{"id":"11ff0c30-d9d4-4993-8002-0c4754158c47","arxiv_id":"2505.02800","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Discrete signature tensors of persistence-landscape critical points give a feature map that is injective up to time-warping and translation and separates knotted-protein structural classes with an ARI of 0.958.","lead":"The paper introduces the discrete landscape feature map (DLFM), which converts the critical-point time-series of a persistence landscape into a discrete signature tensor, a new vector representation of topological summaries. The method is applied to knotted proteins, where it separates nine structural classes and tracks knot depth with statistical significance.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.7's critical-point list omits crossing points of overlapping intervals, so the DLFM time-series may not faithfully represent the landscape.","rationale":"The reader's weakest_assumption correctly identifies that the DLFM relies on the premise that a persistence landscape is completely determined by the time-series of its critical values at a fixed number of levels. My stress-test makes this concern concrete: Lemma 3.7, as stated, is not a correct enumeration of the critical points of a persistence landscape. For two overlapping intervals, the crossing point between the two tent functions is a genuine breakpoint of the landscape but is missed by the lemma's fourth item. Since the DLFM time-series is obtained by evaluating the landscape at exactly these critical points, an omitted breakpoint changes the increments of the time-series and hence the discrete signature. This does not affect the truth of Theorem 3.4 (which concerns the discrete signature on arbitrary time-series and is independently known from [DET22]), but it does affect the central new construction and the experimental results, which depend on the specific time-series produced by I. The concern is not about the discrete signature theory, but about whether the DLFM actually represents the landscape as intended. Other issues identified by the reader—parameter selection, vacuous continuity framing, and lack of variance reporting—are secondary and could be addressed by improved methodology. The proposed concrete test directly checks whether Lemma 3.7's omission changes the DLFM output for a simple barcode. If the test confirms the omission, the authors must correct the lemma, recompute the time-series used in the experiments, and re-evaluate their conclusions. Therefore, the verdict should remain CONDITIONAL, with the added condition of fixing the critical-point characterization.","tokens_in":21571,"tokens_out":20219,"duration_ms":204010,"concrete_test":"Construct B={[0,4],[2,6]}. Compute the landscape λ_1 and its true breakpoints {0,2,3,4,6}. Enumerate the set from Lemma 3.7 as printed; if it equals {0,2,1,4,6} (or otherwise omits 3), the lemma is incorrect. Then compute the DLFM signature for the time-series of values at the printed set and at the full set of breakpoints. If the two signatures differ, the DLFM depends on the erroneous enumeration, confirming the concern.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The DLFM construction in §3.2 defines the time-series x_B as the landscape values at the critical points listed in Lemma 3.7, attributed to [Bub20, Lemma 5.8]. As printed, the fourth class of critical points is 'midpoints (b_i+b_j)/2 of pairs of bars where b_k < b_i < d_j < d_i'. For a barcode with two overlapping intervals, e.g. B={[0,4],[2,6]}, the landscape λ_1(t)=max(min(t,4−t), min(t−2,6−t)) has breakpoints at t=0,2,3,4,6. The point t=3 is the crossing of the right slope of the first tent and the left slope of the second tent, where λ_1 changes slope from −1 to +1. The lemma's item 4, however, only includes the midpoint (0+2)/2=1, which is not a breakpoint of λ_1. Evaluating the landscape only at the lemma's points yields the value sequence (0,2,0,0) instead of the full (0,2,1,0,0), omitting t=3 entirely. Consequently, two barcodes whose landscapes differ only at such omitted crossing points can have identical DLFM signatures, undermining the claim that the DLFM is a faithful discrete representation of the persistence landscape. The error appears to be a mis-specification of the crossing-point formula: the correct breakpoints include (d_i+b_j)/2 and/or (b_i+d_j)/2 for interleaved pairs, not (b_i+b_j)/2. Since the entire feature map I (eq. 3.11) is built on this lemma, the construction's soundness depends on its correctness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21774,"tokens_out":11937,"duration_ms":139865,"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":[{"comment":"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.","section":"§3.4, Eq. (3.14), Theorem 3.15"},{"comment":"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.","section":"§3.2, Lemma 3.7"}],"minor_comments":[{"comment":"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.","section":"§3.2, Lemma 3.7"},{"comment":"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.","section":"§3.2, Eq. (3.11)"},{"comment":"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.","section":"§3.4, proof of Theorem 3.15"},{"comment":"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.","section":"§4, Figures 4.1–4.5"},{"comment":"There are minor typos: 'pourposes' in §3.4 and 'whevener' in §4 should be 'purposes' and 'whenever'.","section":"§3.4 and §4"}],"recommendation":"major_revision","confidential_remarks":"The two blocking issues in the report are the vacuous stability theorem and the incorrect critical-point lemma. As editor, I would also ask the authors to state more prominently that Theorem 3.4 is an independent proof of a known result, so the paper's novelty rests on the DLFM construction and its application; both need to be revalidated after the critical-point list is corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nWho 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.","headline":"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.","tokens_in":22451,"tokens_out":5646,"would_cite":false,"duration_ms":58712,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","68T09","46B85"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["persistent homology","barcodes","persistence landscapes","feature maps","discrete signature tensors","time-warping equivalence","vectorisation","knotted proteins"],"falsifier":"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.","tokens_in":21218,"feed_emoji":"🧬","tokens_out":7534,"duration_ms":83413,"temperature":0.7,"pith_summary":"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.","feed_headline":"Discrete signatures are injective up to time-warping","feed_subtitle":"Applied to persistence landscapes, the map separates protein classes that ordinary landscape signatures conflate.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces discrete signature tensors of time-series and establishes their time-warping invariance, which is the construction the DLFM is built on.","marker":"[DET20b]"},{"why":"Proves in a more general setting that equality of discrete signatures characterises the time-warping and time-translation equivalence, the theorem the paper rediscovers independently.","marker":"[DET22]"},{"why":"Supplies Lemma 3.7 describing the critical points of persistence landscapes and the stability result used for the continuity argument.","marker":"[Bub20]"},{"why":"Defines persistence feature maps via path embeddings and shows that the landscape embedding loses injectivity when composed with the continuous signature.","marker":"[CNO20]"},{"why":"Introduces the bottleneck distance on barcodes used as the domain metric in the uniform-continuity theorem.","marker":"[CEH07]"},{"why":"Provides the software used to compute discrete signatures in the protein experiments.","marker":"[DK24]"},{"why":"Supplies the knotted-protein dataset and the structural-class and knot-depth labels used as ground truth in the application.","marker":"[Ben+23]"},{"why":"Establishes the discrete Chen relation for shifted concatenation, which structures the monoid argument for the kernel of the discrete signature.","marker":"[BP23]"}],"fun_headline_variants":["Injective discrete signatures for persistence landscapes","Time-warping invariance yields unique signatures","Discrete signature map separates protein knots","Signature injectivity up to time-warping and shift","New discrete signature for barcode vectorization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Injective discrete signatures for persistence landscapes","Time-warping invariance yields unique signatures","Discrete signature map separates protein knots","Signature injectivity up to time-warping and shift","New discrete signature for barcode vectorization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000251,"raw_usage":{"total_tokens":1566,"prompt_tokens":964,"completion_tokens":602,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":535}},"tokens_in":580,"tokens_out":602,"duration_ms":7228,"temperature":1.0,"reasoning_tokens":535,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:41:17.382637+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}