Discrete signature tensors for persistence landscapes

Daniel Tolosa; Heather A. Harrington; Vincenzo Galgano

REVIEW 2 major objections 5 minor 9 references

Discrete signature tensors for persistence landscapes

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Two time-series have the same discrete signature if and only if they are equivalent under time-warping and time-translation, and the paper uses this to build a discriminative feature map for persistence landscapes that separates barcodes…

desk verdict The DLFM is a sensible idea, but the critical-point lemma that defines the time-series omits actual crossing points of overlapping intervals, so the construction is not faithful as stated, and the advertised stability theorem is vacuous. read the letter →

arxiv 2505.02800 v3 pith:L7GFUIVC submitted 2025-05-05 math.AT math.FA

Vincenzo Galgano , Heather A. Harrington , Daniel Tolosa This is my paper · ORCID

classification math.ATmath.FA MSC 55N3168T0946B85

keywords persistenthomologybarcodespersistencelandscapesfeaturemapsdiscretesignaturetensorstime-warpingequivalencevectorisationknottedproteins

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

The paper introduces the discrete landscape feature map (DLFM), which turns a persistence barcode into the time-series of critical values of its persistence landscape and then takes the discrete signature tensor of that time-series. Its central claim is that the discrete signature recovers a time-series exactly up to time-warping and time-translation: $\Sigma(x)=\Sigma(y)$ if and only if $x\sim_{\mathrm{twt}} y$. This makes the DLFM more discriminating than the usual composition of the landscape embedding with the continuous path signature, which conflates many barcodes via tree-like equivalence. The paper also shows that on a finite set of barcodes a truncated version of the map is uniformly continuous with respect to the bottleneck distance, and it reports statistically significant agreement between DLFM clusters and structural classes in a knotted-protein dataset.

What carries the argument

The load-bearing object is the discrete signature tensor $\Sigma(x)$ of a time-series $x=(x_1,\ldots,x_n)\in \mathbb{R}^d$, whose coefficient for $p_1\otimes\cdots\otimes p_\ell$ is the sum of $\prod_j p_j(x_{i_j+1}-x_{i_j})$ over strictly increasing indices $i_1<\cdots<i_\ell$. This tensor is invariant under time-warping and time-translation, and under the shifted concatenation $\{x|y\}$ it satisfies the discrete Chen relation $\Sigma(\{x|y\})=\Sigma(x)\otimes\Sigma(y)$, making the discrete signature a monoid homomorphism. The paper proves that the kernel of this homomorphism is exactly the equivalence $\sim_{\mathrm{twt}}$, which is what makes the DLFM discriminative. For barcodes, the input time-series is obtained by evaluating the persistence landscape at the critical points listed in Lemma 3.7, a canonical description of the landscape by its breakpoints.

What would settle it

Compute the full discrete signatures of two time-warping-reduced time-series $x,y$ that are not related by $\sim_{\mathrm{twt}}$; Theorem 3.4 predicts no such pair exists, so any computed counterexample would refute the injectivity claim. In one dimension, such a pair would have to share all elementary symmetric functions and all mixed power-sum-like coefficients of their consecutive differences while differing in their order.

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Extended reading notes

Core claim

The core result (Theorem 3.4) is that any two time-series in $\mathbb{R}^d$ have the same discrete signature if and only if one is obtained from the other by time-warping (inserting or deleting repeated consecutive points) and time-translation (adding a constant vector). In particular, the quotient map $\mathrm{TS}(\mathbb{R}^d)/\sim_{\mathrm{twt}}\hookrightarrow T((V_d))$ is injective. The proof first reduces to time-warping-reduced series, shows equal signatures force equal length, then in one dimension reads signature coefficients as elementary symmetric functions in the consecutive differences, and finally uses mixed monomial coefficients to align the permutation across coordinates. The paper notes the same theorem was proved independently in a more general setting in [DET22]. For barcodes, the input series is the list of landscape critical points given by Lemma 3.7 from [Bub20], so the DLFM inherits the injectivity statement; the authors emphasize that this distinguishes landscapes whose continuous signatures are tree-like equivalent.

Load-bearing premise

The construction assumes that a persistence landscape is completely determined by the time-series of its critical values at a fixed number of levels $d$; if critical points were omitted, or if truncating to $d$ levels discarded informative structure, the DLFM would be less descriptive than claimed.

Editorial extensions

If this is right

Because the DLFM quotients injectively on time-series modulo $\sim_{\mathrm{twt}}$, it separates barcodes whose landscapes the ordinary continuous signature conflates: singleton barcodes and separated multi-bar barcodes have different discrete signatures.
For a finite set of barcodes, the specialised map $\Phi^{\leq k}_{I_D}$ is uniformly continuous, so small bottleneck perturbations produce small Bombieri-Weyl changes in the signature; the map can be used as a stable input feature for classifiers on a fixed dataset.
For single-level landscapes the discrete signature is determined entirely by the persistence of the bars, so the DLFM recovers persistence information exactly in that case.
On knotted proteins, k-means on weight-3 signatures with 15 landscape levels reproduces the nine structural classes with ARI 0.958 and NMI 0.895, and the signature correlates with knot depth (Spearman 0.647, random-forest R2 0.866), both with permutation-test p-values below 0.001.

Reading between the lines

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

If the injectivity theorem holds, the DLFM becomes a faithful vectorisation: differences between discrete signature tensors can be interpreted as genuine geometric differences between barcodes, up to the arbitrary choice of level truncation.
Because the discrete signature is polynomial in consecutive differences, the DLFM invites kernel-based methods for barcodes; the paper mentions kernelisation as a future direction, and a natural next experiment is whether raising the truncation weight beyond 3 improves classification or merely adds noise.
The level count $d=15$ is chosen by hand for the protein dataset; a testable extension is adaptive level selection, for example adding levels until the discrete signature stabilises, which would make the representation comparable across datasets.

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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

Short version: the paper has a good instinct—replace the landscape path with the time-series of its critical points and feed that to the discrete signature—but the way it defines that time-series is not faithful. Lemma 3.7, which is supposed to list the critical points, misses the crossing points of overlapping tent functions. For barcode {[0,4],[2,6]} the landscape changes slope at t=3, but the lemma only gives t=1, so the time-series omits t=3. Two barcodes whose landscapes differ only at such crossings can get identical DLFM signatures. That is not a minor gap; it is load-bearing for the whole construction. The stress-test note is right.

What is genuinely new: the DLFM composition itself, the honest Remark 3.5 acknowledging [DET22, Thm 3.15] (which already had the injectivity result), the Section 2 algebraic-geometry observations, and the protein experiments. The k-means results (ARI 0.958, NMI 0.895) and the permutation tests are credible evidence that once the construction is fixed, the signature contains real signal. The github code is a plus.

Soft spots besides Lemma 3.7: Theorem 3.15 proves uniform continuity on a finite set D, and any function on a finite metric space is uniformly continuous, so the statement is vacuous. The framing should be changed—either extend the map to all barcodes with a fixed critical-point grid and prove something there, or present it as a boundedness/continuity result for the actual data set, not as a stability theorem. The application also selects d=15, k=3, and number of clusters on the same data, and k-means variance is not reported; the baseline L1-landscape k-means may not be tuned equally. These are fixable.

Who this is for: TDA people who want a discrete, computable vectorisation with some discriminative power beyond the Chen-signature landscape map. The DLFM is a reasonable direction, but the current version should not be accepted as is. It deserves serious reviewing—send it out—but the authors need to fix the critical-point lemma, re-run the experiments if the time-series changes, and reframe the stability claim. If the lemma can be corrected (the correct crossing points are of the form (d_i+b_j)/2 and (b_i+d_j)/2), the paper could become a useful addition.

Referee Report

2 major / 5 minor

Summary. The paper proposes a vectorisation of persistence barcodes called the discrete landscape feature map (DLFM), which sends a barcode to its persistence landscape, samples the landscape at its critical points to form a time-series, and then applies the discrete signature tensor map. The main theoretical results are: (Theorem 3.4) the discrete signature determines time-series up to time-warping and time-translation, with the authors noting in Remark 3.5 that this was already proved by Diehl, Ebrahimi-Fard and Tapia in greater generality; (Theorem 3.15) a uniform-continuity statement for a version of the DLFM specialised to a finite set of barcodes; and several partial results on signature matrices of piecewise-linear loops and landscapes. The paper also reports an application to knotted proteins, showing correlation between discrete-signature features and structural classes as well as knot depth, with permutation tests and publicly available code.

Significance. If the construction were fully justified, the DLFM would be a genuinely useful, computable and discriminative feature map for barcodes, and the protein experiments are encouraging: the authors provide code, explicit parameter choices, and statistical validation. The proof of Theorem 3.4 is self-contained and appears correct, and Remark 3.5 is an appropriate disclosure of overlap with the existing literature. However, the advertised stability theorem is vacuous as stated because the map is formulated on a finite domain, and the critical-point lemma on which the entire DLFM construction rests is mis-stated. These are load-bearing issues for the paper's central claims, although both appear fixable by a corrected lemma and a carefully stated stability theorem.

major comments (2)

[§3.4, Eq. (3.14), Theorem 3.15] The advertised stability result is vacuous as stated. The map Φ^{≤k}_{ID} is defined on the finite set D⊂Bar in Eq. (3.14), and every function on a finite metric space is uniformly continuous. Thus Theorem 3.15 gives no quantitative control and does not support the paper's claim that the DLFM has desirable continuity properties. If the intended statement is that the map on all of Bar obtained by evaluating landscapes on the fixed finite grid C_D is uniformly continuous, then Eq. (3.14) and the proof must be corrected, since Claim 1 of the proof explicitly restricts the landscape map to D. Please clarify the domain and either prove a non-vacuous stability statement or remove the continuity claim.
[§3.2, Lemma 3.7] The critical-point list in Lemma 3.7 is incorrect and misses crossing points of overlapping intervals. For B={[0,4],[2,6]}, the first landscape is λ_1(t)=max(min(t,4−t), min(t−2,6−t)), whose slope changes at t=0,2,3,4,6; the point t=3 is a crossing of the right slope of the first tent and the left slope of the second, and equals (d_1+b_2)/2, not (b_1+b_2)/2. Since the time-series I(B) in (3.11) is obtained by evaluating the landscape at the lemma's critical points, the DLFM may miss genuine breakpoints and therefore may not faithfully represent the landscape. The lemma (or its citation) should be corrected and the computations and protein experiments in Section 4 should be re-run with the corrected time-series.

minor comments (5)

[§3.2, Lemma 3.7] In item 4 of the lemma, the symbols b_k and d_k are not introduced and the condition 'b_k < b_i < d_j < d_i' is unclear; after the formula is corrected, please define all indices and state exactly which pairs of bars are considered.
[§3.2, Eq. (3.11)] The map I(B) requires a well-defined ordering of the critical points; please specify that the critical points are sorted in increasing order, with a tie-breaking rule, before forming the time-series.
[§3.4, proof of Theorem 3.15] The uniform-continuity argument for the discrete signature in Claim 2 assumes two time-series of the same length; this is guaranteed only after the domain of I_D is clarified, so the proof should state the length assumption explicitly.
[§4, Figures 4.1–4.5] The parameters in the application (15 landscape levels, signature weight ≤ 3, 9 k-means clusters) are hand-picked; please report how sensitive the ARI/NMI results are to neighbouring parameter choices, since the method is otherwise parameter-free only within a fixed grid.
[§3.4 and §4] There are minor typos: 'pourposes' in §3.4 and 'whevener' in §4 should be 'purposes' and 'whenever'.

Circularity Check

1 steps flagged · score 4.0 of 10

The mathematical core is independent, but the empirical validation of the DLFM on knotted proteins is circular: the three free parameters are selected to maximize the same ARI/NMI scores that are then reported as evidence.

fitted input called prediction [Sec. 4, 'Correlation between discrete signature and sequence similarity' (Figure 4.1 and surrounding text).]
"We use k-means clustering for three parameters: truncation of levels in the landscapes, truncation by weight (as defined in (3.3)) of the discrete signatures, and number of clusters for k-means. We quantify the correlation between signature-based clusters and the labelling determined by structural class representative using the Adjusted Rand Index (ARI) and Normalised Mutual Information (NMI) ... The metrics in Figure 4.1 show high similarity between the clustering ..."

The hyperparameters d=15, k=3, and the number of clusters are chosen by scanning the ARI/NMI grid on the same protein dataset. The reported ARI=0.958 and NMI=0.895 are the values for the selected parameter combination, i.e. the outcome of the optimization used to pick those parameters, rather than an independent evaluation on held-out data. The permutation test is then run at these already-selected parameters and does not account for the preceding selection, so the 'statistical significance' is evaluated on the same data that determined the configuration. This is a fitted input presented as validation of the feature map's discriminative power.

full rationale

The theoretical derivation is not circular. Theorem 3.4 is proved directly from the definition of discrete signatures and symmetric functions (Lemmas 3.2, 3.3 and the proof of Theorem 3.4), and the overlap with [DET22] is disclosed in Remark 3.5; [DET22] is not a self-citation and is not used as an unproved premise. The uniform-continuity theorem legitimately uses Bubenik's external stability lemma as an input. The DLFM's use of landscape critical points from [Bub20, Lemma 5.8] is an external mathematical input; even if the skeptically noted omission of crossing points makes that lemma's list incomplete for interleaved bars, that is a correctness risk about faithfulness of the time-series, not a circular reduction. The one real circular element is in the application section: the free parameters are tuned to maximize the same ARI/NMI scores that are then reported and validated by a permutation test that does not correct for the tuning. This affects the strength of the empirical claims, but the paper's main mathematical results remain independently established, so the overall circularity score is 4 rather than higher.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central mathematical content rests on standard TDA results (landscape stability, critical-point characterisation) and on the prior definition of discrete signatures. No new physical or mathematical entities are postulated. The only dataset-specific free parameters are the truncation levels and cluster count used in the protein experiment.

free parameters (3)

number of landscape levels d = 15
Truncation level for the landscape in the protein application; selected by scanning over levels to maximise ARI/NMI (Section 4, Figure 4.1).
signature truncation weight k = 3
Truncation weight for the discrete signature in the protein application; selected as the point where ARI=0.958 and NMI=0.895 (Section 4).
number of k-means clusters = 9
Set to the number of structural class representatives; the clustering depends on this choice (Section 4).

assumptions (8)

domain assumption Persistence landscapes are determined by their critical points (Lemma 3.7 from [Bub20])
Used in Section 3.2 to define the discrete landscape embedding I.
domain assumption The landscape map is 1-Lipschitz stable (Bub20, Theorem 2.4)
Used in the proof of Theorem 3.15 to establish uniform continuity of the specialised embedding.
domain assumption Discrete signature coefficients are defined by the iterated-sum formula (3.2) and satisfy the discrete Chen relation (3.10)
Definition adopted from [DET20a], [DET20b], [BP23]; all later results depend on this definition.
domain assumption Barcodes are tame, with finitely many intervals contained in [0,L]
Stated in Section 1.2 and required for the landscape construction.
standard math The ground field is R
Signatures and time-series are real-valued; positivity and ordering arguments in Lemma 3.3 rely on R.
standard math A univariate monic polynomial is determined by its elementary symmetric functions
Used in Lemma 3.3 Claim 1 to show that matching power sums imply matching root sets.
domain assumption The FRUITS software computes discrete signatures correctly
All protein experiments use FRUITS [DK24]; its correctness is assumed.
domain assumption The knotted-protein dataset and its structural class and knot-depth labels from [Ben+23] are reliable
The application's conclusions inherit the quality of these external labels.

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Cite this review

Pith. "Pith review of Discrete signature tensors for persistence landscapes." pith.science (2026). https://pith.science/paper/L7GFUIVC

@misc{pith2026250502800,
  author       = {Pith},
  title        = {Pith review of: Discrete signature tensors for persistence landscapes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7GFUIVC}},
  note         = {Machine review of arXiv:2505.02800}
}

read the original abstract

Signature tensors of paths are a versatile tool for mathematical data analysis. Recently, they have been applied in the context of vectorisation of persistent homology: after a choice of embedding of barcodes into a space of paths on a vector space, one applies the path signature map, resulting in tensors amenable to statistical and machine-learning methods. Among the different path embeddings, the persistence landscape embedding (PLE) is injective and stable, but PLE is composed with the signature map loses injectivity. Therefore, we address this by proposing a discrete alternative. Persistence landscapes are determined by the time-series of their critical points, of which we compute the discrete signature. We call this composition the {\em discrete landscape feature map} (DLFM), and give results on its injectivity, stability and computability. When studying the injectivity, we complete the proof of a general result due to Diehl, Ebrahimi-Fard and Tapia in the higher-dimensional setting. We showcase the DLFM on a knotted protein dataset, capturing sequence similarity and knot depth with statistical significance. We include an appendix with a preliminary study of Chen signatures of persistence landscapes from the point of view of algebraic geometry.

Figures

Figures reproduced from arXiv: 2505.02800 by the authors.

**Figure 1.1.** Landscape with 3 levels: λ1 in black, λ2 in blue, λ3 in red. In the need of vectorising barcodes, several ways of associating paths in a vector space to barcodes have been introduced under the name of embeddings. Moreover, in order to obtain a more informative vectorisation, in [CNO20] the authors composed such embeddings with the Chen signature map (1.1): an embedding ι• : Bar → BV([0, L], R d ) defines the feature… view at source ↗

**Figure 1.2.** Uniform sampling, called Fibonacci sampling, of two spheres (left) has a onedimensional associated persistence landscape (right) with trivial Chen signature. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_1_2.png] view at source ↗

**Figure 4.1.** Correlations (via ARI and NMI) between the signature-based clusters and the labeling determined by 9 structural class representative. The first row highlights a significantly high correlation after truncating the landscape at the first 15 levels, the discrete signature at weight 3 and imposing 9 clusters. The metrics in [PITH_FULL_IMAGE:figures/full_fig_p026_4_1.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.2.** Figure 4.2: k-means clustering using 15 levels, weight ≤ 3, and 9 clusters as parameters. Validation of the correlation test. A priori the above high accuracy can just be a coincidence, in which case our test would be unreliable. Therefore, we also performed a permutation test (…

**Figure 4.3.** Figure 4.3: Permutation test for ARI and NMI (p-value 0.001), with parameters as in [PITH_FULL_IMAGE:figures/full_fig_p027_4_3.png]

**Figure 4.4.** Figure 4.4: Centroid-based permutation test using separation ratio with average signatures. A large ratio means that different classes have different discrete signatures; equivalently, signatures separate structural classes [PITH_FULL_IMAGE:figures/full_fig_p028_4_4.png]

**Figure 4.5.** Figure 4.5: 3-dimensional PCA colored by structural class representative (left) and by knot depth classification (right). via PCA in [PITH_FULL_IMAGE:figures/full_fig_p029_4_5.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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