Pith. sign in

REVIEW 3 major objections 5 minor 50 references

Polyra Swarms: A Shape-Based Approach to Machine Learning

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper claims that machine learning can be reframed as shape approximation: a Polyra Swarm of random conditional polytopes can approximate any measurable bounded shape to arbitrary accuracy, and this universality makes the method…

desk verdict Genuinely new non-gradient shape-learning representation with a strong abstraction idea, but the empirical superiority claims rest on an unproven link between an existential theorem and a random heuristic. read the letter →

arxiv 2506.13217 v1 pith:KN4EQJIS submitted 2025-06-16 cs.LG cs.NEcs.SC

Simon Klüttermann , Emmanuel Müller This is my paper · ORCID
classification cs.LGcs.NEcs.SC
keywords shapeapproximationPolyraSwarmconditionalpolytopesanomalydetectionlogicalabstractionuniversalinterpretablemachinelearninggradient-free
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 machine learning can be done by approximating shapes rather than functions. The method, a Polyra Swarm, is an intersection of many conditional polytopes, each of which removes an impossible region from the accepted space; training is random and gradient-free. The central claim is Theorem 1: for every measurable, bounded shape with a finite-length boundary and every $\epsilon > 0$, some Polyra Swarm approximates that shape to within $\epsilon$. From this, the authors argue that general-purpose learning follows, because classification, anomaly detection, regression, and generation can be recast as region modeling. They report that, using a fraction-of-submodels anomaly score, Polyra Swarms achieve the best average rank in a 121-dataset anomaly-detection comparison against seven deep learning baselines, and they introduce an abstraction step that compresses a swarm into a much smaller logical rule set.

What carries the argument

The load-bearing object is the conditional polytope base shape $f_i(x) = (x \in A_i \Rightarrow x \in B_i)$, equivalently the complement of $A_i \setminus B_i$. Each base shape removes one impossible region from the accepted set, and the swarm's conjunction makes the accepted set the complement of the union of all removed regions. The universal approximation proof partitions a bounding box into arbitrarily small $d$-simplices, sets each condition polytope $A_i$ to one simplex, and sets each consequent polytope $B_i$ to match the target shape inside that simplex, so the error is at most the total volume of simplices touching the boundary, which tends to zero. The practical learning algorithm draws random Gaussian constraint directions, sets bounds from the training data, and fixes consequent bounds at the most extreme observed values, which by construction keeps every training sample inside the shape.

What would settle it

Fix a target shape with known volume, such as a disk in $\mathbb{R}^2$, draw training samples inside it, and fit Polyra Swarms with increasing numbers of random submodels under the Equation 5 initialization; measure the $L^1$ volume error against the true shape. If the error does not decrease toward zero as the submodel count grows, the random-initialization version of the universal approximation claim fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that learning a data distribution can be replaced by learning the set of points the distribution occupies, and that this set can be built from simple conditional polytopes. A base shape $f_i$ is the implication $x \in A_i \Rightarrow x \in B_i$, which excludes exactly $A_i \setminus B_i$ from the accepted set; a Polyra Swarm intersects many such base shapes, so its accepted region is the complement of the union of all excluded pieces. Theorem 1 shows that for any measurable bounded shape with finite-length boundary, sufficiently many carefully chosen simplices make the excluded-volume error arbitrarily small. The paper then identifies this universality with general-purpose learning: one swarm per class for classification, the graph of a regression function as a shape in $\mathbb{R}^{d+n}$, a swarm over normal points for anomaly detection, and Hit-and-Run-based sampling for generation. In the 121-dataset anomaly-detection study, the authors report that Polyra Swarms outperform all seven deep learning baselines in average rank.

Load-bearing premise

The load-bearing premise is that the random initialization in Equation 5 yields a swarm close to the target shape, while the proof of Theorem 1 uses carefully chosen simplices rather than random halfspaces and gives no bound on the number of submodels the random procedure needs.

Editorial extensions

If this is right

  • Classification with one swarm per class gives a built-in reject option, so a sample can be labeled 'neither' or 'both' instead of being forced into a class.
  • Regression, modeled as the graph of the function in $\mathbb{R}^{d+n}$, returns a valid interval of outputs for each input, which acts as an uncertainty estimate.
  • The abstraction step can compress a fragmented 2000-submodel fit into a five-halfspace description while raising volumetric intersection-over-union from 0.844 to 0.994, so transparency and generalization improve together.
  • Because no gradient descent is used, the fitted swarm cannot get stuck in local minima, which the authors demonstrate on a two-dimensional 'Polyra' text shape that a neural network fails to recover.
  • Classes can be added or removed without retraining the whole model, since each class is described by its own shape.

Reading between the lines

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

  • If the random initialization converges as assumed, the paper's construction gives a universal approximation theorem with no parameter-count or depth caveats; the unproven step is that random halfspaces behave like the carefully chosen simplices in the proof.
  • The abstraction results suggest that logical simplicity and test-set accuracy can coincide, so model selection for Polyra Swarms could be driven by compression of the logical tree rather than by a validation loss.
  • In higher dimensions, the method's cost and fragmentation make it a natural fit as a geometric post-processor on learned representations, which the authors already use with PCA and autoencoders; a testable extension is whether the same anomaly-detection advantage persists when the input is an embedding from a large pretrained model in a low-dimensional space.
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.

Referee Report

3 major / 5 minor

Summary. The paper introduces Polyra Swarms, a new machine-learning paradigm based on shape approximation rather than function approximation. A Polyra Swarm is a conjunction of base shapes, each a logical implication whose condition and consequent are polytopes; the method carves away regions of space until only the region containing the training samples remains. The authors prove, in Theorem 1 (Appendix B), that every measurable bounded shape with finite-length boundary can be approximated arbitrarily well by some Polyra Swarm, and they present a semi-random initialization procedure (Eqs. 5-6). The paper then applies this framework to classification, anomaly detection, regression with uncertainty, generative sampling, and abstraction, and reports state-of-the-art anomaly detection results on 121 ADBench datasets (Figure 4). The central scientific claim is that shape approximation is a viable general-purpose learning alternative to neural networks in low-to-moderate dimensions, with particular strengths in anomaly detection and interpretability.

Significance. If the central claims hold, Polyra Swarms offer a genuinely different inductive bias from neural networks: they avoid gradient-based optimization, provide explicit logical descriptions after abstraction, and can reach extremely low bias on geometric tasks such as the Mandelbrot fit (Figure 1). The universal approximation theorem is plausible and, at the level of a proof sketch, internally coherent; the abstraction procedure, which reduces a 2,000-submodel swarm to a printed five-inequality model (Figure 7), is a concrete and appealing contribution. The paper also ships links to implementation code and detailed appendices for reproducibility, which is a strength. However, the practical learning algorithm is not connected to the theorem by any convergence or sample-complexity guarantee, and the headline anomaly-detection comparison has protocol weaknesses; these are the load-bearing gaps between the existence result and the reported general-purpose behavior.

major comments (3)
  1. [Section 3, Eqs. (5)-(6); Appendix B] The proof of Theorem 1 constructs an approximating swarm from carefully chosen dim-simplices of volume ν obtained by filling the bounding region (Appendix B), whereas the algorithm actually used in all experiments is the random halfspace procedure of Eqs. (5)-(6), where each consequent polytope is an empirical projection hull. No result in the paper shows that this random procedure converges to the target shape as the number of submodels and samples grow, nor does the paper bound the number of submodels needed for a given error ε. Consequently, the practical claims of "universal shape approximation" as an algorithmic guarantee, including the anomaly-detection and classification results, rest on an unverified equivalence between the theorem's construction and the implemented initialization.
  2. [Section 4.3 and Appendix T, Figure 4, Table 11] The claim that Polyra Swarms "outperforms all our competitors and does so significantly for most of them" is not adequately supported by the reported protocol. The better-performing anomaly score (Eq. 34) and the ensemble size of up to 400,000 submodels were selected after observing performance differences (Table 11), and the comparison reports only average ranks with no per-dataset error bars, no per-dataset variance, and no reported p-values for the Wilcoxon or Friedman tests. The paper should either provide a pre-registered or held-out selection protocol, or present the performance as a heuristic demonstration rather than a statistically supported superiority claim.
  3. [Appendix C versus Appendices E.5 and E.6] Appendix C proves that p(x)=TRUE for every training sample under the default setting, and this property is used in the main text to claim that false negatives are impossible by construction. However, Appendices E.5 and E.6 state that setting the quantile or subsample hyperparameters to nonzero values "breaks the assumption that every training sample is considered inside the learned shape." The text should therefore clearly scope the impossibility-of-false-negatives claim to the quantile=0 and subsample=0 configuration; otherwise the property is presented more generally than the method actually provides.
minor comments (5)
  1. [Title page / affiliations] There is a typo in the affiliation: "TU Dortmund Univerity" should read "TU Dortmund University."
  2. [Section 4.3, Figure 4 caption] The caption refers to "a Friedmann test"; the correct spelling is "Friedman test".
  3. [Appendix J.2, Eqs. (23)-(24)] The distributive-law expansions contain repeated terms: in Eq. (23) the last disjunct should be (b ∧ d), not another (a ∧ d), and in Eq. (24) the last conjunct should be (b ∨ d), not another (a ∨ d).
  4. [Appendix E.8] The sentence "This is shown in Figure 17" appears to reference the wrong figure; the volume-versus-ensemble-size behavior is plotted in Figure 18, while Figure 17 concerns dataset size.
  5. [Appendix M.5] There is a typo: "aswell as the limitations" should be "as well as the limitations."

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the universal approximation theorem is self-contained, and the only self-citation (DeepSVDD baseline) is not load-bearing.

full rationale

The central derivation chain is Theorem 1 (Appendix B) plus the random-initialization algorithm (Eqs. 5-6). Theorem 1 is proved constructively: the swarm is rewritten as the complement of a union of carved-out regions (Eq. 9), bounded boxes are filled with dim-simplices, and the error is bounded by ν·C(ν) → 0. This proof does not assume the target result and does not rely on the random algorithm; it is a self-contained existential construction. The random algorithm's lack of a convergence guarantee (Eqs. 5-6 vs. the carefully chosen simplices of Appendix B) is a gap between existence and practice, not a circularity, because no claim in the paper reduces the theorem to the algorithm's output by definition. The property that all training samples are inside the learned shape is, as the paper states, a construction: Eq. 6 sets (bB) to the maximum over Xtrain, so the inclusion proof in Appendix C follows directly. This is presented as a guarantee, not as a prediction. The anomaly-detection comparison uses DeepSVDD (Ruff et al. 2018a), a baseline co-authored by the present second author; however, the comparison is an external benchmark with implementations from pyod/original papers, and the self-citation only identifies the method rather than justifying Polyra's superiority, so it is not load-bearing. Appendix T's selection of the anomaly score and of up to 400,000 submodels after observing benchmark performance is empirical tuning and a statistical risk, but it does not make the reported average rank equivalent to an input by construction. Overall, no equation-level or citation-level circularity is present.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The paper's central claims rest on several hand-chosen hyperparameters (Adim, minpoi, extend, model count, delta_V) and on unproven assumptions that random initialization finds a good approximation and that abstraction improves generalization. The universal approximation theorem is plausible but assumes regular boundaries.

free parameters (5)
  • Adim = varies (e.g., 2, 6)
    Number of constraints in condition polytope; controls shape complexity and is tuned per experiment (Appendix E.1).
  • minpoi = varies (0 to 7500)
    Minimum points required in a condition polytope; controls convexity/complexity trade-off (Appendix E.4).
  • extend = varies (0 to 0.2)
    Factor extending consequent polytope bounds; reduces fragmentation (Appendix E.3).
  • model count = 100 to 1,000,000
    Number of submodels in the swarm; larger ensembles improve anomaly score stability (Appendices E.8, T).
  • delta_V (abstraction) = 0.05
    Allowed volume overshoot when merging polytopes in abstraction; set by hand, works well in experiments (Appendix L.3).
assumptions (3)
  • domain assumption The target shape has a boundary that is locally hyperplane-like, enabling the counting estimate C(nu) proportional to nu^(-(d-1)/d).
    Used in Appendix B proof; excludes fractal boundaries like the Mandelbrot set, which the paper acknowledges in Appendix Q.
  • ad hoc to paper Random halfspace initialization (Eq. 5) with sufficient submodels will approximate any shape with high probability.
    No proof or sample complexity bound is given; the universal approximation proof constructs simplices deterministically rather than via random search.
  • ad hoc to paper The abstraction procedures (sampling and LP-based) preserve the essential shape and improve generalization.
    Demonstrated on toy datasets (diamond, OldFaithful) but no formal guarantee; the paper itself calls the algorithms 'more heuristic than principled' (Section 7).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Polyra Swarms: A Shape-Based Approach to Machine Learning." pith.science (2026). https://pith.science/paper/KN4EQJIS

@misc{pith2026250613217,
  author       = {Pith},
  title        = {Pith review of: Polyra Swarms: A Shape-Based Approach to Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KN4EQJIS}},
  note         = {Machine review of arXiv:2506.13217}
}
read the original abstract

We propose Polyra Swarms, a novel machine-learning approach that approximates shapes instead of functions. Our method enables general-purpose learning with very low bias. In particular, we show that depending on the task, Polyra Swarms can be preferable compared to neural networks, especially for tasks like anomaly detection. We further introduce an automated abstraction mechanism that simplifies the complexity of a Polyra Swarm significantly, enhancing both their generalization and transparency. Since Polyra Swarms operate on fundamentally different principles than neural networks, they open up new research directions with distinct strengths and limitations.

Figures

Figures reproduced from arXiv: 2506.13217 by the authors.

Figure 1
Figure 1. Example Polyra Swarm fit to a dataset inspired by the mandelbrot set (purple). To see [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Example of how to solve matching using shape approximation. Given a pattern between [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Classification results on the two moons dataset using various models. The neural network [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Anomaly detection critical difference plot comparing Polyra swarms to recent/common deep [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Example of a function (sin(x) + 5% error) approximated by a Polyra swarm. For x = π, the valid range of the model (green region) lies between −0.070 and 0.054, closely following the expectation of 0 ± 0.05. Please note that the generated range is exact, but requires th…
Figure 6
Figure 6. Figure 6: Example of samples generated from a Polyra swarm. We first fit a swarm to represent the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Example of the benefits abstraction can provide. We fit a Polyra swarm to samples uniformly [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Equivalent of Figure 7 on a real-world dataset. Left: Shape approximation of the OldFaithful [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Comparison of the precision of a Polyra swarm to a neural network. We search for a [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: From left to right: The ground truth of our toy dataset, the area of a Polyra Swarm with [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Hyperparameter influence of the Adim hyperparameter. E.2 Bdim Next, we consider the number of conditions in the consequent polytope in each submodel (B in Equation 4) in [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: Hyperparameter influence of the Bdim hyperparameter. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Hyperparameter influence of the extend hyperparameter. E.4 Minpoi An alternative to reduce the fragmentation of the learned shape is to demand that a minimum number of samples (minpoi) are included in the condition polytope. This removes small condition polytopes, whi…
Figure 14
Figure 14. Figure 14: Hyperparameter influence of the minpoi hyperparameter. E.5 Quantile The opposite effect of extend is achieved with the quantile parameter. A quantile > 0 means that only a fraction of 1 − quantile is used to select the most extreme values in Equation 6. This hyperpara…
Figure 15
Figure 15. Figure 15: Hyperparameter influence of the quantile hyperparameter. E.6 Subsample A classical approach for one-class classification/outlier detection ensemble methods is to use subsam￾pling Zimek et al. [2013] (See [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Hyperparameter influence of the subsample hyperparameter. E.7 Dataset Size While not strictly a hyperparameter, we also study the effect a different training set has. As [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]
Figure 17
Figure 17. Figure 17: Influence of the dataset size/sample count. [PITH_FULL_IMAGE:figures/full_fig_p017_17.png]
Figure 18
Figure 18. Figure 18: Influence of the number of submodels. F Linearisation In the previous appendix, we mainly studied the effect singular hyperparameter changes have on a learned model. However, interesting effects also happen when we combine multiple hyperparameters. In [PITH_FULL_IMAG…
Figure 19
Figure 19. Figure 19: Example of a hyperparameter combination that we name [PITH_FULL_IMAGE:figures/full_fig_p018_19.png]
Figure 20
Figure 20. Figure 20: ROC AUC score on the cardio dataset Halder [2020] with an isolation forest. In red, you [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: We interpret this effect as indicator that Polyra Swarms don’t work well in high-dimensional data but still have the precision to learn complicated shapes (See Chapter 6.1). We can further improve the performance, by considering anomaly scores (See Chapter 4.3) in the…
Figure 21
Figure 21. Figure 21: Distribution of binary classification task ( [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 22
Figure 22. Figure 22: Toy experiment to show that abstraction can be used for clustering. We generate data [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: Images of MNIST zeros generated by a Polyra Swarm. Because both Polyra Swarms and [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]
Figure 24
Figure 24. Figure 24: Example of how to use a Polyra Swarm to find solutions to Equation 31. We show the [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 35 canonical work pages

  1. [1]

    Adam, Martin Piccolrovazzi, Sebastian Eger, and Eckehard Steinbach

    Michael G. Adam, Martin Piccolrovazzi, Sebastian Eger, and Eckehard Steinbach. Bounding box disparity: 3d metrics for object detection with full degree of freedom. In 2022 IEEE International Conference on Image Processing (ICIP), page 1491–1495. IEEE, October 2022. doi:10.1109/icip46576.2022.9897588. URL http://dx.doi.org/10.1109/ICIP46576.2022.9897588

    arXiv 2022

  2. [2]

    Does deep learning learn to abstract? a systematic probing framework

    Shengnan An, Zhen Lin, Beichen Chen, Qi Fu, Nian Zheng, and Jianguo Lou. Does deep learning learn to abstract? a systematic probing framework. In Proceedings of the International Conference on Learning Representations (ICLR), 2023. URL https://openreview.net/forum?id=QB1dMPEXau5

    work page 2023

  3. [3]

    Azzalini and A

    A. Azzalini and A. W. Bowman. A look at some data on the old faithful geyser. Journal of the Royal Statistical Society. Series C (Applied Statistics), 39 0 (3): 0 357--365, 1990. ISSN 00359254, 14679876. URL http://www.jstor.org/stable/2347385

    work page arXiv 1990

  4. [4]

    Dynamic programming

    Richard Bellman. Dynamic programming. Science, 153 0 (3731): 0 34--37, 1966

    work page 1966

  5. [5]

    Classification-based anomaly detection for general data

    Liron Bergman and Yedid Hoshen. Classification-based anomaly detection for general data. In ICLR. OpenReview.net, 2020. URL http://dblp.uni-trier.de/db/conf/iclr/iclr2020.html#BergmanH20

    work page 2020

  6. [6]

    Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Occam's razor. Information Processing Letters, 24 0 (6): 0 377--380, 1987. ISSN 0020-0190. doi:https://doi.org/10.1016/0020-0190(87)90114-1. URL https://www.sciencedirect.com/science/article/pii/0020019087901141

    arXiv 1987

  7. [7]

    Analysis of explainers of black box deep neural networks for computer vision: A survey

    Vanessa Buhrmester, David Münch, and Michael Arens. Analysis of explainers of black box deep neural networks for computer vision: A survey. Machine Learning and Knowledge Extraction, 3 0 (4): 0 966--989, 2021. ISSN 2504-4990. doi:10.3390/make3040048. URL https://www.mdpi.com/2504-4990/3/4/48

    doi:10.3390/make3040048 2021

  8. [8]

    Giordano, Michele Pagano, and Teresa Pepe

    Christian Callegari, Loris Gazzarrini, S. Giordano, Michele Pagano, and Teresa Pepe. A novel pca-based network anomaly detection. pages 1 -- 5, 07 2011. doi:10.1109/icc.2011.5962595

    arXiv 2011

Show all 50 references
  1. [9]

    Topology and data

    Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46 0 (2): 0 255--308, 2009. doi:10.1090/S0273-0979-09-01249-X

    2009 doi

  2. [10]

    Fast mcmc sampling algorithms on polytopes

    Yuansi Chen, Raaz Dwivedi, Martin Wainwright, and Bin Yu. Fast mcmc sampling algorithms on polytopes. Journal of Machine Learning Research, 19, 10 2017. doi:10.48550/arXiv.1710.08165

    2017 doi arXiv

  3. [11]

    The mnist database of handwritten digit images for machine learning research

    Li Deng. The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29, 2012

    2012

  4. [12]

    The Reflex Arc Concept in Psychology, volume 3

    John Dewey. The Reflex Arc Concept in Psychology, volume 3. The Psychological Review, 1896. doi:10.1037/h0070601

    doi

  5. [13]

    Diffusion models beat gans on image synthesis

    Prafulla Dhariwal and Alex Nichol. Diffusion models beat gans on image synthesis. In Proceedings of the 35th International Conference on Neural Information Processing Systems, NIPS '21, Red Hook, NY, USA, 2021. Curran Associates Inc. ISBN 9781713845393

    2021

  6. [14]

    Benoit Richard Umbert Dherin, Micheal Munn, and David G. T. Barrett. The geometric occam's razor implicit in deep learning. ArXiv, abs/2111.15090, 2021

    2021 arXiv

  7. [15]

    Empirical comparison between autoencoders and traditional dimensionality reduction methods

    Quentin Fournier and Daniel Aloise. Empirical comparison between autoencoders and traditional dimensionality reduction methods. In 2019 IEEE Second International Conference on Artificial Intelligence and Knowledge Engineering (AIKE), pages 211--214, 2019. doi:10.1109/AIKE.2019.00044

    2019

  8. [16]

    A comparison of alternative tests of significance for the problem of \ m\ rankings

    Milton Friedman. A comparison of alternative tests of significance for the problem of \ m\ rankings. Annals of Mathematical Statistics, 11: 0 86--92, 1940

    1940

  9. [17]

    Cardiovascular disease dataset, 2020

    Rajib Kumar Halder. Cardiovascular disease dataset, 2020. URL https://dx.doi.org/10.21227/7qm5-dz13

    2020 doi

  10. [18]

    Adbench: Anomaly detection benchmark

    Songqiao Han, Xiyang Hu, Hailiang Huang, Mingqi Jiang, and Yue Zhao. Adbench: Anomaly detection benchmark. In NeurIPS, 2022

    2022

  11. [19]

    Machine learning with a reject option: A survey, 2024

    Kilian Hendrickx, Lorenzo Perini, Dries Van der Plas, Wannes Meert, and Jesse Davis. Machine learning with a reject option: A survey, 2024. URL https://arxiv.org/abs/2107.11277

    2024 arXiv

  12. [20]

    Deep learning with topological signatures

    Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl. Deep learning with topological signatures. In Advances in Neural Information Processing Systems (NeurIPS), volume 30, 2017

    2017

  13. [21]

    A simple sequentially rejective multiple test procedure

    Sture Holm. A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6 0 (2): 0 65--70, 1979. ISSN 03036898, 14679469. URL http://www.jstor.org/stable/4615733

    1979

  14. [22]

    Khan and Michael G

    Shehroz S. Khan and Michael G. Madden. A survey of recent trends in one class classification. In Lorcan Coyle and Jill Freyne, editors, Artificial Intelligence and Cognitive Science, pages 188--197, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. ISBN 978-3-642-17080-5

    2010

  15. [23]

    Gregory R. Koch. Siamese neural networks for one-shot image recognition. 2015

    2015

  16. [24]

    The self-organizing map

    Teuvo Kohonen. The self-organizing map. Proceedings of the IEEE, 78 0 (9): 0 1464--1480, 1990. doi:10.1109/5.58325

    1990 doi

  17. [25]

    Siegelmann

    Robert Kozma, Roman Ilin, and Hava T. Siegelmann. Evolution of abstraction across layers in deep learning neural networks. Procedia Computer Science, 144: 0 203--213, 2018. ISSN 1877-0509. doi:https://doi.org/10.1016/j.procs.2018.10.520. URL https://www.sciencedirect.com/scien...

    2018 doi

  18. [26]

    On diffusion modeling for anomaly detection

    Victor Livernoche, Vineet Jain, Yashar Hezaveh, and Siamak Ravanbakhsh. On diffusion modeling for anomaly detection. In ICLR 2024, 2024. URL https://openreview.net/forum?id=lR3rk7ysXz

    2024

  19. [27]

    Mitchell

    Tom M. Mitchell. Machine Learning. McGraw-Hill, New York, 1997. ISBN 9780070428072

    1997

  20. [28]

    Meta-survey on outlier and anomaly detection

    Madalina Olteanu, Fabrice Rossi, and Florian Yger. Meta-survey on outlier and anomaly detection. Neurocomputing, 555: 0 126634, 2023. ISSN 0925-2312. doi:https://doi.org/10.1016/j.neucom.2023.126634. URL https://www.sciencedirect.com/science/article/pii/S0925231223007579

    2023

  21. [29]

    Pardalos

    Panos M. Pardalos. Linear programmingLinear Programming, pages 1276--1278. Springer US, Boston, MA, 2001. ISBN 978-0-306-48332-5. doi:10.1007/0-306-48332-7_261. URL https://doi.org/10.1007/0-306-48332-7_261

    2001 doi

  22. [30]

    Kepler’s goat herd: An exact solution to kepler’s equation for elliptical orbits

    Oliver H E Philcox, Jeremy Goodman, and Zachary Slepian. Kepler’s goat herd: An exact solution to kepler’s equation for elliptical orbits. Monthly Notices of the Royal Astronomical Society, 506 0 (4): 0 6111–6116, May 2021. ISSN 1365-2966. doi:10.1093/mnras/stab1296. URL http:...

    2021 doi

  23. [31]

    Functionalism as a philosophical theory of the cognitive sciences

    Thomas Polger. Functionalism as a philosophical theory of the cognitive sciences. Wiley Interdisciplinary Reviews: Cognitive Science, 3, 05 2012. doi:10.1002/wcs.1170

    2012 doi

  24. [32]

    Interior Point Methods for Nonlinear Optimization, pages 215--276

    Imre P \'o lik and Tam \'a s Terlaky. Interior Point Methods for Nonlinear Optimization, pages 215--276. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. ISBN 978-3-642-11339-0. doi:10.1007/978-3-642-11339-0_4. URL https://doi.org/10.1007/978-3-642-11339-0_4

    2010 doi

  25. [33]

    Neural transformation learning for deep anomaly detection beyond images, 2022

    Chen Qiu, Timo Pfrommer, Marius Kloft, Stephan Mandt, and Maja Rudolph. Neural transformation learning for deep anomaly detection beyond images, 2022. URL https://arxiv.org/abs/2103.16440

    2022 arXiv

  26. [34]

    Ross Quinlan

    J. Ross Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufmann, 1993. ISBN 978-1-55860-238-0

    1993

  27. [35]

    Occam s razor

    Carl Rasmussen and Zoubin Ghahramani. Occam s razor. In T. Leen, T. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems, volume 13. MIT Press, 2000. URL https://proceedings.neurips.cc/paper_files/paper/2000/file/0950ca92a4dcf426067cfd2246bb5ff3...

    2000

  28. [36]

    Variational inference with normalizing flows

    Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, ICML'15, page 1530–1538. JMLR.org, 2015

    2015

  29. [37]

    The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain, volume 65

    Frank Rosenblatt. The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain, volume 65. Psychological Review, 1958. doi:10.1037/h0042519

    1958 doi

  30. [38]

    Deep one-class classification

    Lukas Ruff, Robert Vandermeulen, Nico Goernitz, Lucas Deecke, Shoaib Ahmed Siddiqui, Alexander Binder, Emmanuel M \"u ller, and Marius Kloft. Deep one-class classification. In ICML, 2018 a

    2018

  31. [39]

    Deep one-class classification

    Lukas Ruff, Robert Vandermeulen, Nico Goernitz, Lucas Deecke, Shoaib Ahmed Siddiqui, Alexander Binder, Emmanuel M \"u ller, and Marius Kloft. Deep one-class classification. In Proceedings of the 35th International Conference on Machine Learning, 2018 b

    2018

  32. [40]

    Kauffmann, Robert A

    Lukas Ruff, Jacob R. Kauffmann, Robert A. Vandermeulen, Gregoire Montavon, Wojciech Samek, Marius Kloft, Thomas G. Dietterich, and Klaus-Robert Muller. A unifying review of deep and shallow anomaly detection. Proceedings of the IEEE, 109 0 (5): 0 756–795, May 2021. ISSN 1558-2256

    2021

  33. [41]

    Titchener

    Edward B. Titchener. An Outline of Psychology. Macmillan, 1896

  34. [42]

    Santosh S. Vempala. Hit-and-run is fast and fun. 2003. URL https://api.semanticscholar.org/CorpusID:8398546

    2003

  35. [43]

    Individual comparisons by ranking methods

    Frank Wilcoxon. Individual comparisons by ranking methods. Biometrics Bulletin, 1 0 (6): 0 80--83, 1945. ISSN 00994987

    1945

  36. [44]

    K. V. Wilkes. Functionalism, psychology, and the philosophy of mind. Philosophical Topics, 12 0 (1): 0 147--167, 1981. ISSN 02762080, 2154154X. URL http://www.jstor.org/stable/43153849

    1981

  37. [45]

    Mang Ye, Jianbing Shen, Gaojie Lin, Tao Xiang, Ling Shao, and Steven C. H. Hoi. Deep Learning for Person Re-Identification: A Survey and Outlook . IEEE Transactions on Pattern Analysis & Machine Intelligence, 44 0 (06): 0 2872--2893, June 2022. ISSN 1939-3539. doi:10.1109/TPAM...

    2022

  38. [46]

    Zelle and Raymond J

    John M. Zelle and Raymond J. Mooney. Learning to parse database queries using inductive logic programming. In Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI), pages 1050--1055, Portland, OR, USA, 1996. AAAI Press

    1996

  39. [47]

    Pyod: A python toolbox for scalable outlier detection

    Yue Zhao, Zain Nasrullah, and Zheng Li. Pyod: A python toolbox for scalable outlier detection. Journal of Machine Learning Research, 20 0 (96): 0 1--7, 2019. URL http://jmlr.org/papers/v20/19-011.html

    2019

  40. [48]

    Feature Engineering for Machine Learning: Principles and Techniques for Data Scientists

    Alice Zheng and Amanda Casari. Feature Engineering for Machine Learning: Principles and Techniques for Data Scientists. O'Reilly Media, Inc., 1st edition, 2018. ISBN 1491953241

    2018

  41. [49]

    Campello, and J\" o rg Sander

    Arthur Zimek, Matthew Gaudet, Ricardo J.G.B. Campello, and J\" o rg Sander. Subsampling for efficient and effective unsupervised outlier detection ensembles. In Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '13, page 42...

    2013

  42. [50]

    Deep autoencoding gaussian mixture model for unsupervised anomaly detection

    Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Dae ki Cho, and Haifeng Chen. Deep autoencoding gaussian mixture model for unsupervised anomaly detection. In International Conference on Learning Representations, 2018. URL https://api.semanticscholar.org/Co...

    2018

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.