3 would make the experimental pipeline complete.
[Introduction and Related Work] The claim that Algorithm 2 'appears to be the only approach capable of being extended' to collections of multivariate time series is not supported by a systematic literature comparison; the cited related work is a small set, and the phrasing should be softened or substantiated.
[Throughout] There are several typographical and notational slips, such as 'actual' for 'actually' in the second future-work item, the unusual phrase 'R2 > 0.67 (for a model quality insurance)', and inconsistent use of k in Algorithm 2 versus K. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the predicted hypergraph is evaluated against an independently generated ground-truth adjacency, and the self-citation to the authors' Algorithm 2 is provenance, not proof.

full rationale

The paper's derivation chain is not circular. Algorithm 3 predicts hyper-adjacency tensors from signature/Lasso regressions (Eq. 1) and then averages over random time subsamples; the experimental target is the independently defined adjacency A2 = Id + D_{-1} + D_{1} of the synthetic ODE system in Eq. (4). No fitted parameter is relabeled as a prediction: hyperparameters are fixed 'at values that seemed correct with respect to literature', and the ground truth is not used to select Lasso coefficients or thresholds. The core method is inherited from the authors' [3], but it is restated in this paper as Algorithm 1/2 and Eq. (1), so the citation is provenance rather than a load-bearing authority. The paper's admitted limitations (only 1-dimensional complexes tested, only LASSO used, future hyperparameter optimization) are correctness and completeness concerns, not circularity. In particular, Eq. (1)'s unproven sparse-linear-signature ansatz and its dimensional ambiguity for higher-order hyperedges are modeling risks, but they do not make the output equivalent to the input by construction. The evaluation against an external ground truth keeps the claim self-contained; the appropriate finding is no circularity, not a reduction to input.

Assumptions & free parameters 6 free parameters · 4 assumptions · 1 invented entities

The method leans on a sparse linear regression model for signatures (Equation 1) and on several hand-chosen thresholds and hyperparameters. The synthetic benchmark adds generative ODE parameters that influence the reported accuracy but are not varied. No new physical entity is introduced; probability tensors are a bookkeeping summary of repeated runs.

free parameters (6)

R^2 threshold > 0.67 = 0.67
Used in Algorithm 1 to decide whether a regression coefficient corresponds to an edge; chosen by hand with no sensitivity analysis.
Signature truncation order = not reported
Order of truncated signature fixed at values that seemed correct from literature (Section 3), but not listed for the experiments.
Subsample size l = not reported
Number of time points selected per trial in Algorithm 3; required input not specified in the experiments.
Number of tries N_tries = not reported
Number of random subsamples in Algorithm 3; required input not specified in the experiments.
Lasso sparsity coefficient = not reported
Regularization weight for variable selection in Algorithm 1; mentioned as a future hyperparameter to optimize, not given.
Synthetic ODE parameters (c, h, sigma_epsilon, sigma_startId) = not reported
Equation (4) parameters chosen by hand for the synthetic benchmark, with no sensitivity analysis or range reported.

assumptions (4)

domain assumption Equation (1): the signature of a vertex is a sparse linear combination of signatures of other vertex subsets.
Algorithm 1 assumes this regression represents dependency structure; no proof or consistency result is given.
domain assumption The signature transform captures enough path information for Lasso to recover interaction structure.
The paper invokes signature properties from [2,4,5] without stating which property justifies the link-recovery regression.
domain assumption The ground-truth adjacency for Equation (4) is exactly Identity plus nearest-neighbor shifts.
Used to score Table 1; delayed term B1 Y_i(t-h) and noise epsilon can in principle produce non-adjacent or self lagged dependencies not reflected in A2.
standard math Bounded-variation path and signature existence.
Standard setting for signature transforms; stated in Section 1.3.

invented entities (1)

Probability tensors (mu_{A_i})
purpose: Summarize the average hyperedge probability over random time subsamples in Algorithm 3.
They are a constructed summary of the algorithm's own outputs; the paper provides no external falsifiable prediction tied to them.

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Cite this review

Pith. "Pith review of Hypergraphs on high dimensional time series sets using signature transform." pith.science (2026). https://pith.science/paper/T7DP4TAF

@misc{pith2026250715802,
  author       = {Pith},
  title        = {Pith review of: Hypergraphs on high dimensional time series sets using signature transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7DP4TAF}},
  note         = {Machine review of arXiv:2507.15802}
}

read the original abstract

In recent decades, hypergraphs and their analysis through Topological Data Analysis (TDA) have emerged as powerful tools for understanding complex data structures. Various methods have been developed to construct hypergraphs -- referred to as simplicial complexes in the TDA framework -- over datasets, enabling the formation of edges between more than two vertices. This paper addresses the challenge of constructing hypergraphs from collections of multivariate time series. While prior work has focused on the case of a single multivariate time series, we extend this framework to handle collections of such time series. Our approach generalizes the method proposed in Chretien and al. by leveraging the properties of signature transforms to introduce controlled randomness, thereby enhancing the robustness of the construction process. We validate our method on synthetic datasets and present promising results.

Figures

Figures reproduced from arXiv: 2507.15802 by the authors.

**Figure 2.** The explicit graph between each univariate time series [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Trajectories Yij for all i, j. dimensional structure over this type of data. Our preliminary results demonstrate promising performance, with accuracy rates ranging from 66% to 71% across different network sizes. – In future works, metrics must be refined. – We need to optimize hyperparameters: order of truncation for signature, sparsity coefficient for variable selection algorithm, etc. – Only LASSO was used as var… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Y1 and Y2 only. Number of vertices n = 5 n = 6 n = 7 n = 8 TP 0.72 0.75 0.67 0.68 TN 0.62 0.68 0.68 0.72 Accuracy 0.66 0.71 0.67 0.71 Precision 0.56 0.54 0.45 0.45 Recall 0.72 0.75 0.67 0.68 F1-score 0.63 0.63 0.54 0.54 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 8 canonical work pages

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, " * write output.state after.block = add.period write

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[12]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

The annals of applied statistics 9(3), 1103 (2015)

Bogdan, M., Van Den Berg, E., Sabatti, C., Su, W., Cand \`e s, E.J.: Slope—adaptive variable selection via convex optimization. The annals of applied statistics 9(3), 1103 (2015)

work page 2015

[2] [2]

arXiv preprint arXiv:1603.03788 (2016)

Chevyrev, I., Kormilitzin, A.: A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016)

arXiv 2016

[3] [3]

In: Complex Networks & Their Applications XII: Proceedings of The Twelfth International Conference on Complex Networks and their Applications: COMPLEX NETWORKS 2023 Volume 1

Chr \'e tien, S., Gao, B., Guiochon, A.T., Vaucher, R.: Leveraging the power of signatures for the construction of topological complexes for the analysis of multivariate complex dynamics. In: Complex Networks & Their Applications XII: Proceedings of The Twelfth International Conference on Complex Networks and their Applications: COMPLEX NETWORKS 2023 Volu...

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[4] [4]

Fermanian, A.: Learning time-dependent data with the signature transform. Ph.D. thesis, Sorbonne Universit \'e (2021)

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[5] [5]

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Lyons, T., McLeod, A.D.: Signature methods in machine learning. arXiv preprint arXiv:2206.14674 (2022)

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[6] [6]

Journal of The Royal Society Interface 11(101), 20140873 (2014)

Petri, G., Expert, P., Turkheimer, F., Carhart-Harris, R., Nutt, D., Hellyer, P.J., Vaccarino, F.: Homological scaffolds of brain functional networks. Journal of The Royal Society Interface 11(101), 20140873 (2014)

work page 2014

[7] [7]

Sparse Graphical Modelling via the Sorted L$_1$-Norm

Riccobello, R., Bogdan, M., Bonaccolto, G., Kremer, P.J., Paterlini, S., Sobczyk, P.: Sparse graphical modelling via the sorted 1-norm. arXiv preprint arXiv:2204.10403 (2022)

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Nature Physics 19(2), 221--229 (2023)

Santoro, A., Battiston, F., Petri, G., Amico, E.: Unveiling the higher-order organization of multivariate time series. Nature Physics 19(2), 221--229 (2023)

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