pith. sign in

arxiv: 2606.17782 · v1 · pith:Y237FTE5new · submitted 2026-06-16 · 💻 cs.LG

Blind Recovery of Latent Domains via Unsupervised Symmetry Discovery

Onur Efe , Arkadas Ozakin This is my paper

Pith reviewed 2026-06-27 02:01 UTC · model grok-4.3

classification 💻 cs.LG
keywords symmetry discoveryblind inverse problemsgroup convolutional networkslatent domain recoveryunsupervised structure learningstationarity regularizationlocality regularization
0
0 comments X

The pith

A shallow group-convolutional network recovers latent domains and signals from linear measurements by discovering the underlying symmetries of the data distribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to recover signals and their domains from observations corrupted by unknown linear transformations. It models the observations as linear measurements of signals drawn from a latent random field and trains a shallow group-convolutional network to find the field's symmetries while enforcing stationarity and locality at the output. If this works, the network learns both a symmetry action and a filter that together produce a representation in which the latent signals become visible. A reader would care because many practical blind inverse problems involve general linear obfuscations that standard deconvolution cannot handle, such as shuffled images or scrambled neural recordings.

Core claim

The central claim is that optimizing a shallow group-convolutional network by imposing stationarity and locality regularization at the model output allows the model to learn a latent symmetry action and an appropriate filter, thereby mapping unstructured observations to a symmetry-based representation that reveals latent signals.

What carries the argument

A shallow group-convolutional network optimized under stationarity and locality regularization at the output to learn the latent symmetry action and filter.

If this is right

  • The learned symmetry action maps the observations back onto the latent domain structure.
  • Latent signals become recoverable from stochastic processes and Ising models without knowledge of the measurement process.
  • The same procedure succeeds on shuffled images, bit-scrambled images, and neural recordings.
  • Symmetry discovery supplies a general route to unsupervised structure learning in blind inverse problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on data whose linear measurement operator is known in advance to verify that the learned filter matches the true corruption.
  • If the latent symmetry group is discrete, the method might extend to recovering categorical latent variables in other high-dimensional datasets.
  • Approximate or noisy symmetries could be handled by relaxing the exact stationarity constraint, broadening applicability to real recordings.

Load-bearing premise

Observations are linear measurements of signals sampled from a latent random field whose symmetries can be recovered by the network optimization.

What would settle it

Generate observations from a latent field that has no recoverable symmetry group, such as completely unstructured random noise, apply the method, and check whether it still produces a symmetry action that maps the data to a coherent recovered signal.

Figures

Figures reproduced from arXiv: 2606.17782 by Arkadas Ozakin, Onur Efe.

Figure 1
Figure 1. Figure 1: Bit-scrambled MNIST recovery. The model is trained on randomly shuffled bits obtained from image pixels. The underlying digit, up to reflection, is recovered by discovering the translation symmetry action and an appropriate filter. Across all experiments, the model operates directly on unordered vector observations. First, we demonstrate domain recovery and symmetry discovery under dense linear transformat… view at source ↗
Figure 2
Figure 2. Figure 2: Symmetry-based domain recovery. Latent signals are superpositions of translated band-limited point sources. Under the augmented measurement, these become orbit elements ρ(s)h0, whose superpositions form samples z. Learning this symmetry orbit reveals the latent domain structure. 3 Problem setting Using a translation symmetry group as a domain surrogate. Our formulation accesses the latent domain through tr… view at source ↗
Figure 3
Figure 3. Figure 3: Training scheme. (a) The lifting network update optimizes E, Lˆ, and w0 using stationarity, resolution, and InfoMax losses. (b) The auxiliary network update fits statistical divergence and entropy estimators. Homogeneous case. In some experimental settings, we encounter a case where f(t) is not a general superposition of orbit elements in OδB , but a single orbit element. We refer to this as the homogeneou… view at source ↗
Figure 4
Figure 4. Figure 4: Waveform recovery experiments. (a) Native domain: learned actions and resolving filters match one￾dimensional translations and point responses. (b) DST-I domain: learned actions and filters transform accordingly, with point responses represented by sinusoidal filters. (c) The learned lifting approximately inverts the DST-I transform, yielding an identity-like composition. We evaluate FFT-based fractional c… view at source ↗
Figure 5
Figure 5. Figure 5: Learning rotation equivariance from neural signals. Samples are color-coded by stimulus orientation. (a–b) The learned output forms a sinusoidal code whose phase recovers the circular orientation order, while amplitude captures response strength. (c) In the first two principal components, real and synthetically shifted samples show substantial overlap. correspond to the embedding coordinates for the i-th s… view at source ↗
Figure 7
Figure 7. Figure 7: At each layer, we either use strides or pooling layer to downsample the features along spatial axis. Additionally, [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Invariant Data Distributions: Gaussian vs. Legendre [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Lifting map and bitstream recomposition. (a) The flattened lifting map displays parallel diagonal lines consistently across each bit channel. Since vertical and horizontal pixel shifts correspond to index increments of 7 and 1 respectively for flattened 7 × 7 patch, this pattern forms a banded Toeplitz structure functioning as a local averaging operator with a receptive field of ≈ 3 pixels. (b) The bit wei… view at source ↗
read the original abstract

Primary motivation in blind inverse problems is to recover signals of interest from corrupted observations without knowing the obfuscating mechanism. Blind deconvolution is a prominent approach when the corruption is convolutional, but it is not applicable when general linear transformations obfuscate the domain structure. In this work, we propose an unsupervised framework for recovering latent domains and signals by discovering symmetries of the data distribution. Our framework models observations as linear measurements of signals sampled from a latent random field, and optimizes a shallow group-convolutional network by imposing stationarity and locality regularization at the model output. The model learns a latent symmetry action and an appropriate filter, thereby mapping unstructured observations to a symmetry-based representation that reveals latent signals. Experiments on stochastic processes, Ising models, shuffled and bit-scrambled images, and neural recordings show that the method recovers latent domains and signals from unstructured observations, suggesting symmetry discovery as a new direction for unsupervised structure learning and blind inverse problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes an unsupervised framework for recovering latent domains and signals from linear measurements of observations by discovering symmetries of the data distribution. It models observations as linear measurements of signals from a latent random field and optimizes a shallow group-convolutional network under stationarity and locality regularization at the output to learn a latent symmetry action and filter, with experiments on stochastic processes, Ising models, shuffled images, and neural recordings demonstrating recovery of latent structures.

Significance. If the central claim holds, the work could establish symmetry discovery as a viable direction for unsupervised structure learning and blind inverse problems, extending beyond convolutional assumptions in traditional blind deconvolution. The multi-domain experiments provide initial empirical support for practical utility in recovering signals from unstructured data.

major comments (2)
  1. [Abstract] Abstract (modeling premise): the claim that optimizing the shallow group-convolutional network under output stationarity and locality regularization recovers the unknown latent symmetry group and linear measurement operator lacks any identifiability analysis, uniqueness proof, or conditions on the loss landscape guaranteeing that the true group action is recovered rather than a trivial or alternative solution satisfying the same constraints.
  2. [Abstract] Abstract (experiments): the reported recoveries on stochastic processes, Ising models, shuffled/bit-scrambled images, and neural recordings are presented as evidence, but without accompanying analysis of how the regularization isolates the latent symmetry (e.g., ablation on the stationarity/locality terms or checks against degenerate group parameterizations), the results do not confirm that recovery occurs for the stated theoretical reason.
minor comments (1)
  1. [Abstract] The abstract refers to 'shallow group-convolutional network' and 'symmetry-based representation' without defining the group parameterization or network architecture details that would allow reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address the two major comments point by point below, acknowledging the absence of formal identifiability analysis and ablation studies in the current version. We plan revisions to strengthen the presentation of the framework's claims and empirical support.

read point-by-point responses

  1. Referee: [Abstract] Abstract (modeling premise): the claim that optimizing the shallow group-convolutional network under output stationarity and locality regularization recovers the unknown latent symmetry group and linear measurement operator lacks any identifiability analysis, uniqueness proof, or conditions on the loss landscape guaranteeing that the true group action is recovered rather than a trivial or alternative solution satisfying the same constraints.

    Authors: The abstract summarizes the optimization objective and its intended outcome based on the stationarity and locality constraints derived from the latent random field model. The manuscript does not contain a formal identifiability analysis, uniqueness proof, or detailed characterization of the loss landscape. We agree this is a substantive gap: without such analysis it remains possible that alternative or degenerate solutions satisfy the constraints. In the revised manuscript we will add a dedicated discussion paragraph addressing the loss landscape under the group-convolutional parameterization and stating the modeling assumptions (locality of the filter and stationarity of the field) under which recovery of the true action is expected. We will also revise the abstract wording to indicate that recovery is observed empirically rather than theoretically guaranteed. revision: yes

  2. Referee: [Abstract] Abstract (experiments): the reported recoveries on stochastic processes, Ising models, shuffled/bit-scrambled images, and neural recordings are presented as evidence, but without accompanying analysis of how the regularization isolates the latent symmetry (e.g., ablation on the stationarity/locality terms or checks against degenerate group parameterizations), the results do not confirm that recovery occurs for the stated theoretical reason.

    Authors: The experiments are designed to demonstrate that the framework recovers plausible latent domains and signals across synthetic and real data. We acknowledge that the current presentation does not include ablations on the stationarity or locality terms nor explicit verification against degenerate group actions, making it harder to attribute success specifically to the regularization isolating the latent symmetry. We will add these analyses in the revision: (i) ablation experiments that disable each regularization term individually and report degradation in recovery metrics, and (ii) checks that the learned group actions are non-trivial (e.g., by comparing against fixed-identity or random group parameterizations). These additions will directly address how the constraints drive the observed recoveries. revision: yes

Circularity Check

0 steps flagged

Proposed framework with empirical validation; no reduction to self-definition or fitted inputs

full rationale

The paper defines a modeling premise (observations as linear measurements of a latent random field) and a method (optimize shallow group-convolutional network under output stationarity and locality regularization to recover symmetry action and filter). This is presented as a proposed unsupervised approach, not a derivation claiming first-principles uniqueness. No equations or steps reduce by construction to their inputs; the central claim rests on experimental recoveries across multiple datasets rather than tautological identities or self-citation chains. Per rules, this qualifies as self-contained against external benchmarks with score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only; ledger entries extracted directly from stated modeling choices.

axioms (1)
  • domain assumption Observations are linear measurements of signals sampled from a latent random field
    Core modeling assumption stated in the abstract.
invented entities (1)
  • latent symmetry action no independent evidence
    purpose: Maps unstructured observations to a symmetry-based representation revealing latent signals
    Learned component of the model; no independent evidence provided in abstract.

pith-pipeline@v0.9.1-grok · 5683 in / 1080 out tokens · 32491 ms · 2026-06-27T02:01:00.797112+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 17 canonical work pages · 4 internal anchors

  1. [1]

    Blind deconvolution in autocorrelation inversion for multiview light-sheet microscopy.Microscopy Research and Technique, 85(6):2349–2358, 2022

    S Corbetta, D Ancora, and A Bassi. Blind deconvolution in autocorrelation inversion for multiview light-sheet microscopy.Microscopy Research and Technique, 85(6):2349–2358, 2022

    2022

  2. [2]

    Blind calibration of sensor networks

    Laura Balzano and Robert Nowak. Blind calibration of sensor networks. InProceedings of the 6th International Conference on Information Processing in Sensor Networks, pages 79–90. ACM, 2007

    2007

  3. [3]

    Image scrambling encryption algorithm of pixel bit based on chaos map.Pattern Recognition Letters, 31(5):347–354, 2010

    Guodong Ye. Image scrambling encryption algorithm of pixel bit based on chaos map.Pattern Recognition Letters, 31(5):347–354, 2010

    2010

  4. [4]

    Non-invasive imaging through opaque scattering layers.Nature, 491(7423):232–234, 2012

    Jacopo Bertolotti, Elbert G Van Putten, Christian Blum, Ad Lagendijk, Willem L V os, and Allard P Mosk. Non-invasive imaging through opaque scattering layers.Nature, 491(7423):232–234, 2012

    2012

  5. [5]

    Learning the 2-d topology of images.Advances in Neural Information Processing Systems, 20, 2007

    Nicolas Roux, Yoshua Bengio, Pascal Lamblin, Marc Joliveau, and Balázs Kégl. Learning the 2-d topology of images.Advances in Neural Information Processing Systems, 20, 2007

    2007

  6. [6]

    Symmetry discovery with deep learning.Physical Review D, 105(9):096031, 2022

    Krish Desai, Benjamin Nachman, and Jesse Thaler. Symmetry discovery with deep learning.Physical Review D, 105(9):096031, 2022

    2022

  7. [7]

    Latent space symmetry discovery.arXiv preprint arXiv:2310.00105, 2023

    Jianke Yang, Nima Dehmamy, Robin Walters, and Rose Yu. Latent space symmetry discovery.arXiv preprint arXiv:2310.00105, 2023

    work page arXiv 2023

  8. [8]

    Metzler, Richard G

    Gregory Ongie, Ajil Jalal, Christopher A. Metzler, Richard G. Baraniuk, Alexandros G. Dimakis, and Rebecca Willett. Deep learning techniques for inverse problems in imaging, 2020. URL https://arxiv.org/abs/2005. 06001

    2020

  9. [9]

    Understanding and evaluating blind deconvolution algorithms

    Anat Levin, Yair Weiss, Fredo Freeman, and Fredo Durand. Understanding and evaluating blind deconvolution algorithms. InIEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 1964–1971. IEEE, 2009

    1964

  10. [10]

    Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations.Nature Photonics, 8(10):784–790, 2014

    Ori Katz, Pierre Heidmann, Mathias Fink, and Sylvain Gigan. Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations.Nature Photonics, 8(10):784–790, 2014

    2014

  11. [11]

    Multichannel blind deconvolution of seismic signals.Geophysics, 63(6): 2093–2107, 1998

    Kjetil F Kaaresen and Torfinn Taxt. Multichannel blind deconvolution of seismic signals.Geophysics, 63(6): 2093–2107, 1998

    2093

  12. [12]

    Independent component analysis of electroencephalographic data

    Scott Makeig, Anthony J Bell, Tzyy-Ping Jung, and Terrence J Sejnowski. Independent component analysis of electroencephalographic data. InAdvances in Neural Information Processing Systems (NIPS), volume 8, pages 145–151, 1996

    1996

  13. [13]

    Learning partial equivariances from data.arXiv preprint arXiv:2110.10211, 2021

    David W Romero and Suhas Lohit. Learning partial equivariances from data.arXiv preprint arXiv:2110.10211, 2021

    work page arXiv 2021

  14. [14]

    Learning equivariant models by discovering symmetries with learnable augmentations, 2025

    Eduardo Santos-Escriche and Stefanie Jegelka. Learning equivariant models by discovering symmetries with learnable augmentations, 2025. URLhttps://arxiv.org/abs/2506.03914

    work page arXiv 2025

  15. [15]

    Deep learning symmetries and their lie groups, algebras, and subalgebras from first principles.Machine Learning: Science and Technology, 4(2):025027, 2023

    Roy T Forestano, Konstantin T Matchev, Katia Matcheva, Alexander Roman, Eyup B Unlu, and Sarunas Verner. Deep learning symmetries and their lie groups, algebras, and subalgebras from first principles.Machine Learning: Science and Technology, 4(2):025027, 2023

    2023

  16. [16]

    Liegg: Studying learned lie group generators

    Artem Moskalev, Anna Sepliarskaia, Ivan Sosnovik, and Arnold Smeulders. Liegg: Studying learned lie group generators. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors,Advances in Neural Information Processing Systems, volume 35, pages 25212–25223. Curran Associates, Inc., 2022. URL https://proceedings.neurips.cc/paper_files/pa...

    2022

  17. [17]

    Atlasd: Automatic local symmetry discovery.arXiv preprint arXiv:2504.10777, 2025

    Manu Bhat, Jonghyun Park, Jianke Yang, Nima Dehmamy, Robin Walters, and Rose Yu. Atlasd: Automatic local symmetry discovery.arXiv preprint arXiv:2504.10777, 2025

    work page arXiv 2025

  18. [18]

    Meta-learning symmetries by reparameterization.arXiv preprint arXiv:2007.02933, 2020

    Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization.arXiv preprint arXiv:2007.02933, 2020

    work page arXiv 2007

  19. [19]

    Hamiltonian neural networks.Advances in neural information processing systems, 32, 2019

    Samuel Greydanus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks.Advances in neural information processing systems, 32, 2019

    2019

  20. [20]

    Noether networks: meta-learning useful conserved quantities.Advances in Neural Information Processing Systems, 34:16384–16397, 2021

    Ferran Alet, Dylan Doblar, Allan Zhou, Josh Tenenbaum, Kenji Kawaguchi, and Chelsea Finn. Noether networks: meta-learning useful conserved quantities.Advances in Neural Information Processing Systems, 34:16384–16397, 2021

    2021

  21. [21]

    Flow map learning for unknown dynamical systems: Overview, implementation, and benchmarks.Journal of Machine Learning for Modeling and Computing, 4(2):173–201, 2023

    Victor Churchill and Dongbin Xiu. Flow map learning for unknown dynamical systems: Overview, implementation, and benchmarks.Journal of Machine Learning for Modeling and Computing, 4(2):173–201, 2023. ISSN 2689- 3967. 11 Blind Recovery of Latent Domains via Unsupervised Symmetry DiscoveryA PREPRINT

    2023

  22. [22]

    An Unsupervised Algorithm For Learning Lie Group Transformations

    Jascha Sohl-Dickstein, Ching Ming Wang, and Bruno A Olshausen. An unsupervised algorithm for learning lie group transformations.arXiv preprint arXiv:1001.1027, 2010

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    Nathan and Brunton, Steven L

    Kathleen Champion, Bethany Lusch, J. Nathan Kutz, and Steven L. Brunton. Data-driven discovery of coordinates and governing equations.Proceedings of the National Academy of Sciences, 116(45):22445–22451, October 2019. ISSN 1091-6490. doi: 10.1073/pnas.1906995116. URLhttp://dx.doi.org/10.1073/pnas.1906995116

    work page doi:10.1073/pnas.1906995116 2019

  24. [24]

    Automatic symmetry discovery with lie algebra convolutional network.Advances in Neural Information Processing Systems, 34:2503–2515, 2021

    Nima Dehmamy, Robin Walters, Yanchen Liu, Dashun Wang, and Rose Yu. Automatic symmetry discovery with lie algebra convolutional network.Advances in Neural Information Processing Systems, 34:2503–2515, 2021

    2021

  25. [25]

    Neural fourier transform: A general approach to equivariant representation learning.arXiv preprint arXiv:2305.18484, 2023

    Masanori Koyama, Kenji Fukumizu, Kohei Hayashi, and Takeru Miyato. Neural fourier transform: A general approach to equivariant representation learning.arXiv preprint arXiv:2305.18484, 2023

    work page arXiv 2023

  26. [26]

    Unsupervised learning of equivariant structure from sequences.Advances in Neural Information Processing Systems, 35:768–781, 2022

    Takeru Miyato, Masanori Koyama, and Kenji Fukumizu. Unsupervised learning of equivariant structure from sequences.Advances in Neural Information Processing Systems, 35:768–781, 2022

    2022

  27. [27]

    Learning symmetric embeddings for equivariant world models.arXiv preprint arXiv:2204.11371, 2022

    Jung Yeon Park, Ondrej Biza, Linfeng Zhao, Jan Willem van de Meent, and Robin Walters. Learning symmetric embeddings for equivariant world models.arXiv preprint arXiv:2204.11371, 2022

    work page arXiv 2022

  28. [28]

    Neural isometries: Taming transformations for equivariant ml.Advances in Neural Information Processing Systems, 37:7311–7338, 2024

    Thomas Mitchel, Michael J Taylor, and Vincent Sitzmann. Neural isometries: Taming transformations for equivariant ml.Advances in Neural Information Processing Systems, 37:7311–7338, 2024

    2024

  29. [29]

    Learning invariances in neural networks from training data.Advances in neural information processing systems, 33:17605–17616, 2020

    Gregory Benton, Marc Finzi, Pavel Izmailov, and Andrew G Wilson. Learning invariances in neural networks from training data.Advances in neural information processing systems, 33:17605–17616, 2020

    2020

  30. [30]

    A method to challenge symmetries in data with self-supervised learning

    Rupert Tombs and Christopher G Lester. A method to challenge symmetries in data with self-supervised learning. Journal of Instrumentation, 17(08):P08024, 2022

    2022

  31. [31]

    Generative adversarial symmetry discovery

    Jianke Yang, Robin Walters, Nima Dehmamy, and Rose Yu. Generative adversarial symmetry discovery. In International Conference on Machine Learning, pages 39488–39508. PMLR, 2023

    2023

  32. [32]

    Symmetrylens: Unsupervised symmetry learning via locality and density preservation.Symmetry, 17(3), 2025

    Onur Efe and Arkadas Ozakin. Symmetrylens: Unsupervised symmetry learning via locality and density preservation.Symmetry, 17(3), 2025. doi: 10.3390/sym17030425

    work page doi:10.3390/sym17030425 2025

  33. [33]

    Ben Shaw, Abram Magner, and Kevin R. Moon. Symmetry discovery beyond affine transformations, 2024. URL https://arxiv.org/abs/2406.03619

    work page arXiv 2024

  34. [34]

    Learning discrete structures for graph neural networks

    Luca Franceschi, Mathias Niepert, Massimiliano Pontil, and Xiao He. Learning discrete structures for graph neural networks. InInternational conference on machine learning, pages 1972–1982. PMLR, 2019

    1972

  35. [35]

    A survey on graph structure learning: Progress and opportunities.arXiv preprint arXiv:2103.03036, 2021

    Yanqiao Zhu, Weizhi Xu, Jinghao Zhang, Yuanqi Du, Jieyu Zhang, Qiang Liu, Carl Yang, and Shu Wu. A survey on graph structure learning: Progress and opportunities.arXiv preprint arXiv:2103.03036, 2021

    work page arXiv 2021

  36. [36]

    A global geometric framework for nonlinear dimen- sionality reduction.science, 290(5500):2319–2323, 2000

    Joshua B Tenenbaum, Vin de Silva, and John C Langford. A global geometric framework for nonlinear dimen- sionality reduction.science, 290(5500):2319–2323, 2000

    2000

  37. [37]

    UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

    Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction, 2020. URLhttps://arxiv.org/abs/1802.03426

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  38. [38]

    Fast, expressive se(n) equivariant networks through weight-sharing in position-orientation space, 2024

    Erik J Bekkers, Sharvaree Vadgama, Rob D Hesselink, Putri A van der Linden, and David W Romero. Fast, expressive se(n) equivariant networks through weight-sharing in position-orientation space, 2024. URL https: //arxiv.org/abs/2310.02970

    work page arXiv 2024

  39. [39]

    Mathematical analysis of random noise.The Bell System Technical Journal, 23(3):282–332, 1944

    Stephen O Rice. Mathematical analysis of random noise.The Bell System Technical Journal, 23(3):282–332, 1944

    1944

  40. [40]

    The ising model: highlights and perspectives: C

    Christof Külske. The ising model: highlights and perspectives: C. külske.Mathematical Physics, Analysis and Geometry, 28(3):20, 2025

    2025

  41. [41]

    Survey of spiking in the mouse visual system reveals functional hierarchy.Nature, 592(7852):86–92, 2021

    Joshua H Siegle, Xiaoxuan Jia, Séverine Durand, Sam Gale, Corbett Bennett, Nile Graddis, Greggory Heller, Tamina K Ramirez, Hannah Choi, Jennifer A Luviano, et al. Survey of spiking in the mouse visual system reveals functional hierarchy.Nature, 592(7852):86–92, 2021

    2021

  42. [42]

    A correlation coefficient for circular data.Biometrika, 70(2):327–332, 1983

    Nick I Fisher and Alan J Lee. A correlation coefficient for circular data.Biometrika, 70(2):327–332, 1983

    1983

  43. [43]

    world scientific, 2001

    S Rao Jammalamadaka and Ambar Sengupta.Topics in circular statistics, volume 5. world scientific, 2001

    2001

  44. [44]

    Neural estimation of statistical divergences, 2022

    Sreejith Sreekumar and Ziv Goldfeld. Neural estimation of statistical divergences, 2022. URL https://arxiv. org/abs/2110.03652

    work page arXiv 2022

  45. [45]

    Asymptotic evaluation of certain markov process expectations for large time, i.Communications on pure and applied mathematics, 28(1):1–47, 1975

    Monroe D Donsker and SR Srinivasa Varadhan. Asymptotic evaluation of certain markov process expectations for large time, i.Communications on pure and applied mathematics, 28(1):1–47, 1975. 12 Blind Recovery of Latent Domains via Unsupervised Symmetry DiscoveryA PREPRINT

    1975

  46. [46]

    A differential entropy estimator for training neural networks

    Georg Pichler, Pierre Jean A Colombo, Malik Boudiaf, Günther Koliander, and Pablo Piantanida. A differential entropy estimator for training neural networks. InInternational Conference on Machine Learning, pages 17691– 17715. PMLR, 2022

    2022

  47. [47]

    Stochastic Pooling for Regularization of Deep Convolutional Neural Networks

    Matthew D Zeiler and Rob Fergus. Stochastic pooling for regularization of deep convolutional neural networks. arXiv preprint arXiv:1301.3557, 2013

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  48. [48]

    mixup: Beyond Empirical Risk Minimization

    Hongyi Zhang, Moustapha Cissé, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization.CoRR, abs/1710.09412, 2017. URLhttp://arxiv.org/abs/1710.09412. 13 Blind Recovery of Latent Domains via Unsupervised Symmetry DiscoveryA PREPRINT A Additional theoretical analysis A.1 Resolution term and Markov structure In this section, we analy...

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  49. [49]

    Partition they-representation into patches along each axis

  50. [50]

    amount of gradient flow

    Randomly select one component within each patch. This operation is formalized in Equation 19, where the coarse-grained representation ycoarse is indexed by positive integers (n1, n2, . . . , nk). For each sample, we independently generate random integer offsets{ri} uniformly distributed in {0,1, . . . , qi −1} , where qi denotes the dilation factor along ...

    2000

  51. [51]

    Bounds: [0.75,1.25]

    Resolution:The regularization coefficient for the total correlation minimization objective. Bounds: [0.75,1.25]

  52. [52]

    most influential

    InfoMax:The regularization coefficient for the joint entropy maximization objective. Bounds: [0.5625,0.9375]. Using 6 trajectories and an 8-level grid structure, we generated 36 distinct experimental configurations. Each experiment was trained for 1500 epochs with a batch size of 500. We evaluate the framework’s sensitivity using two primary metrics: • In...