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Multiset Transformer: Advancing Representation Learning in Persistence Diagrams

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Multiset Transformer treats persistence diagrams as multisets, adding a learnable multiplicity bias to attention to preserve permutation invariance and cut complexity.

desk verdict Useful multiset attention variant with clean proofs, but the 'outperforms existing methods' claim rests on a single copied baseline and is contradicted by the paper's own appendix. read the letter →

arxiv 2411.14662 v1 pith:M6JS2CDT submitted 2024-11-22 cs.LG

Minghua Wang , Ziyun Huang , Jinhui Xu This is my paper · ORCID
classification cs.LG MSC 68T0755N31
keywords multisettransformerpersistencediagramstopologicaldataanalysispermutationinvarianceattentionmechanismgraphclassificationrepresentationlearningmultiplicitybias
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

Persistence diagrams record the births and deaths of topological features, and they naturally contain duplicate points; treating them as lists loses this multiplicity and inflates cost. This paper proposes the Multiset Transformer, an attention architecture that keeps a base set and a multiplicity vector separate, adds a learnable multiplicity bias to attention, and proves the resulting self-attention is permutation equivariant while the learned-query attention is permutation invariant. If correct, it is the first attention network built for multisets rather than sets, it cuts the quadratic cost of a duplicated input by a factor of the maximum multiplicity, and it gives a neural vectorization of persistence diagrams that beats the PersLay baseline on most of the tested graph datasets. The practical payoff is a representation learning pipeline that can also cluster a persistence diagram first and still keep accuracy, making large diagrams feasible.

What carries the argument

The load-bearing mechanism is the multiplicity-bias term $B$ injected inside the softmax attention. It is built only from the multiplicity vectors $(M_Q-\mathbf{1})(M_X-\mathbf{1})^\top$, normalized by its Frobenius norm, so it vanishes when all multiplicities are one and the model reduces to ordinary set attention. This term is what makes the equivariance and invariance proofs go through while directing extra attention to high-multiplicity points. Around it, the architecture follows the pool-decomposition scheme: permutation-equivariant Multiset Attention Blocks (SAB and IMAB) carry the base set through hidden layers, and an invariant block with learned queries (MAB$_Q$) produces the final representation, with multiplicities re-entering at the invariant layer.

What would settle it

Rerun PersLay and MST from scratch on the same ten folds, the same number of repetitions, and the same train/test protocol for all nine graph datasets; if PersLay equals or beats MST on a majority of entries, the central superiority claim fails.

Watch

Extended reading notes

Core claim

The central claim is that a persistence diagram should be encoded as a multiset $(X, M_X)$ — a base set of distinct points plus their multiplicities — and processed by attention with a dedicated multiplicity-bias term. The multiset-enhanced attention is $A(Q,X)=(\mathrm{softmax}(QX^\top/\sqrt{d})+\alpha B)X$ with $B=(M_Q-\mathbf{1})(M_X-\mathbf{1})^\top/(\|(M_Q-\mathbf{1})(M_X-\mathbf{1})^\top\|_F+\varepsilon)$, and the learned-query variant uses a learnable vector in place of $\alpha$. The paper proves Theorem 5.1 (permutation equivariance of multiset self-attention) and Theorem 5.2 (permutation invariance of learned-query attention), then stacks equivariant blocks before an invariant pooling layer. Because the base set is not expanded by multiplicities, the complexity is $O(n^2)$ and $O(nq)$ instead of $O(n^2m^2)$ and $O(nmq)$. Empirically, on ordinary and extended persistence diagrams of nine graph classification datasets, the Multiset Transformer reports accuracies above the PersLay baseline on a majority of entries, and an ablation shows the multiplicity term drives most of the gain.

Load-bearing premise

The empirical superiority claim depends on assuming that the PersLay numbers quoted from an earlier paper were produced under the same ten-fold, repeated-run protocol as the new Multiset Transformer results, so the two columns can be compared directly.

Editorial extensions

If this is right

  • On multiset inputs, MST has $O(n^2)$ time and space for SAB and $O(nq)$ for IMAB, versus $O(n^2m^2)$ and $O(nmq)$ for Set Transformer, so diagrams with high duplication become feasible.
  • Because clustering preprocessing maps a large diagram to few representative points with absorbed multiplicities, MST can process very large persistence diagrams with only small accuracy loss.
  • The ablation results imply multiplicities are not incidental: removing them lowers accuracy on most real-world datasets, for example by 3.62% on IMDB-B extended diagrams.
  • The synthetic experiments show MST without multiplicity is near random on 5- and 11-class tasks, while MST with multiplicity reaches 88.86% and 41.14%, establishing that the design goal of allocating attention by frequency is met.

Reading between the lines

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

  • I would expect the same multiplicity-bias attention to transfer to bag-of-words and multiple-instance learning, where multiplicities are already available; a quick test would be replacing set pooling in those pipelines and measuring accuracy versus complexity.
  • The paper's Appendix A notes that on the 11-class synthetic task the Set Transformer slightly edges out MST, which suggests the duplication-expanded model can still win when expressiveness matters more than cost; a hybrid that starts from multiplicities but widens the base set on demand is an untested middle ground.
  • The clustering results raise a question the paper leaves open: if clustering to 1% of the input can preserve or even improve accuracy, then some high-multiplicity points carry most of the signal; one could test this directly by ranking clusters by lifespan and ablating low-lifespan clusters.
  • A natural next benchmark is comparing MST against Persformer and other PD vectorizers on the same folds, since the paper only compares against PersLay numbers and notes Set Transformer is infeasible on real 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.

Referee Report

4 major / 5 minor

Summary. The paper proposes the Multiset Transformer (MST), an attention-based architecture for representation learning on multisets, with applications to persistence diagram (PD) vectorization. MST separates a multiset into a base set and a multiplicity vector, and adds a learnable multiplicity bias to the attention weights, preserving multiplicities across equivariant layers and using a pool-decomposition scheme for invariance. The paper proves permutation equivariance of the multiset self-attention block and permutation invariance of the multiset attention with learnable queries, and gives a complexity analysis showing that MST avoids the duplicate-expansion cost of the Set Transformer. Experiments include a synthetic multiset classification task and graph classification on nine real-world datasets, where MST is compared with PersLay as the only neural baseline, plus an ablation study of the multiplicity terms.

Significance. If the claims are properly supported, the MST is a useful architectural contribution: it is one of the first attention mechanisms that explicitly consumes multiset structure, and the multiplicity bias is a principled way to respect the distinction between unique items and their multiplicities. The theoretical proofs (Theorems 5.1 and 5.2) are clean for the single-head case, and the complexity reduction relative to the Set Transformer (Section 5.5) is real and potentially important for large persistence diagrams. The paper also provides source code and detailed hyperparameters, which aids reproducibility. However, the empirical contribution is currently not at the level of the theoretical one: the only neural baseline is PersLay with copied numbers, and the paper's own Appendix A concedes that a stronger attention-based baseline may outperform MST on real data. The overall significance is therefore moderate, contingent on a more careful empirical comparison.

major comments (4)
  1. [Abstract and Section 6.2.4] The abstract and Section 6.2.4 claim that MST outperforms existing neural network methods for persistence diagram representation learning, but Table 2 compares only against PersLay. Appendix A (Table 4) reports that the Set Transformer matches or exceeds MST on the 11-class synthetic task (ST 42.02±1.85 vs. MST 41.14±2.24) and the authors state that they 'hypothesize that the ST, owing to its higher model complexity, is likely to outperform the MST in most real-world scenarios.' This internal admission shows that the general superiority claim is not established; the claim must be narrowed to 'outperforms PersLay' or additional attention-based baselines must be evaluated on the real-world datasets.
  2. [Section 6.2.4, Table 2] The PersLay baseline values are copied from Carrière et al. (2020, Table 7) without standard deviations, so the reader cannot assess whether the reported differences are statistically significant. The paper states that it 'adopts the experimental settings described in Carrière et al. (2020)' (Section 6.2.3), but does not demonstrate that the folds, seeds, diagram types, and hyperparameter tuning are identical across the two studies. Since the performance differences in some rows are small (e.g., NCI1 Ordinary: 69.12±0.17 vs. PersLay 68.9), the claim that MST 'exceeds' PersLay requires either rerunning PersLay under the same protocol or providing a statistical comparison that accounts for the variability of both methods.
  3. [Section 6.2.5, Table 3] The text claims that 'consistently across all datasets, the MST model, when fully equipped with PD and its associated multiplicities, demonstrates superior performance over its counterparts,' but Table 3 shows the opposite for IMDB-M, where MST (w/o mult.) achieves 50.64±0.41 versus 50.33±0.17 for the full MST. In addition, the NCI1 row reports the MST value as 68.65±1.23 in Table 2 but 68.65±0.14 in Table 3, despite the statement that the MST column in Table 3 is the same as in Table 2. The overclaim and the numerical inconsistency should be corrected.
  4. [Section 6.1] The synthetic experiment defines class labels by the element with the highest frequency in each multiset, so the multiplicity-aware attention has an advantage by construction. This makes the experiment a valid sanity check of the mechanism, but it does not provide external evidence of general performance superiority. The text should frame it as such and avoid suggesting that the synthetic gains transfer to real-world tasks without additional support.
minor comments (5)
  1. [Title] The title contains a stray space: 'Persistence Diagram s' should read 'Persistence Diagrams'.
  2. [Section 5.2.3 heading] The heading contains a typo: 'Multset Attention with Learnable Queries' should be 'Multiset Attention with Learnable Queries'.
  3. [Appendix B.1] The proof states that 'the softmax function is applied element-wise (or row-wise)'; standard softmax is row-wise, so the parenthetical should be removed or corrected to avoid confusion.
  4. [Section 5.5] The notation is inconsistent: n denotes the number of unique elements in the multiset here, while in Section 5.2 n and m denote numbers of queries and keys. Please unify the notation to prevent ambiguity.
  5. [Table 6] The DBSCAN eps values are listed per dataset, but no sensitivity analysis is provided; since clustering is a preprocessing step that affects both accuracy and complexity, a brief discussion of the sensitivity to eps (or a reference to the chosen values) would strengthen the presentation.

Circularity Check

1 steps flagged · score 2.0 of 10

The core architecture proofs and complexity analysis are self-contained; the only self-confirming element is the synthetic benchmark, which defines labels by the same high-multiplicity property the model is designed to exploit.

  1. self definitional [Section 6.1, Synthetic Experiments, Table 1]
    "In the synthetic experiments, our goal was to demonstrate the ability of the MST to highlight elements that appear with the highest frequency within a multiset. ... The true label is determined based on the element that appears with the highest frequency."

    The synthetic classification target is defined as exactly the property that the multiset-enhanced attention bias is constructed to encode: Equations (7) and (9) explicitly allocate extra attention weight to elements with larger multiplicities. Therefore, comparing MST with multiplicity inputs against MST without multiplicity inputs on labels that are literally 'which element has the highest multiplicity' measures the model against its own design premise rather than against an independent external signal. The near-perfect or perfect accuracies in Table 1 are substantially forced by this label construction, so the claim that this experiment 'validates' the multiplicity mechanism is self-confirming.

full rationale

The paper's load-bearing theoretical claims are derived in the text rather than imported from the authors' prior work: Theorem 5.1 and Theorem 5.2 are proved directly in Appendix B, and the O(n^2)/O(nq) versus O(n^2 m^2)/O(nmq) complexity comparison follows from the definitions of the attention blocks. There are no self-citations used to justify uniqueness or to forbid alternative architectures. The one genuinely self-confirming element is the synthetic experiment: the label is defined by the highest-frequency element, which is precisely the quantity the multiplicity bias terms in Equations (7) and (9) are designed to emphasize, so the strong improvement shown in Table 1 is built into the evaluation rather than independently demonstrated. The real-data comparison against PERSLAY uses baseline numbers copied from Carrière et al. (2020, Table 7) without standard deviations, and Appendix A concedes that Set Transformer was not benchmarked on real persistence diagrams and that the authors hypothesize it 'is likely to outperform the MST in most real-world scenarios.' These are correctness and completeness concerns about the scope of the empirical superiority claim, especially the abstract's phrase 'outperforms existing neural network methods,' but they are not circularity: they do not make the derivation equivalent to its inputs. The overall circularity burden is therefore low and confined to the synthetic validation.

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

The invariance theorems are self-contained; the empirical comparisons depend on an external baseline copied from a prior paper and on hand-set DBSCAN epsilons. No new physical or conceptual entities are postulated. The synthetic experiment's labels are defined by multiplicity, so it does not independently test the value of multiplicity information.

free parameters (4)
  • Learnable multiplicity bias scale alpha = not reported; trained
    Controls how strongly the multiset bias term B enters attention in Equation 6; learned from data.
  • Learnable multiplicity query M_alpha = not reported; trained
    Learnable per-query multiplicity coefficients in Equation 9 for the invariant layer; learned from data.
  • DBSCAN eps per dataset = 0.01, 0.04, 0.1, or 0.5 depending on dataset
    Hand-set neighborhood parameter for clustering preprocessing (Table 6). It controls how much a PD is compressed; the paper does not analyze sensitivity.
  • Architecture hyperparameters (heads, hidden size, learning rate, epochs, batch) = listed in Table 6
    Standard training choices made by hand; they affect the reported accuracy but are not part of the theoretical guarantees.
assumptions (4)
  • domain assumption Persistence diagrams are preprocessed into bounded, finite multisets.
    Invoked in Section 4 ('it is presupposed that PDs are treated as bounded and finite multisets'); real diagrams can be unbounded, so preprocessing is needed.
  • domain assumption Copied PersLay results from Carrière et al. (2020, Table 7) are comparable to the authors' own 5-run, 10-fold protocol.
    The paper says it adopts Carrière et al.'s settings (Section 6.2.3) and copies baseline numbers (Table 2 note) without rerunning or reporting baseline variance.
  • domain assumption DBSCAN clustering with fixed eps maps each cluster to one representative point while preserving classification-relevant information in multiplicities.
    Appendix D.3 states clustered information is incorporated into multiplicities; no error analysis quantifies information loss, and eps values are fixed by hand (Table 6).
  • domain assumption Attention should be biased toward elements with higher multiplicity.
    Design premise stated in Section 5.2; synthetic experiments construct labels from the highest-frequency element, so they do not independently validate this premise.

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Cite this review

Pith. "Pith review of Multiset Transformer: Advancing Representation Learning in Persistence Diagrams." pith.science (2026). https://pith.science/paper/M6JS2CDT

@misc{pith2026241114662,
  author       = {Pith},
  title        = {Pith review of: Multiset Transformer: Advancing Representation Learning in Persistence Diagrams},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M6JS2CDT}},
  note         = {Machine review of arXiv:2411.14662}
}
read the original abstract

To improve persistence diagram representation learning, we propose Multiset Transformer. This is the first neural network that utilizes attention mechanisms specifically designed for multisets as inputs and offers rigorous theoretical guarantees of permutation invariance. The architecture integrates multiset-enhanced attentions with a pool-decomposition scheme, allowing multiplicities to be preserved across equivariant layers. This capability enables full leverage of multiplicities while significantly reducing both computational and spatial complexity compared to the Set Transformer. Additionally, our method can greatly benefit from clustering as a preprocessing step to further minimize complexity, an advantage not possessed by the Set Transformer. Experimental results demonstrate that the Multiset Transformer outperforms existing neural network methods in the realm of persistence diagram representation learning.

Figures

Figures reproduced from arXiv: 2411.14662 by the authors.

Figure 1
Figure 1. Persistence diagram examples. PDs are represente [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. MST architecture. Base set X with multiplicities MX is processed by equivariant layers, preserving permutation order. Representation output R is generated, with multiplicities MX used as input to an invariant layer. Within the given figure, a multiset is denoted as (X, MX), where X represents the base set, and MX indicates its corresponding multiplicities. The architecture consists of two main components: the Equiva… view at source ↗
Figure 3
Figure 3. Sythetic data classification pipeline. A multiset [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Graph classification architecture. Given a graph [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Multiset Transformer architecture with clusteri [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams

    cs.LG 2026-07 conditional novelty 5.5 of 10

    Voronoi-cell histograms of normalized Expected Persistence Diagrams give a stable, adaptive EPD vectorization that is competitive on topology-sensitive classification and scales better with the number of subsampled di...

  2. Dynamical Persistent Homology via Wasserstein Gradient Flow

    math.AT 2024-12 reject novelty 4.0 of 10

    The paper combines McCann interpolation and JKO Wasserstein gradient flow with differentiable persistent homology to iteratively retarget persistence diagrams and update filtrations, but it provides only qualitative 2...

Reference graph

Works this paper leans on

42 extracted references · 28 canonical work pages · cited by 2 Pith papers

  1. [1]

    Persistence images: A stable vector representation of persistent homology

    Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18, 2017

    work page 2017

  2. [2]

    A survey of vectorization methods in topological data analysis

    Dashti Ali, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgo, and Manuel Soriano-Trigueros. A survey of vectorization methods in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023

    work page 2023

  3. [3]

    Layer normalization

    Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016

    arXiv 2016

  4. [4]

    Robust object tracking with online multiple instance learning

    Boris Babenko, Ming-Hsuan Yang, and Serge Belongie. Robust object tracking with online multiple instance learning. IEEE transactions on pattern analysis and machine intelligence, 33 0 (8): 0 1619--1632, 2010

    work page 2010

  5. [5]

    Statistical topological data analysis using persistence landscapes

    Peter Bubenik et al. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., 16 0 (1): 0 77--102, 2015

    work page 2015

  6. [6]

    Zigzag persistence

    Gunnar Carlsson and Vin De Silva. Zigzag persistence. Foundations of computational mathematics, 10: 0 367--405, 2010

    work page 2010

  7. [7]

    Perslay: A neural network layer for persistence diagrams and new graph topological signatures

    Mathieu Carri \`e re, Fr \'e d \'e ric Chazal, Yuichi Ike, Th \'e o Lacombe, Martin Royer, and Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topological signatures. In International Conference on Artificial Intelligence and Statistics, pp.\ 2786--2796. PMLR, 2020

    work page 2020

  8. [8]

    Extending persistence using poincar \'e and lefschetz duality

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Extending persistence using poincar \'e and lefschetz duality. Foundations of Computational Mathematics, 9 0 (1): 0 79--103, 2009

    work page 2009

Show all 42 references
  1. [9]

    Histograms of oriented gradients for human detection

    Navneet Dalal and Bill Triggs. Histograms of oriented gradients for human detection. In 2005 IEEE computer society conference on computer vision and pattern recognition (CVPR'05), volume 1, pp.\ 886--893. Ieee, 2005

    2005

  2. [10]

    Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds

    Asim Kumar Debnath, Rosa L Lopez de Compadre, Gargi Debnath, Alan J Shusterman, and Corwin Hansch. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity. Journal of medicinal che...

    1991

  3. [11]

    Computational topology for data analysis

    Tamal Krishna Dey and Yusu Wang. Computational topology for data analysis. Cambridge University Press, 2022

    2022

  4. [12]

    Solving the multiple instance problem with axis-parallel rectangles

    Thomas G Dietterich, Richard H Lathrop, and Tom \'a s Lozano-P \'e rez. Solving the multiple instance problem with axis-parallel rectangles. Artificial intelligence, 89 0 (1-2): 0 31--71, 1997

    1997

  5. [13]

    Distinguishing enzyme structures from non-enzymes without alignments

    Paul D Dobson and Andrew J Doig. Distinguishing enzyme structures from non-enzymes without alignments. Journal of molecular biology, 330 0 (4): 0 771--783, 2003

    2003

  6. [14]

    Topological persistence and simplification

    Edelsbrunner, Letscher, and Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry, 28: 0 511--533, 2002

    2002

  7. [15]

    Computational topology: an introduction

    Herbert Edelsbrunner and John L Harer. Computational topology: an introduction. American Mathematical Society, 2022

    2022

  8. [16]

    Two’s company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data

    Chad Giusti, Robert Ghrist, and Danielle S Bassett. Two’s company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data. Journal of computational neuroscience, 41: 0 1--14, 2016

    2016

  9. [17]

    Deep learning with topological signatures

    Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl. Deep learning with topological signatures. Advances in neural information processing systems, 30, 2017

    2017

  10. [18]

    Learning representations of persistence barcodes

    Christoph D Hofer, Roland Kwitt, and Marc Niethammer. Learning representations of persistence barcodes. J. Mach. Learn. Res., 20 0 (126): 0 1--45, 2019

    2019

  11. [19]

    Topological graph neural networks

    Max Horn, Edward De Brouwer, Michael Moor, Yves Moreau, Bastian Rieck, and Karsten Borgwardt. Topological graph neural networks. arXiv preprint arXiv:2102.07835, 2021

    2021 arXiv

  12. [20]

    Stable and informative spectral signatures for graph matching

    Nan Hu, Raif M Rustamov, and Leonidas Guibas. Stable and informative spectral signatures for graph matching. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.\ 2305--2312, 2014

    2014

  13. [21]

    Transformers in vision: A survey

    Salman Khan, Muzammal Naseer, Munawar Hayat, Syed Waqas Zamir, Fahad Shahbaz Khan, and Mubarak Shah. Transformers in vision: A survey. ACM computing surveys (CSUR), 54 0 (10s): 0 1--41, 2022

    2022

  14. [22]

    Set transformer: A framework for attention-based permutation-invariant neural networks

    Juho Lee, Yoonho Lee, Jungtaek Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh. Set transformer: A framework for attention-based permutation-invariant neural networks. In International conference on machine learning, pp.\ 3744--3753. PMLR, 2019

    2019

  15. [23]

    A survey of transformers

    Tianyang Lin, Yuxin Wang, Xiangyang Liu, and Xipeng Qiu. A survey of transformers. AI Open, 2022

    2022

  16. [24]

    Persistent homology and many-body atomic structure for medium-range order in the glass

    Takenobu Nakamura, Yasuaki Hiraoka, Akihiko Hirata, Emerson G Escolar, and Yasumasa Nishiura. Persistent homology and many-body atomic structure for medium-range order in the glass. Nanotechnology, 26 0 (30): 0 304001, 2015

    2015

  17. [25]

    Persistence theory: from quiver representations to data analysis, volume 209

    Steve Y Oudot. Persistence theory: from quiver representations to data analysis, volume 209. American Mathematical Soc., 2017

    2017

  18. [26]

    Pointnet: Deep learning on point sets for 3d classification and segmentation

    Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.\ 652--660, 2017

    2017

  19. [27]

    Multiple-instance learning for medical image and video analysis

    Gwenol \'e Quellec, Guy Cazuguel, B \'e atrice Cochener, and Mathieu Lamard. Multiple-instance learning for medical image and video analysis. IEEE reviews in biomedical engineering, 10: 0 213--234, 2017

    2017

  20. [28]

    Deep learning with sets and point clouds

    Siamak Ravanbakhsh, Jeff Schneider, and Barnabas Poczos. Deep learning with sets and point clouds. arXiv preprint arXiv:1611.04500, 2016

    2016 arXiv

  21. [29]

    Persformer: A transformer architecture for topological machine learning

    Raphael Reinauer, Matteo Caorsi, and Nicolas Berkouk. Persformer: A transformer architecture for topological machine learning. arXiv preprint arXiv:2112.15210, 2021

    2021 arXiv

  22. [30]

    Weisfeiler-lehman graph kernels

    Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels. Journal of Machine Learning Research, 12 0 (9), 2011

    2011

  23. [31]

    A concise and provably informative multi-scale signature based on heat diffusion

    Jian Sun, Maks Ovsjanikov, and Leonidas Guibas. A concise and provably informative multi-scale signature based on heat diffusion. In Computer graphics forum, volume 28, pp.\ 1383--1392. Wiley Online Library, 2009

    2009

  24. [32]

    Spline-fitting with a genetic algorithm: A method for developing classification structure- activity relationships

    Jeffrey J Sutherland, Lee A O'brien, and Donald F Weaver. Spline-fitting with a genetic algorithm: A method for developing classification structure- activity relationships. Journal of chemical information and computer sciences, 43 0 (6): 0 1906--1915, 2003

    1906

  25. [33]

    Attention is all you need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, ukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017

    2017

  26. [34]

    Comparison of descriptor spaces for chemical compound retrieval and classification

    Nikil Wale, Ian A Watson, and George Karypis. Comparison of descriptor spaces for chemical compound retrieval and classification. Knowledge and Information Systems, 14: 0 347--375, 2008

    2008

  27. [35]

    Loss functions for multiset prediction

    Sean Welleck, Zixin Yao, Yu Gai, Jialin Mao, Zheng Zhang, and Kyunghyun Cho. Loss functions for multiset prediction. Advances in Neural Information Processing Systems, 31, 2018

    2018

  28. [36]

    On layer normalization in the transformer architecture

    Ruibin Xiong, Yunchang Yang, Di He, Kai Zheng, Shuxin Zheng, Chen Xing, Huishuai Zhang, Yanyan Lan, Liwei Wang, and Tieyan Liu. On layer normalization in the transformer architecture. In International Conference on Machine Learning, pp.\ 10524--10533. PMLR, 2020

    2020

  29. [37]

    Deep graph kernels

    Pinar Yanardag and SVN Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining, pp.\ 1365--1374, 2015

    2015

  30. [38]

    Deep sets

    Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. Advances in neural information processing systems, 30, 2017

    2017

  31. [39]

    Lipton, Mu Li, and Alexander J

    Aston Zhang, Zachary C. Lipton, Mu Li, and Alexander J. Smola. Dive into Deep Learning. Cambridge University Press, 2023. https://D2L.ai

    2023

  32. [40]

    Multiset-equivariant set prediction with approximate implicit differentiation

    Yan Zhang, David W Zhang, Simon Lacoste-Julien, Gertjan J Burghouts, and Cees GM Snoek. Multiset-equivariant set prediction with approximate implicit differentiation. arXiv preprint arXiv:2111.12193, 2021

    2021 arXiv

  33. [41]

    Understanding bag-of-words model: a statistical framework

    Yin Zhang, Rong Jin, and Zhi-Hua Zhou. Understanding bag-of-words model: a statistical framework. International journal of machine learning and cybernetics, 1: 0 43--52, 2010

    2010

  34. [42]

    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...

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