Topological Sequence Analysis of Genomes: Delta Complex approaches

Dong Chen; Guo-Wei Wei; Jian Liu; Li Shen

REVIEW 3 major objections 5 minor 1 cited by

Topological Sequence Analysis of Genomes: Delta Complex approaches

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Genome sequences can be turned into $\Delta$-complexes whose persistent homology and Laplacian spectral gaps give fast, alignment-free phylogenetic classification.

desk verdict A useful but uneven paper: the TSA pipeline and code are real, but the central Delta-closure proposition is false as stated and the phylogenetics validation is thinner than advertised. read the letter →

arxiv 2507.05452 v1 pith:NGNQ5ZDY submitted 2025-07-07 math.AT q-bio.QM

Jian Liu , Li Shen , Dong Chen , Guo-Wei Wei This is my paper · ORCID

classification math.ATq-bio.QM MSC 55N3162R4068Q07

keywords topologicalsequenceanalysisDelta-complexclassifyingspacepersistenthomologyLaplacianpathgenomephylogenetics

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 establish that a DNA or RNA sequence can be studied topologically by declaring its overlapping segments to be simplices of a $\Delta$-complex, with repetitions allowed. It defines filtrations by functions such as the shortest subsequence length $\ell$ and the occurrence count $N$, then derives persistent homology, persistent path homology, and $\Delta$-complex persistent Laplacians from them. The practical payoff is that spectral-gap curves of the 1-dimensional Laplacian yield phylogenetic trees that largely separate 59 Ebola genomes into five families and 30 bacterial genomes into nine families, while processing the bacterial set in about 179 seconds compared with thousands of seconds for k-mer topology. A sympathetic reader would take the paper's contribution to be a general, efficient topological fingerprint for sequential data that does not require alignment or k-mer enumeration.

What carries the argument

The central object is the $\Delta$-complex $EX$ whose $n$-simplices are all $(n+1)$-tuples over the finite alphabet $X$, with face maps $d_i$ deleting the $i$-th entry; because repeats are allowed, $EX$ is a genuine $\Delta$-complex rather than a simplicial complex. The carrying filtrations come from $\ell(\sigma)=\inf\{\mathrm{length}(P(\sigma))\}$ over subsequences $P(\sigma)$ of $\xi$, which is face-preserving, and from the occurrence count $N$, whose superlevel sets form a path complex. For arbitrary functions the paper uses the $\Delta$-closure $\Delta S$, which adds every face of every simplex in $S$, and claims in Proposition 3.1 that this closure filtration has the same persistent homology as the face-preserving envelope $\tilde{f}(\sigma)=\max_{\tau} f(\tau)$ over faces of $\sigma$. The persistent Laplacian machinery then turns the smallest positive eigenvalue $\lambda(t)$ of the 1-dimensional Laplacian into the practical clustering feature.

What would settle it

Take a two-letter alphabet with $f(x)=f(y)=1$ and $f(x,y)=0$ at threshold $a=0.5$: under sublevel sets the envelope $\tilde{f}^{-}(0.5)$ is empty, while the $\Delta$-closure $\Delta f^{-}(0.5)$ contains the edge and both vertices, so their homology groups differ and Proposition 3.1 fails under the paper's stated convention; switching both sides to superlevel sets makes the two sets equal, pinpointing the repair.

Watch

Extended reading notes

Core claim

The paper's central claim is that for any finite alphabet $X$ and sequence $\xi$, the graded set $EX$ of all finite tuples over $X$ is a $\Delta$-complex whose face maps delete entries. The length function $\ell(\sigma)$, defined as the length of the shortest subsequence of $\xi$ realizing $\sigma$, is face-preserving and therefore defines a filtration of sub-$\Delta$-complexes; the occurrence count $N$ is not face-preserving, but its superlevel sets $N^{+}(a)$ are path complexes, yielding persistent path homology. When the alphabet is a finite group $G$, averaging any face-preserving function over the group orbit gives a face-preserving function on the classifying space $BG$ and hence persistent homology that records symmetry information. The paper further defines $\Delta$-complex persistent Laplacians whose kernel is isomorphic to the persistent homology group, and uses the smallest positive eigenvalue (the spectral gap) of the 1-dimensional Laplacian as a sequence feature. On the Ebola and bacterial datasets, Manhattan distances between these spectral-gap curves produce phylogenetic trees that separate the annotated families with minor discrepancies, and the whole 30-genome computation runs roughly 30 to 100 times faster than the k-mer topology baseline.

Load-bearing premise

The load-bearing premise is that closing a sublevel set under faces does not change persistent homology; Proposition 3.1 asserts this, but its proof appears to run with superlevel sets instead of the paper's stated sublevel convention, and with sublevel sets the claimed equality is false, so if the proof cannot be repaired the general $\Delta$-closure framework for arbitrary functions is unsupported even though the concrete $\ell$ and $N$ constructions remain well defined.

Editorial extensions

If this is right

Persistent homology, persistent path homology, and persistent Laplacians are defined directly on sequences, so no pairwise alignment between genomes is required.
A 5-million-base-pair genome yields 0- to 3-dimensional Betti and spectral-gap curves in about 6 seconds on a laptop, so the method scales to whole-genome collections.
The dimension-1 spectral-gap curve separates the five Ebola virus families and separates the nine bacterial families with only minor discrepancies, two of which the paper identifies explicitly.
Because only segment frequency and position are used, the same construction transfers to any character stream, including linguistic, musical, and social data.
Spectral-gap curves encode more than Betti numbers, since the persistent Laplacian kernel recovers persistent homology while the nonzero spectrum adds an independent fingerprint.

Reading between the lines

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

Editorial inference: the speed advantage comes mostly from avoiding k-mer enumeration, so the gap over k-mer topology should widen for larger alphabets and longer genomes; running the comparison on protein sequences would test this directly.
Editorial inference: the Proposition 3.1 convention mismatch disappears if the entire filtration theory is restated with superlevel sets, which would also unify the persistent-homology and path-homology conventions in one framework.
Editorial inference: because $\ell(\sigma)$ measures how soon a segment appears as a subsequence, the spectral-gap curve can be read as a compressed motif spectrum, pointing toward anomaly detection in streams such as logs, texts, or event histories rather than only phylogeny.

Share X Bluesky LinkedIn Reddit HN

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

Colleague,

Quick take: this is a workable TSA pipeline with public code and a real speed advantage, but the main new math is in worse shape than the applications. The l-filtration in Sec. 3.1 is, by the authors' own admission, just the path complex of [8,30]. The genuinely new pieces are the Delta-closure and the classifying-space persistence. The classifying-space construction is interesting, and the worked Z/4 example checks out; the Delta-closure, though, has a false proposition at its center.

Prop. 3.1 is proved with superlevel sets f^{-(a)} = {f >= a}, while the paper's own definition in Secs. 3.1 and 3.2 is sublevel: f^{-(a)} = {f <= a}. Under the stated convention the equality ef^{-(a)} = Delta f^{-(a)} is false; the stress-test example with a triangle and an edge value works. Even under the proof's superlevel convention, the inclusion step reverses the face relation. So the advertised equivalence between Delta-closure persistence and the face-preserving envelope is unsupported. To be fair, the closure construction itself is still well-defined as a filtration, so the paper could drop or repair the faulty comparison. The concrete l and N functions used later do not depend on Prop. 3.1.

Also worth flagging: Example 3.1 reports beta^{1,2}_1 = 10, but by its own annihilation list (AA, CC, GG, TT, AG+GA, CT+TC, GC+CT+TG all die at l = 2, and only AT+TA persists to 3), the image from level 1 to level 2 should be 1, not 10. That inconsistency needs fixing.

What earns credit: code and data are public; the timing comparison (179 s vs 5k-17k s for k-mer topology on 30 bacterial genomes) is suggestive and independently checkable, though the k-mer runs are not described on the same hardware. The phylogenetic trees look plausible, and the authors honestly admit the clustering accuracy is somewhat inferior. But there is no quantitative comparison against FFP, NVM, or the original k-mer topology--no cluster purity, no tree-distance numbers. For a paper advertising phylogenetics, that is thin. The self-citations are appropriate, particularly where they identify prior path-complex work.

Bottom line: the TSA pipeline is potentially useful for large-genome work, and the classifying-space idea deserves follow-up, but the math needs repair and the validation needs numbers, not just pictures. A serious referee should engage rather than desk-reject.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a framework for topological sequence analysis (TSA) in which the set E_X of all finite tuples over an alphabet X is treated as a Delta-complex, and functions on E_X (sequence length, occurrence count, group-averaged length) induce filtrations. The authors define persistent homology and persistent Laplacians for these filtrations, introduce a Delta-closure construction intended to handle non-face-preserving functions, and extend the construction to classifying spaces of finite groups. The methods are demonstrated on two datasets: 59 Ebola virus sequences and 30 whole bacterial genomes, using spectral-gap curves of the 1-dimensional persistent Laplacian as features for distance-based clustering. The paper claims that the resulting TSA is substantially faster than the k-mer topology method of Hozumi and Wei.

Significance. If the mathematical statements were correct, this would provide an alignment-free topological approach to genome comparison that avoids systematic k-mer enumeration, with a public GitHub implementation and a clear runtime advantage over k-mer topology. The construction of persistent path homology from occurrence counts and the classifying-space variant are interesting and fit a growing literature on algebraic topology for sequence data. However, the main general-purpose Delta-closure proposition is false as stated, and the primary worked example contains an inconsistent persistent Betti number, so the advertised generality of the framework is not currently established. The specific length-based and count-based constructions used in the applications are more credible and may survive revision.

major comments (3)

[Section 3.3, Proposition 3.1] The proof of the equality ef^-(a) = Delta f^-(a) uses the superlevel convention f^-(a) = {sigma : f(sigma) >= a}, whereas Definition 3.1 and the surrounding text in Sections 3.1 and 3.2 use the sublevel convention f^-(a) = {sigma : f(sigma) <= a}. Under the stated sublevel convention the equality is false: for X={0,1,2}, f(0,1,2)=1, f(0,1)=10, and f=0 on all other tuples, with a=2, Delta f^-(2) contains (0,1) because (0,1) is a face of (0,1,2), but ef(0,1)=10, so (0,1) is not in ef^-(2). Reading f^-(a) as superlevel does not repair the proof, because if sigma is in Delta f^-(a) then some face tau of an element of f^-(a) is involved, not necessarily a face of sigma with f(tau)>=a. The claimed isomorphism H^{a,b}_n(f) is isomorphic to H^{a,b}_n(ef) is therefore unsupported, and the general Delta-closure framework for arbitrary functions, advertised in the Introduction, needs either a corrected proposition (for sublevel sets, a lower envelope over cofaces is the appropriate construction) or a clear restriction of the scope.
[Example 3.1, persistent Betti numbers] The reported value beta^{1,2}_1 = 10 is inconsistent with Definition 3.1 and with the annihilation data in the same example. According to the text, the classes AA, CC, GG, TT, AG+GA, CT+TC, and GC+CT+TG are annihilated at l=2, while AC+CT+TA and TG+GA+AT are homologous to AT+TA in H_1(f^{-1}(2)). Hence the image of H_1(f^{-1}(1)) in H_1(f^{-1}(2)) has rank at most one, so beta^{1,2}_1 should be 1 (the class of AT+TA), not 10. The barcode in Figure 2a must be reconciled with the definition of beta^{a,b} as a rank of an image, or the definition must be amended.
[Section 3.2, persistent Laplacian definition] With the preceding convention d^{a,b}_m : C^{a,b}_{m+1} -> C^a_m, the displayed formula Delta^{a,b}_n = d^{a,b}_{n+1} (d^{a,b}_{n+1})^* + (d^a_n)^* d^a_n is not an operator on C^a_n: the first summand acts on C^a_{n+1} (after composing with the adjoint), while the second acts on C^a_n. The standard persistent Laplacian on n-chains is d^{a,b}_n (d^{a,b}_n)^* + (d^a_n)^* d^a_n with d^{a,b}_n as defined in the paper. Because the spectral-gap curves in Section 4 are computed from this operator, the indexing error must be corrected and the kernel-isomorphism claim restated for the corrected operator.

minor comments (5)

[Section 2.2, face-map identity] The identity d_i d_j = d_j d_{i-1} for i<j is not the standard simplicial identity; the usual relation is d_i d_j = d_{j-1} d_i. The tuple complex satisfies the standard identity, so this appears to be a typo, but it should be corrected because it is used to assert that E_X is a Delta-complex.
[Section 3.4, proof of Proposition 3.2] The inequality N(x0...x_{n-1}) >= N(x1...x_n) is false in general (for example, the sequence BAB has N(A)=1 and N(B)=2). The needed inequalities are N(x0...x_{n-1}) >= N(x0...x_n) and N(x1...x_n) >= N(x0...x_n), which do imply the conclusion. The proof should be rewritten accordingly.
[Section 3.5, notation] In the definition of the sublevel set for the averaged function, the text writes {[sigma] in BG | f([sigma]) <= a}; the function should be \bar f([sigma]), not f([sigma]).
[Section 4, phylogenetic evaluation] The phylogenetic evaluation is qualitative. No external cluster-quality measure (for example, adjusted Rand index, F1 score, or Robinson-Foulds distance) is reported, and the text acknowledges discrepancies for CP000578.1 and CP001151.1. Adding quantitative measures would support the claim that the trees closely align with the expected classifications.
[Table 1, runtime comparison] The runtime comparison should state whether the k-mer topology timings were obtained on the same machine and implementation, since the TSA timing is reported for a specific laptop while the k-mer timings are cited from reference [15].

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the TSA constructions are defined directly from sequences and benchmarked on external datasets; same-group citations are attribution and comparison, not load-bearing derivation.

full rationale

The paper's central derivation is not circular. The Delta-complex EX and classifying-space BG constructions are defined explicitly in Sections 3.1-3.5; the face-preserving functions ell and N are constructed directly from the input sequence, and the persistent homology and persistent Laplacian definitions are built from the paper's own chain-complex formalism. The phylogenetic applications use unsupervised Manhattan distances between spectral-gap curves and are checked against external family labels; no parameter is fitted from those labels, so the clustering result is an independent empirical outcome. The self-citations ([8,15,30]) appear as attribution to prior persistent path homology, persistent path Laplacian, and k-mer topology work, but none of these citations is used to justify a claim that would otherwise be unsupported: the paper defines the persistent Laplacians itself, and the timing comparison with k-mer topology is a direct measurement. A separate mathematical correctness concern, not a circularity, is that Proposition 3.1's proof switches from the sublevel convention used in Sections 3.1 and 3.2 to superlevel sets, and the asserted equality of filtrations fails under the paper's stated convention. That affects the generality of the claimed Delta-closure framework for arbitrary functions, but the concrete ell- and N-based constructions used in the applications do not depend on Proposition 3.1. Because the paper repeatedly leans on same-group prior work for context and baselines, a mild score of 2 is assigned, but no circular reduction of the central claim to its own inputs was found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The framework relies on standard algebraic topology facts, on the proven face-preserving property of the path-length function l and the path-complex property of the occurrence-count function N, and on the untested domain assumption that spectral-gap curves separate biological families. No free parameters are fitted to the target labels; the filtration threshold is a scale parameter, not a fitted constant.

assumptions (4)

standard math Standard results in algebraic topology: homology of Delta-complexes, and the classifying space BG of a finite group has the homotopy type K(G,1)
Used in Sections 2.2 and 2.3 to justify homology computations for sequence-built complexes.
standard math The kernel of the persistent Laplacian is isomorphic to the persistent homology group
Invoked in Section 3.2 to equate zero eigenvalues with persistent Betti numbers; cited from Refs. [7,19,29].
domain assumption Spectral gap curves of the 1-dimensional persistent Laplacian built from occurrence counts carry enough signal to separate biological families
This is the load-bearing modeling assumption in Section 4; no quantitative validation or theoretical justification is given.
domain assumption Frequency counts N(sigma) of sequence fragments are sufficient features for phylogenetic inference
Stated in Section 4: this study primarily aims to demonstrate the feasibility and effectiveness of employing frequency-based features; not established against baselines.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Topological Sequence Analysis of Genomes: Delta Complex approaches." pith.science (2026). https://pith.science/paper/NGNQ5ZDY

@misc{pith2026250705452,
  author       = {Pith},
  title        = {Pith review of: Topological Sequence Analysis of Genomes: Delta Complex approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGNQ5ZDY}},
  note         = {Machine review of arXiv:2507.05452}
}

abstract

Algebraic topology has been widely applied to point cloud data to capture geometric shapes and topological structures. However, its application to genome sequence analysis remains rare. In this work, we propose topological sequence analysis (TSA) techniques by constructing $\Delta$-complexes and classifying spaces, leading to persistent homology, and persistent path homology on genome sequences. We also develop $\Delta$-complex-based persistent Laplacians to facilitate the topological spectral analysis of genome sequences. Finally, we demonstrate the utility of the proposed TSA approaches in phylogenetic analysis using Ebola virus sequences and whole bacterial genomes. The present TSA methods are more efficient than earlier TSA model, k-mer topology, and thus have a potential to be applied to other time-consuming sequential data analyses, such as those in linguistics, literature, music, media, and social contexts.

Figures

Figures reproduced from arXiv: 2507.05452 by the authors.

**Figure 1.** a Illustrating the triangulation of a 2-dimensional sphere. b The ∆-complex of the triangulation of a 2-dimensional sphere. In this way, we obtain a ∆-complex with the following structure: K0 = {v0, v1, v2}, K1 = {e0, e1, e2}, K2 = {σ, τ}. The face maps are given by: ∂0σ = e0, ∂1σ = ∂2σ = e1, ∂0τ = e0, ∂1τ = ∂2τ = e2, ∂0e0 = ∂1e0 = v0, ∂0e1 = v0, ∂1e1 = v1, ∂0e2 = v0, ∂1e2 = v2. Recall that if we represent a 2-dimen… view at source ↗

**Figure 2.** a The corresponding barcode based on the function ℓ : EG → Z on the sequence of “N-China-F” primer. b The corresponding barcode based on the function N : EG → Z on the sequence of “N-China-F” primer. For a very long random sequence ξ of length L in a set X, consider a tuple σ = (x0, x1, . . . , xk) ∈ EX, where k << L. There almost always exists a contiguous subsequence x0x1 · · · xk in ξ. In this case, we can focus … view at source ↗

**Figure 3.** a The 0-dimensional Betti curve and spectral gap curve for the “N-China-F” sequence. b The 1-dimensional Betti curve and spectral gap curve for the “N-China-F” sequence. c The 2-dimensional Betti curve and spectral gap curve for the “N-China-F” sequence. d The 3-dimensional Betti curve and spectral gap curve for the “N-China-F” sequence. “N1-U.S.-P,” “N2-U.S.-P,” and “N3-U.S.-P” sequences. These curves reveal notice… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: a The 0-dimensional Betti curve and spectral gap curve for the “N1-U.S.-P” sequence. b The 1-dimensional Betti curve and spectral gap curve for the “N1-U.S.-P” sequence. c The 0-dimensional Betti curve and spectral gap curve for the “N2-U.S.-P” sequence. d The 1-dimens…

**Figure 5.** Figure 5: The corresponding barcode based on the function [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Phylogenetic tree for the Ebola virus generated using the TSA algorithm. [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Phylogenetic tree for the whole bacterial genomes generated using the TSA algorithm. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review
math.HO 2025-07 conditional novelty 3.0 of 10

A survey organizing recent TDA and TDL methods beyond persistent homology and connecting them to data structures and vectorization.

Reference graph

Works this paper leans on

32 extracted references · 29 canonical work pages · cited by 1 Pith paper

[1]

Quantum persistent homology

Bernardo Ameneyro, Vasileios Maroulas, and George Siopsis. Quantum persistent homology. Journal of Applied and Computational Topology , 8(7):1961–1980, 2024

work page 1961
[2]

Topologynet: Topology based deep convolutional and multi- task neural networks for biomolecular property predictions

Zixuan Cang and Guo-Wei Wei. Topologynet: Topology based deep convolutional and multi- task neural networks for biomolecular property predictions. PLoS computational biology , 13(7):e1005690, 2017

work page 2017
[3]

Topology and data

Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society , 46(2):255–308, 2009

2009
[4]

Topology of viral evolution

Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan. Topology of viral evolution. Proceedings of the National Academy of Sciences , 110(46):18566–18571, 2013

work page 2013
[5]

Multiscale topology-enabled structure-to-sequence transformer for protein–ligand interaction predictions

Dong Chen, Jian Liu, and Guo-Wei Wei. Multiscale topology-enabled structure-to-sequence transformer for protein–ligand interaction predictions. Nature Machine Intelligence , 6(7):799– 810, 2024

work page 2024
[6]

Path topology in molecular and materials sciences

Dong Chen, Jian Liu, Jie Wu, Guo-Wei Wei, Feng Pan, and Shing-Tung Yau. Path topology in molecular and materials sciences. Journal of Physical Chemistry Letters , 14(4):954–964, 2023

work page 2023
[7]

Evolutionary de Rham-Hodge method

Jiahui Chen, Rundong Zhao, Yiying Tong, and Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete and continuous dynamical systems. Series B , 26(7):3785, 2021

work page 2021
[8]

Persistent path homology of directed networks

Samir Chowdhury and Facundo M´ emoli. Persistent path homology of directed networks. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 1152–1169. SIAM, 2018

work page 2018

Show all 32 references

[9]

A novel method of characterizing genetic sequences: genome space with biological distance and applications

Mo Deng, Chenglong Yu, Qian Liang, Rong L He, and Stephen S-T Yau. A novel method of characterizing genetic sequences: genome space with biological distance and applications. PloS one , 6(3):e17293, 2011

2011
[10]

Combinatorial topological models for phylogenetic networks and the mergegram invariant

Pawe l D lotko, Jan Felix Senge, and Anastasios Stefanou. Combinatorial topological models for phylogenetic networks and the mergegram invariant. Foundations of data science (Springfield, Mo.), 7(2):617–670, 2025

2025
[11]

Persistent homology-a survey

Herbert Edelsbrunner, John Harer, et al. Persistent homology-a survey. Contemporary math- ematics, 453(26):257–282, 2008

2008
[12]

Homologies of path complexes and digraphs

Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Homologies of path complexes and digraphs. arXiv preprint arXiv:1207.2834 , 2012

2012 arXiv
[13]

Cohomology of digraphs and (undirected) graphs

Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Cohomology of digraphs and (undirected) graphs. Asian Journal of Mathematics , 19(5):887–932, 2015

2015
[14]

A new method to cluster dna sequences using fourier power spectrum

Tung Hoang, Changchuan Yin, Hui Zheng, Chenglong Yu, Rong Lucy He, and Stephen S- T Yau. A new method to cluster dna sequences using fourier power spectrum. Journal of theoretical biology, 372:135–145, 2015. 22

2015
[15]

Revealing the shape of genome space via k-mer topology

Yuta Hozumi and Guo-Wei Wei. Revealing the shape of genome space via k-mer topology. arXiv preprint arXiv:2412.20202 , 2024

2024 arXiv
[16]

Whole-proteome phylogeny of prokaryotes by feature frequency profiles: An alignment-free method with optimal feature resolution

Se-Ran Jun, Gregory E Sims, Guohong A Wu, and Sung-Hou Kim. Whole-proteome phylogeny of prokaryotes by feature frequency profiles: An alignment-free method with optimal feature resolution. Proceedings of the National Academy of Sciences , 107(1):133–138, 2010

2010
[17]

Persistent interaction topology in data analysis

Jian Liu, Dong Chen, and Guo-Wei Wei. Persistent interaction topology in data analysis. Foundations of data science (Springfield, Mo.) , page in press, 2025

2025
[18]

The algebraic stability for persistent Laplacians

Jian Liu, Jingyan Li, and Jie Wu. The algebraic stability for persistent Laplacians. Homology, Homotopy and Applications , 26:297–323, 2024

2024
[19]

Persistent laplacians: Properties, algo- rithms and implications

Facundo M´ emoli, Zhengchao Wan, and Yusu Wang. Persistent laplacians: Properties, algo- rithms and implications. SIAM Journal on Mathematics of Data Science , 4(2):858–884, 2022

2022
[20]

A topological characterization of dna sequences based on chaos geometry and persistent homology

Dong Quan Ngoc Nguyen, Phuong Dong Tan Le, Lin Xing, and Lizhen Lin. A topological characterization of dna sequences based on chaos geometry and persistent homology. In 2022 International Conference on Computational Science and Computational Intelligence (CSCI) , pages 1591–15...

2022
[21]

Position: Topo- logical deep learning is the new frontier for relational learning

Theodore Papamarkou, Tolga Birdal, Michael M Bronstein, Gunnar E Carlsson, Justin Curry, Yue Gao, Mustafa Hajij, Roland Kwitt, Pietro Lio, Paolo Di Lorenzo, et al. Position: Topo- logical deep learning is the new frontier for relational learning. In Forty-first International C...

2024
[22]

Persistent-homology-based machine learning: a survey and a comparative study

Chi Seng Pun, Si Xian Lee, and Kelin Xia. Persistent-homology-based machine learning: a survey and a comparative study. Artificial Intelligence Review , 55(7):5169–5213, 2022

2022
[23]

Whole proteome prokaryote phylogeny without sequence alignment: Ak-string composition approach

Ji Qi, Bin Wang, and Bai-Iin Hao. Whole proteome prokaryote phylogeny without sequence alignment: Ak-string composition approach. Journal of molecular evolution , 58:1–11, 2004

2004
[24]

Alignment-free genome comparison with feature frequency profiles (ffp) and optimal resolutions

Gregory E Sims, Se-Ran Jun, Guohong A Wu, and Sung-Hou Kim. Alignment-free genome comparison with feature frequency profiles (ffp) and optimal resolutions. Proceedings of the National Academy of Sciences , 106(8):2677–2682, 2009

2009
[25]

Revealing hidden medium-range order in amorphous materials using topological data analysis

Søren S Sørensen, Christophe AN Biscio, Mathieu Bauchy, Lisbeth Fajstrup, and Morten M Smedskjaer. Revealing hidden medium-range order in amorphous materials using topological data analysis. Science Advances, 6(37):eabc2320, 2020

2020
[26]

Persistent dirac of paths on digraphs and hypergraphs

Faisal Suwayyid and Guo-Wei Wei. Persistent dirac of paths on digraphs and hypergraphs. Foundations of data science (Springfield, Mo.) , 6(2):124, 2024

2024
[27]

Alignment-free sequence comparison—a review

Susana Vinga and Jonas Almeida. Alignment-free sequence comparison—a review. Bioinfor- matics, 19(4):513–523, 2003

2003
[28]

Mutations on covid-19 diag- nostic targets

Rui Wang, Yuta Hozumi, Changchuan Yin, and Guo-Wei Wei. Mutations on covid-19 diag- nostic targets. Genomics, 112(6):5204–5213, 2020. 23

2020
[29]

Persistent spectral graph

Rui Wang, Duc Duy Nguyen, and Guo-Wei Wei. Persistent spectral graph. International journal for numerical methods in biomedical engineering , 36(9):e3376, 2020

2020
[30]

Persistent path laplacian

Rui Wang and Guo-Wei Wei. Persistent path laplacian. Foundations of data science (Spring- field, Mo.) , 5(1):26, 2023

2023
[31]

Persistent dirac for molecular representation

JunJie Wee, Ginestra Bianconi, and Kelin Xia. Persistent dirac for molecular representation. Scientific Reports, 13(1):11183, 2023

2023
[32]

Real time classification of viruses in 12 dimensions

Chenglong Yu, Troy Hernandez, Hui Zheng, Shek-Chung Yau, Hsin-Hsiung Huang, Rong Lucy He, Jie Yang, and Stephen S-T Yau. Real time classification of viruses in 12 dimensions. PloS one, 8(5):e64328, 2013. 24

2013

Pith tools

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

[1] [1]

Quantum persistent homology

Bernardo Ameneyro, Vasileios Maroulas, and George Siopsis. Quantum persistent homology. Journal of Applied and Computational Topology , 8(7):1961–1980, 2024

work page 1961

[2] [2]

Topologynet: Topology based deep convolutional and multi- task neural networks for biomolecular property predictions

Zixuan Cang and Guo-Wei Wei. Topologynet: Topology based deep convolutional and multi- task neural networks for biomolecular property predictions. PLoS computational biology , 13(7):e1005690, 2017

work page 2017

[3] [3]

Topology and data

Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society , 46(2):255–308, 2009

2009

[4] [4]

Topology of viral evolution

Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan. Topology of viral evolution. Proceedings of the National Academy of Sciences , 110(46):18566–18571, 2013

work page 2013

[5] [5]

Multiscale topology-enabled structure-to-sequence transformer for protein–ligand interaction predictions

Dong Chen, Jian Liu, and Guo-Wei Wei. Multiscale topology-enabled structure-to-sequence transformer for protein–ligand interaction predictions. Nature Machine Intelligence , 6(7):799– 810, 2024

work page 2024

[6] [6]

Path topology in molecular and materials sciences

Dong Chen, Jian Liu, Jie Wu, Guo-Wei Wei, Feng Pan, and Shing-Tung Yau. Path topology in molecular and materials sciences. Journal of Physical Chemistry Letters , 14(4):954–964, 2023

work page 2023

[7] [7]

Evolutionary de Rham-Hodge method

Jiahui Chen, Rundong Zhao, Yiying Tong, and Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete and continuous dynamical systems. Series B , 26(7):3785, 2021

work page 2021

[8] [8]

Persistent path homology of directed networks

Samir Chowdhury and Facundo M´ emoli. Persistent path homology of directed networks. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 1152–1169. SIAM, 2018

work page 2018