pith. sign in

arxiv: 2605.18931 · v1 · pith:MPG2JXIMnew · submitted 2026-05-18 · 📊 stat.ML · cs.AI· cs.LG

Markov Chain Decoders Overcome the Heavy-Tail Limitations of Lipschitz Generative Models

Abdelhakim Ziani (MICS) , Andras Horvath (UNITO) , Paolo Ballarini (MICS) This is my paper

Pith reviewed 2026-05-20 08:29 UTC · model grok-4.3

classification 📊 stat.ML cs.AIcs.LG
keywords heavy-tailed distributionsphase-type distributionsvariational autoencodersMarkov chainsLipschitz continuitygenerative modelstail approximationPareto distributions
0
0 comments X

The pith

Replacing Gaussian decoders with Phase-Type Markov chain distributions allows Lipschitz VAEs to generate heavy-tailed outputs.

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

Standard VAEs with Gaussian decoder likelihoods and Lipschitz-constrained networks cannot produce heavy-tailed data, as the exponential Gaussian tail decay cannot be overcome by bounded amplification from the latent space. The paper replaces only the decoder likelihood with a Phase-Type distribution represented by a continuous-time Markov chain, which can approximate any positive distribution to arbitrary accuracy while preserving the encoder, latent space, and training procedure. Controlled experiments on synthetic Pareto data with tail indices from 2 to 30 and dimensions up to 10 show the change reduces tail Kolmogorov-Smirnov distance by factors of up to 6 and extreme quantile error by factors of up to 10. This establishes a practical route to heavy-tail generation in otherwise standard generative models used for performance evaluation, network traffic, and risk modeling.

Core claim

Heavy-tailed distributions pose a fundamental challenge for modern deep generative models because Gaussian tails decay exponentially and Lipschitz continuity prevents the decoder from amplifying rare latent events sufficiently. Replacing the Gaussian decoder with a Phase-Type distribution based on Markov chains, while keeping the encoder, latent space, and training identical, overcomes this structural limitation since Phase-Type distributions approximate any positive-valued distribution, including heavy-tailed families, to arbitrary precision. On synthetic Pareto data across tail indices alpha in {2, 3, 5, 30} and dimensions d in {1, 5, 10}, the Phase-Type decoder reduces tail Kolmogorov-Sm,

What carries the argument

Phase-Type distribution modeled by a continuous-time Markov chain that serves as the decoder likelihood for positive outputs.

If this is right

  • Generative models can now produce accurate samples from heavy-tailed distributions common in performance evaluation and risk modeling.
  • The same encoder and latent space can be reused for both light- and heavy-tailed data by swapping only the decoder distribution.
  • Training remains end-to-end differentiable without additional constraints or changes to the optimization procedure.
  • The approach directly addresses the structural mismatch between Lipschitz networks and heavy tails without sacrificing model capacity.

Where Pith is reading between the lines

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

  • The same decoder substitution could be tested in other Lipschitz-constrained architectures such as Wasserstein GANs to check whether the heavy-tail limitation is decoder-specific.
  • Multivariate extensions of Phase-Type distributions might enable joint heavy-tail modeling in higher-dimensional settings where marginal approximations alone are insufficient.
  • Because the Markov chain representation is explicit, one could inspect the inferred chain parameters to diagnose which phases capture the heavy-tail behavior.

Load-bearing premise

Phase-Type distributions can approximate any positive-valued distribution including heavy-tailed families to arbitrary precision and can be integrated as the decoder likelihood while leaving the encoder, latent space, and training procedure unchanged.

What would settle it

Running the Phase-Type decoder model on real heavy-tailed datasets such as network traffic traces or financial returns and finding that tail Kolmogorov-Smirnov distance or extreme quantile error shows no reduction or an increase relative to the Gaussian baseline would falsify the practical effectiveness claim.

Figures

Figures reproduced from arXiv: 2605.18931 by Abdelhakim Ziani (MICS), Andras Horvath (UNITO), Paolo Ballarini (MICS).

Figure 1
Figure 1. Figure 1: A degree 3 PH distribu￾tion. PH distributions form a dense family on (0, ∞): any positive-valued distribution can be approximated arbitrarily well by a PH distri￾bution. Although PH distributions are asymp￾totically light-tailed (their tails ultimately de￾cay exponentially due to the finite Markov chain), they can closely approximate heavy￾tailed behavior over any bounded, data-relevant range by using mult… view at source ↗
Figure 2
Figure 2. Figure 2: Series canonical form Phase-Type Distribution [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Log-log CCDF of true Pareto data, Gaussian VAE generations, and PH [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tail KS distance at the 99th percentile for Gaussian VAE and PH-VAE [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: A degree 3 PH distribu￾tion. PH distributions form a dense family on (0, ∞): any positive-valued distribution can be approximated arbitrarily well by a PH distri￾bution. Although PH distributions are asymp￾totically light-tailed (their tails ultimately de￾cay exponentially due to the finite Markov chain), they can closely approximate heavy￾tailed behavior over any bounded, data-relevant range by using mult… view at source ↗
Figure 2
Figure 2. Figure 2: Series canonical form Phase-Type Distribution [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Log-log CCDF of true Pareto data, Gaussian VAE generations, and PH [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tail KS distance at the 99th percentile for Gaussian VAE and PH-VAE [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
read the original abstract

Heavy-tailed distributions are prevalent in performance evaluation, network traffic, and risk modeling. This behavior poses a fundamental challenge for modern deep generative models. Standard Variational Autoencoders (VAEs) employ Gaussian decoder likelihoods and Lipschitz-constrained neural networks, a combination that is structurally incapable of producing heavy-tailed outputs: the Gaussian tail decays exponentially, and Lipschitz continuity prevents the decoder from amplifying rare events from the latent space input to sufficiently overcome this decay. We provide both a theoretical characterization of this limitation and a controlled empirical demonstration using synthetic Pareto data across a grid of tail indices $\alpha$ $\in$ {2, 3, 5, 30} and dimensions d $\in$ {1, 5, 10}. As a solution, we replace the Gaussian decoder with a Phase-Type (PH) distribution based on Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions allow for arbitrarily precise approximations of any positive-valued distributions, including heavy-tailed families. Experiments showed that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to x6 and extreme quantile error by up to x10 compared to the Gaussian baseline for heavy-tailed data. These results demonstrate that integrating Markov chain-based distributions into the decoder of a generative model institutes a principled and practically effective solution to the heavy-tail generation problem.

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 / 2 minor

Summary. The manuscript claims that standard VAEs with Gaussian decoders and Lipschitz-constrained networks cannot generate heavy-tailed outputs, as Gaussian tails decay exponentially and Lipschitz continuity bounds the amplification of rare latent events. It provides a theoretical characterization of this limitation and proposes replacing the Gaussian decoder with a Phase-Type (PH) distribution parameterized via continuous-time Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions are argued to approximate any positive-valued distribution arbitrarily well. Controlled experiments on synthetic Pareto data across tail indices α ∈ {2, 3, 5, 30} and dimensions d ∈ {1, 5, 10} report that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to a factor of 6 and extreme quantile error by up to a factor of 10 relative to the Gaussian baseline.

Significance. If the central claims hold, the work offers a principled approach to heavy-tailed generation in deep models, relevant to performance evaluation, network traffic, and risk modeling. Strengths include the controlled synthetic experimental grid, direct baseline comparison, and the flexibility of PH approximations for positive support. The theoretical characterization of the Lipschitz-Gaussian limitation is a useful contribution if rigorously derived.

major comments (2)
  1. [§3 (Decoder Architecture)] §3 (Decoder Architecture): The assertion that the encoder, latent space, and training procedure remain identical is undermined by the algebraic constraints required for a valid continuous PH distribution. The decoder must output a subgenerator matrix T (negative diagonals, non-negative off-diagonals, non-positive row sums) and initial vector α (non-negative, sums to 1). This necessitates specialized activations (e.g., softplus on rates), output reshaping, or projection steps absent from standard Gaussian decoders, which only require unconstrained mean and positive variance. These changes alter the decoder head, gradient computation, and numerical stability (via matrix exponential), weakening the 'drop-in replacement' claim.
  2. [Experiments section] Experiments section: The quantitative claims of up to ×6 reduction in tail KS distance and ×10 in extreme quantile error lack supporting details on the number of phases used for the PH approximation, variance across random seeds, or statistical significance tests. Without these, it is difficult to evaluate robustness, especially for the heaviest tails (α=2) where approximation quality depends critically on phase count and parameterization.
minor comments (2)
  1. [Abstract] Abstract: The 'up to ×6' and 'up to ×10' improvements should specify the exact (α, d) pair at which the maxima occur, rather than leaving the range implicit.
  2. [Notation] Notation: Define explicitly how the neural network outputs the Markov chain parameters (rates, initial probabilities) and any normalization or constraint enforcement applied during forward passes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for major revision. We address each major comment point by point below, outlining planned changes to improve clarity and completeness while preserving the core contributions of the work.

read point-by-point responses

  1. Referee: [§3 (Decoder Architecture)] The assertion that the encoder, latent space, and training procedure remain identical is undermined by the algebraic constraints required for a valid continuous PH distribution. The decoder must output a subgenerator matrix T (negative diagonals, non-negative off-diagonals, non-positive row sums) and initial vector α (non-negative, sums to 1). This necessitates specialized activations (e.g., softplus on rates), output reshaping, or projection steps absent from standard Gaussian decoders, which only require unconstrained mean and positive variance. These changes alter the decoder head, gradient computation, and numerical stability (via matrix exponential), weakening the 'drop-in replacement' claim.

    Authors: We appreciate the referee's careful reading of the implementation requirements. The manuscript's statement that the encoder, latent space, and training procedure remain identical is accurate in the sense that these components are unchanged from the Gaussian baseline; only the decoder likelihood is replaced. However, we agree that the PH decoder requires specific output constraints and activations to produce a valid subgenerator matrix T and probability vector α. In the revised manuscript we will expand §3 with an explicit description of the decoder head, including the use of softplus on the diagonal and off-diagonal entries of T (to enforce sign constraints) and softmax on α (to ensure non-negativity and summation to one). We will also briefly discuss the matrix-exponential computation and its effect on gradient propagation. These additions clarify the localized nature of the decoder changes without altering the experimental protocol or the central claim. revision: partial

  2. Referee: [Experiments section] The quantitative claims of up to ×6 reduction in tail KS distance and ×10 in extreme quantile error lack supporting details on the number of phases used for the PH approximation, variance across random seeds, or statistical significance tests. Without these, it is difficult to evaluate robustness, especially for the heaviest tails (α=2) where approximation quality depends critically on phase count and parameterization.

    Authors: We concur that additional experimental details are required for reproducibility and to demonstrate robustness. In the revised version we will state that a fixed number of 10 phases was employed for all PH approximations. We will augment the reported metrics with means and standard deviations computed over five independent random seeds and will include paired t-test p-values comparing the PH and Gaussian models on the tail KS and extreme-quantile errors. These updates will appear in the Experiments section and in revised tables and figures, directly addressing concerns about the heaviest tails (α=2). revision: yes

Circularity Check

0 steps flagged

No circularity: modeling substitution and empirical comparison are independent of fitted inputs.

full rationale

The paper advances a modeling substitution (Gaussian decoder to Phase-Type/Markov-chain decoder) plus a theoretical characterization of Lipschitz+Gaussian limitations, followed by controlled experiments on synthetic Pareto data. No derivation step reduces a claimed prediction or result to a fitted parameter or self-citation by construction. The 'identical training procedure' statement is an empirical claim about implementation, not a definitional equivalence. External benchmarks (KS distance, quantile error) are measured against a separate baseline, satisfying the self-contained criterion.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on the known approximation power of Phase-Type distributions and the structural analysis of Gaussian-Lipschitz limitations; no new free parameters or invented entities are introduced beyond standard VAE components.

free parameters (1)
  • Phase-Type distribution parameters
    Parameters of the Markov chain (number of phases, transition rates) are fitted as part of decoder training.
axioms (1)
  • domain assumption Phase-Type distributions can approximate any positive-valued distribution to arbitrary precision
    Invoked to justify that the decoder can represent heavy-tailed families.

pith-pipeline@v0.9.0 · 5791 in / 1351 out tokens · 45987 ms · 2026-05-20T08:29:08.040441+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

111 extracted references · 111 canonical work pages · 1 internal anchor

  1. [1]

    2017 , primaryClass=

    Adam: A Method for Stochastic Optimization , author=. 2017 , primaryClass=

    work page 2017

  2. [2]

    Proceedings of the 10th EAI International Conference on Performance Evaluation Methodologies and Tools , year =

    Gábor Horváth and Miklós Telek , title =. Proceedings of the 10th EAI International Conference on Performance Evaluation Methodologies and Tools , year =

    work page

  3. [3]

    Computer Performance Evaluation: Modelling Techniques and Tools (TOOLS 2002) , series =

    András Horváth and Miklós Telek , title =. Computer Performance Evaluation: Modelling Techniques and Tools (TOOLS 2002) , series =. 2002 , doi =

    work page 2002

  4. [4]

    International Workshop on Timed Petri Nets , pages =

    Cumani, Aldo , title =. International Workshop on Timed Petri Nets , pages =. 1985 , isbn =

    work page 1985

  5. [5]

    1981 , publisher=

    Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach , author=. 1981 , publisher=

    work page 1981

  6. [6]

    1999 , publisher=

    Introduction to Matrix Analytic Methods in Stochastic Modeling , author=. 1999 , publisher=

    work page 1999

  7. [7]

    Bobbio and M

    A. Bobbio and M. Telek , title =. Stochastic Models , volume =

    work page

  8. [8]

    Fitting phase-type distributions via the

    Asmussen, S. Fitting phase-type distributions via the. Scandinavian Journal of Statistics , volume=. 1996 , publisher=

    work page 1996

  9. [9]

    , journal=

    Okamura, Hiroyuki and Dohi, Tadashi and Trivedi, Kishor S. , journal=. A refined. 2011 , publisher=

    work page 2011

  10. [10]

    2024 , publisher=

    Phase Type Distributions: Theory and Application , author=. 2024 , publisher=

    work page 2024

  11. [11]

    and Taaffe, Michael R

    Johnson, Mark A. and Taaffe, Michael R. , journal=. Matching moments to phase distributions: Mixtures of. 1989 , publisher=

    work page 1989

  12. [12]

    , title =

    Buchholz, Peter and Kriege, Jan. , title =. Proceedings of the 2009 Sixth International Conference on the Quantitative Evaluation of Systems , pages =. 2009 , isbn =. doi:10.1109/QEST.2009.36 , abstract =

    work page doi:10.1109/qest.2009.36 2009

  13. [13]

    2025 , eprint=

    An Unconstrained Optimization Approach to Moment Fitting with Phase Type Distributions , author=. 2025 , eprint=

    work page 2025

  14. [14]

    Stochastic Models , volume=

    Matching three moments with minimal acyclic phase type distributions , author=. Stochastic Models , volume=. 2005 , publisher=

    work page 2005

  15. [15]

    Stochastic Models , volume=

    Matching more than three moments with acyclic phase type distributions , author=. Stochastic Models , volume=. 2007 , publisher=

    work page 2007

  16. [16]

    Application and Theory of Petri Nets 1997 , editor =

    Serge Haddad and Patrice Moreaux and Giovanni Chiola , title =. Application and Theory of Petri Nets 1997 , editor =. 1997 , doi =

    work page 1997

  17. [17]

    1994 , publisher =

    Introduction to the Numerical Solution of Markov Chains , author =. 1994 , publisher =

    work page 1994

  18. [18]

    Horv\'ath AND M

    A. Horv\'ath AND M. Telek , title =. Proc. of 3rd International Conference on Matrix-Analytic Methods in Stochastic models , year = 2000, pages=

    work page 2000

  19. [19]

    Aldous and L

    D. Aldous and L. Shepp. The least variable phase type distribution is E rlang. Stochastic Models. 1987

    work page 1987

  20. [20]

    Approximation of Cumulative Distribution Functions by Bernstein Phase-Type Distributions

    Horv \'a th, Andr \'a s and Horv \'a th, Ill \'e s and Paolieri, Marco and Telek, Mikl \'o s and Vicario, Enrico. Approximation of Cumulative Distribution Functions by Bernstein Phase-Type Distributions. Quantitative Evaluation of Systems and Formal Modeling and Analysis of Timed Systems. 2024

    work page 2024

  21. [21]

    M. Neuts. Probability distributions of phase type. Liber Amicorum Prof. Emeritus H. Florin. 1975

    work page 1975

  22. [22]

    2025 , issn =

    Approximation of cumulative distribution functions by Bernstein phase-type distributions , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.peva.2025.102480 , author =

    work page doi:10.1016/j.peva.2025.102480 2025

  23. [23]

    Construction of Phase Type Distributions by

    Andr. Construction of Phase Type Distributions by. Proceedings of

    work page

  24. [24]

    Feldman and W

    A. Feldman and W. Whitt. Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation. 1998

    work page 1998

  25. [25]

    2011 , publisher=

    An introduction to heavy-tailed and subexponential distributions , author=. 2011 , publisher=

    work page 2011

  26. [26]

    Wiley StatsRef: Statistics Reference Online , pages=

    Pareto distribution , author=. Wiley StatsRef: Statistics Reference Online , pages=. 2014 , publisher=

    work page 2014

  27. [27]

    Queuing theory with heavy tails and network traffic modeling , author=. , year=

    work page

  28. [28]

    2022 , school=

    Extended Entropy Maximisation and Queueing Systems with Heavy-Tailed Distributions , author=. 2022 , school=

    work page 2022

  29. [29]

    2003 , publisher=

    Handbook of heavy tailed distributions in finance: Handbooks in finance, Book 1 , author=. 2003 , publisher=

    work page 2003

  30. [30]

    New Journal of Physics , volume=

    Scaling laws and fluctuations in the statistics of word frequencies , author=. New Journal of Physics , volume=. 2014 , publisher=

    work page 2014

  31. [31]

    arXiv preprint arXiv:2410.11985 , year=

    The fair language model paradox , author=. arXiv preprint arXiv:2410.11985 , year=

    work page arXiv

  32. [32]

    (No Title) , year=

    Adam: A method for stochastic optimization , author=. (No Title) , year=

    work page

  33. [33]

    2021 , note =

    A short review on queuing theory as a deterministic tool in sustainable telecommunication system , journal =. 2021 , note =. doi:https://doi.org/10.1016/j.matpr.2021.01.092 , author =

    work page doi:10.1016/j.matpr.2021.01.092 2021

  34. [34]

    Queueing Systems , year =

    Mor Harchol-Balter , title =. Queueing Systems , year =. doi:10.1007/s11134-020-09684-6 , issn =

    work page doi:10.1007/s11134-020-09684-6

  35. [35]

    Heavy-tailed Distributions and Risk Management of Equity Market Tail Events , journal=

    Zi-Yi,Guo , year=. Heavy-tailed Distributions and Risk Management of Equity Market Tail Events , journal=

    work page

  36. [36]

    , year =

    Steven T. Piantadosi , title =. Psychonomic Bulletin & Review , year =. doi:https://doi.org/10.3758/s13423-014-0585-6 , issn =

    work page doi:10.3758/s13423-014-0585-6

  37. [37]

    Evaluating Large Language Models on Twitter Based on Hashtag Dynamics and Scaling Properties

    Su, Shilan and Zhang, Hongzhong and Wang, Ziye. Evaluating Large Language Models on Twitter Based on Hashtag Dynamics and Scaling Properties. Intelligent Multilingual Information Processing. 2025

    work page 2025

  38. [38]

    Computer Communication Review , volume=

    Fitting heavy tailed distributions to internet traffic data , author=. Computer Communication Review , volume=. 1998 , publisher=

    work page 1998

  39. [39]

    2003 , publisher=

    Applied Probability and Queues , author=. 2003 , publisher=

    work page 2003

  40. [40]

    Stewart , publisher =

    William J. Stewart , publisher =. Introduction to the Numerical Solution of Markov Chains , urldate =

    work page

  41. [41]

    1981 , publisher =

    Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach , author =. 1981 , publisher =

    work page 1981

  42. [42]

    , title =

    Neuts, Marcel F. , title =. Liber Amicorum Professor Emeritus H. Florin , year =

    work page

  43. [43]

    Superconductivity in Tetragonal LaPt_{2-x}Ge_{2+x}

    Vogel, Richard M and Papalexiou, Simon Michael and Lamontagne, Jonathan R and Dolan, Flannery C , title =. The American … , publisher =. 2025 , abstract =. doi:10.1080/00031305.2024.2402898 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1080/00031305.2024.2402898 2025

  44. [44]

    Journal of Statistical Computation and Simulation , publisher =

    Wu, Shuli and Peng, Zuoxiang and Hu, Shuang , title =. Journal of Statistical Computation and Simulation , publisher =. 2025 , abstract =. doi:10.1080/00949655.2025.2566415 , url =

    work page doi:10.1080/00949655.2025.2566415 2025

  45. [45]

    2024 35th Irish Signals … , publisher =

    Dunne, Jonathan and Leech, Sonya and Muller, Markus and Manotas, Irene and Swift, Mary , title =. 2024 35th Irish Signals … , publisher =. 2024 , abstract =

    work page 2024

  46. [46]

    arXiv preprint arXiv:2410.14171 , year=

    Heavy-tailed diffusion models , author=. arXiv preprint arXiv:2410.14171 , year=

    work page arXiv

  47. [47]

    2020 , publisher=

    Probability and random processes , author=. 2020 , publisher=

    work page 2020

  48. [48]

    Estimation of extreme quantiles from heavy-tailed distributions with neural networks , journal =

    M Allouche and S Girard and E Gobet , type =. Estimation of extreme quantiles from heavy-tailed distributions with neural networks , journal =. 2024 , abstract =. doi:10.1007/s11222-023-10331-2 , url =

    work page doi:10.1007/s11222-023-10331-2 2024

  49. [49]

    Heavy-tailed phase-type distributions: a unified approach , journal =

    M Bladt and J Yslas , type =. Heavy-tailed phase-type distributions: a unified approach , journal =. 2022 , abstract =. doi:10.1007/s10687-022-00436-8 , url =

    work page doi:10.1007/s10687-022-00436-8 2022

  50. [50]

    Quantitative Science Studies , publisher =

    R Delabays and M Tyloo , title =. Quantitative Science Studies , publisher =. 2022 , abstract =

    work page 2022

  51. [51]

    Wiley Interdisciplinary Reviews: Computational Statistics , volume=

    Financial modeling with heavy-tailed stable distributions , author=. Wiley Interdisciplinary Reviews: Computational Statistics , volume=. 2014 , publisher=

    work page 2014

  52. [52]

    Powerlaw: a Python package for analysis of heavy-tailed distributions , journal =

    J Alstott and E Bullmore and D Plenz , type =. Powerlaw: a Python package for analysis of heavy-tailed distributions , journal =. 2014 , abstract =

    work page 2014

  53. [53]

    Heavy-tailed phenomena and tail index inference , publisher =

    M Jia , type =. Heavy-tailed phenomena and tail index inference , publisher =. 2014 , abstract =

    work page 2014

  54. [54]

    How extreme is extreme?

    Papalexiou, SM and Koutsoyiannis, D and Makropoulos, C , journal=. How extreme is extreme?. 2013 , publisher=

    work page 2013

  55. [55]

    Computational Intelligence, Cyber Security and Computational Models: Proceedings of ICC3, 2013 , pages=

    Modeling heavy tails in traffic sources for network performance evaluation , author=. Computational Intelligence, Cyber Security and Computational Models: Proceedings of ICC3, 2013 , pages=. 2013 , publisher=

    work page 2013

  56. [56]

    An introduction to heavy-tailed and subexponential distributions , publisher =

    S Foss and D Korshunov and S Zachary , type =. An introduction to heavy-tailed and subexponential distributions , publisher =. 2011 , abstract =. doi:10.1007/978-1-4614-7101-1 , url =

    work page doi:10.1007/978-1-4614-7101-1 2011

  57. [57]

    Heavy-tailed distributions in disaster analysis , publisher =

    V Pisarenko and M Rodkin , type =. Heavy-tailed distributions in disaster analysis , publisher =

    work page

  58. [58]

    Heavy-tail phenomena: probabilistic and statistical modeling , publisher =

    SI Resnick , type =. Heavy-tail phenomena: probabilistic and statistical modeling , publisher =. 2007 , abstract =. doi:10.1007/978-0-387-45024-7_9 , url =

    work page doi:10.1007/978-0-387-45024-7_9 2007

  59. [59]

    A practical guide to heavy tails: statistical techniques and applications , publisher =

    R Adler and R Feldman and M Taqqu , type =. A practical guide to heavy tails: statistical techniques and applications , publisher =

    work page

  60. [60]

    Dean Malmgren and Daniel B

    R. Dean Malmgren and Daniel B. Stouffer and Adilson E. Motter and Luís A. N. Amaral , title =. Proceedings of the National Academy of Sciences , volume =. 2008 , doi =

    work page 2008

  61. [61]

    Journal of Quantitative Linguistics , publisher =

    R Feng and C Yang and Y Qu , title =. Journal of Quantitative Linguistics , publisher =. 2022 , abstract =. doi:10.1080/09296174.2020.1767481 , url =

    work page doi:10.1080/09296174.2020.1767481 2022

  62. [62]

    IEEE Access , volume=

    Searching for heavy-tailed probability distributions for modeling real-world complex networks , author=. IEEE Access , volume=. 2022 , publisher=

    work page 2022

  63. [63]

    arXiv preprint arXiv:2511.21060 , publisher =

    V Berman , title =. arXiv preprint arXiv:2511.21060 , publisher =. 2025 , abstract =

    work page arXiv 2025

  64. [64]

    2023 IEEE MIT Undergraduate Research Technology Conference (URTC) , pages=

    From bits to insights: Exploring network traffic, traffic matrices, and heavy-tailed data , author=. 2023 IEEE MIT Undergraduate Research Technology Conference (URTC) , pages=. 2023 , organization=

    work page 2023

  65. [65]

    Internet traffic volumes are not

    Alasmar, Mohammed and Clegg, Richard and Zakhleniuk, Nickolay and Parisis, George , journal=. Internet traffic volumes are not. 2021 , publisher=

    work page 2021

  66. [66]

    2018 , abstract =

    Y Li , title =. 2018 , abstract =

    work page 2018

  67. [67]

    CoRR , year=

    Auto-Encoding Variational Bayes , author=. CoRR , year=

    work page

  68. [68]

    The Eleventh International Conference on Learning Representations , year=

    The Tilted Variational Autoencoder: Improving Out-of-Distribution Detection , author=. The Eleventh International Conference on Learning Representations , year=

    work page

  69. [69]

    NeurIPS 2024 Workshop on Bayesian Decision-making and Uncertainty , year=

    Capturing Extreme Events in Turbulence using an Extreme Variational Autoencoder , author=. NeurIPS 2024 Workshop on Bayesian Decision-making and Uncertainty , year=

    work page 2024

  70. [70]

    Autoencoder and Its Various Variants , year=

    Zhai, Junhai and Zhang, Sufang and Chen, Junfen and He, Qiang , booktitle=. Autoencoder and Its Various Variants , year=

    work page

  71. [71]

    arXiv preprint arXiv:1606.05908

    C Doersch , title =. arXiv preprint arXiv:1606.05908 , publisher =. 2016 , abstract =

    work page arXiv 2016

  72. [72]

    2018 IEEE international conference on systems, man, and cybernetics (SMC) , pages=

    Autoencoder and its various variants , author=. 2018 IEEE international conference on systems, man, and cybernetics (SMC) , pages=. 2018 , organization=

    work page 2018

  73. [73]

    ACM Computing Surveys , publisher =

    S Liang and Z Pan and W Liu and J Yin and M De Rijke , title =. ACM Computing Surveys , publisher =. 2024 , abstract =. doi:10.1145/3663364 , url =

    work page doi:10.1145/3663364 2024

  74. [74]

    ArXiv , year=

    t3-Variational Autoencoder: Learning Heavy-tailed Data with Student's t and Power Divergence , author=. ArXiv , year=

    work page

  75. [75]

    ArXiv , year=

    Conditional-t3VAE: Equitable Latent Space Allocation for Fair Generation , author=. ArXiv , year=

    work page

  76. [76]

    2025 , eprint=

    Capturing Extreme Events in Turbulence using an Extreme Variational Autoencoder (xVAE) , author=. 2025 , eprint=

    work page 2025

  77. [77]

    2025 , eprint=

    Modeling Spatio-temporal Extremes via Conditional Variational Autoencoders , author=. 2025 , eprint=

    work page 2025

  78. [78]

    2023 , eprint=

    A VAE Approach to Sample Multivariate Extremes , author=. 2023 , eprint=

    work page 2023

  79. [79]

    Physical Review E , publisher =

    S Sormunen and L Leskelä and J Saramäki , title =. Physical Review E , publisher =. 2024 , abstract =. doi:10.1103/PhysRevE.109.054308 , url =

    work page doi:10.1103/physreve.109.054308 2024

  80. [80]

    Physical Review E , publisher =

    J Del Castillo and P Puig , title =. Physical Review E , publisher =. 2023 , abstract =. doi:10.1103/PhysRevE.107.064113 , url =

    work page doi:10.1103/physreve.107.064113 2023

Showing first 80 references.