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arxiv: 2510.08364 · v2 · submitted 2025-10-09 · 💻 cs.IT · math.IT

Exponential Error Bounds for Information Bottleneck Source Coding Problems

Han Wu , Hamdi Joudeh This is my paper

Pith reviewed 2026-05-18 08:18 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords information bottlenecksource codingerror exponentstrong converselogarithmic lossauxiliary random variablesWyner-Ahlswede-Körner problemrate-distortion function
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The pith

Exact error and strong converse exponents are established for information bottleneck source coding via optimizations over auxiliary random variables.

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

The paper seeks to determine the precise exponential rates at which the excess distortion probability in information bottleneck source coding either vanishes or approaches one, depending on whether the available rate exceeds or falls short of the fundamental rate-distortion limit. A sympathetic reader would care because these rates quantify the reliability of remote soft estimation from rate-limited noisy observations under logarithmic loss, which arises in many distributed inference settings. Matching upper and lower bounds are obtained by adapting sphere packing and single-letterization arguments to handle the auxiliary variables that appear in the exponent expressions. The work further links every valid code for the Wyner-Ahlswede-Körner problem to a valid code for the information bottleneck problem, thereby recovering the best known sphere-packing bound for the former with a direct operational meaning.

Core claim

We establish both the exact error exponent and the exact strong converse exponent for IB source coding by deriving matching upper and lower exponential bounds. The obtained exponents involve optimizations over auxiliary random variables. The matching converse bounds are derived through non-trivial extensions of existing sphere packing and single-letterization techniques, which we adapt to incorporate auxiliary random variables. In the second part of this paper, we establish a code-level connection between IB source coding and source coding with a helper, also known as the Wyner-Ahlswede-Körner problem, and re-derive the best known sphere packing exponent of the WAK problem with an additional

What carries the argument

Optimizations over auxiliary random variables that define the error exponent and strong converse exponent, obtained by extending sphere packing and single-letterization techniques to the information bottleneck setting under logarithmic loss.

If this is right

  • When the coding rate lies above the information bottleneck rate-distortion function, the excess distortion probability decays to zero at the exact error exponent rate.
  • When the coding rate lies below the information bottleneck rate-distortion function, the excess distortion probability grows to one at the exact strong converse exponent rate.
  • Any code constructed for the Wyner-Ahlswede-Körner problem with a helper at both encoder and decoder is automatically a valid code for information bottleneck source coding.
  • The sphere-packing exponent previously known for the Wyner-Ahlswede-Körner problem acquires an operational interpretation as the strong converse exponent of the corresponding information bottleneck problem.

Where Pith is reading between the lines

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

  • The code-level equivalence may allow finite-blocklength refinements developed for one problem to be imported into the other.
  • The auxiliary-variable formulation suggests that similar exponent derivations could be attempted for other remote source coding problems with different distortion measures.
  • Explicit computation of the optimized exponents for simple channels could yield closed-form reliability curves useful for system dimensioning in distributed estimation networks.

Load-bearing premise

The non-trivial extensions of sphere packing and single-letterization techniques remain valid when auxiliary random variables are introduced for the information bottleneck source coding problem under logarithmic loss and excess distortion probability.

What would settle it

Numerical evaluation of the optimized exponent expression for a binary symmetric source followed by a binary symmetric observation channel, compared against the measured exponential decay rate of excess distortion in random coding simulations at large block lengths.

Figures

Figures reproduced from arXiv: 2510.08364 by Hamdi Joudeh, Han Wu.

Figure 1
Figure 1. Figure 1: Information Bottleneck Source The distortion between a source sequence x n and its soft reconstruction Pˆ is measured by the logarithmic loss defined in (1), which we restate here for convenience: ¯d(x n , Pˆ) ≜ − 1 n log Pˆ(x n ). (5) Since Pˆ(x n ) ≤ 1 for every Pˆ, we have ¯d(x n , Pˆ) ≥ 0. It follows that ¯d(x n , Pˆ) = 0 if and only if Pˆ(x n ) = 1, i.e., the receiver produces a correct “hard” reconst… view at source ↗
Figure 2
Figure 2. Figure 2: Source Coding with a Helper at Both Sides [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

We study the information bottleneck (IB) source coding problem, also known as remote lossy source coding under logarithmic loss. Based on a rate-limited description of noisy observations, the receiver produces a soft estimate for the remote source, i.e., a probability distribution, evaluated under the logarithmic loss. We focus on the excess distortion probability of IB source coding and investigate how fast it converges to 0 or 1, depending on whether the rate is above or below the rate-distortion function. The latter case is also known as the exponential strong converse. We establish both the exact error exponent and the exact strong converse exponent for IB source coding by deriving matching upper and lower exponential bounds. The obtained exponents involve optimizations over auxiliary random variables. The matching converse bounds are derived through non-trivial extensions of existing sphere packing and single-letterization techniques, which we adapt to incorporate auxiliary random variables. In the second part of this paper, we establish a code-level connection between IB source coding and source coding with a helper, also known as the Wyner-Ahlswede-K\"orner (WAK) problem. We show that every code for the WAK problem is a code for IB source coding. This requires noticing that IB source coding, under the excess distortion criterion, is equivalent to source coding with a helper available at both the transmitter and the receiver; the latter in turn relates to the WAK problem. Through this connection, we re-derive the best known sphere packing exponent of the WAK problem, and provide it with an operational interpretation.

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

1 major / 2 minor

Summary. The paper studies the information bottleneck (IB) source coding problem, equivalent to remote lossy source coding under logarithmic loss with excess distortion probability. It claims to establish both the exact error exponent and the exact strong converse exponent by deriving matching upper and lower exponential bounds, expressed as optimizations over auxiliary random variables. The converse bounds rely on non-trivial extensions of sphere-packing and single-letterization techniques adapted to auxiliary variables. The paper also establishes a code-level equivalence between IB source coding and the Wyner-Ahlswede-Körner (WAK) problem, re-deriving the WAK sphere-packing exponent with an operational interpretation.

Significance. If the matching bounds hold, the work provides the first exact exponential characterizations for the IB source coding problem, which is a core remote source coding setting with soft reconstruction. This advances understanding of convergence rates for excess distortion probability above and below the rate-distortion function. The WAK connection supplies a new operational view of an existing exponent. The exponent expressions are optimizations over auxiliaries and thus parameter-free in the classical sense; this is a strength when the derivations are verified.

major comments (1)
  1. The central claim of exact matching exponents rests on the validity of the extended sphere-packing and single-letterization arguments that incorporate an auxiliary random variable Z while preserving the remote observation channel, the Markov chain X–Y–Z, and the logarithmic-loss excess-distortion criterion. A concrete verification that the packing-volume argument correctly accounts for the geometry induced by log-loss (without reducing to a fitted parameter or circular definition) is needed to confirm tightness; this is load-bearing for both the IB exponent and the re-derived WAK exponent.
minor comments (2)
  1. Notation for the auxiliary variable and the precise statement of the Markov chain should be introduced earlier in the problem formulation to improve readability of the exponent optimizations.
  2. The abstract and introduction could explicitly reference the section containing the detailed extension of the sphere-packing argument for easier navigation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: The central claim of exact matching exponents rests on the validity of the extended sphere-packing and single-letterization arguments that incorporate an auxiliary random variable Z while preserving the remote observation channel, the Markov chain X–Y–Z, and the logarithmic-loss excess-distortion criterion. A concrete verification that the packing-volume argument correctly accounts for the geometry induced by log-loss (without reducing to a fitted parameter or circular definition) is needed to confirm tightness; this is load-bearing for both the IB exponent and the re-derived WAK exponent.

    Authors: We thank the referee for this observation. The sphere-packing argument in Sections IV–V is constructed by defining the packing region over the typical sets of the joint distribution P_{XYZ} induced by the auxiliary Z. Under logarithmic loss the excess-distortion event is equivalent to the empirical cross-entropy exceeding a threshold that is a direct function of the rate and the rate-distortion function; this yields a volume that is exactly the exponential of the relevant conditional entropy term without any auxiliary fitting parameter. The Markov chain X–Y–Z is preserved throughout, and the single-letterization follows from standard application of the AEP to the remote source coding setting. The derivation is therefore not circular: the geometry is dictated by the explicit form of the log-loss distortion measure. To make the volume calculation fully explicit and self-contained, we will add a dedicated lemma together with a step-by-step expansion of the packing argument in a new appendix of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations extend standard sphere-packing and single-letterization techniques independently

full rationale

The paper derives matching upper and lower exponential bounds for the IB source coding error exponent and strong converse by adapting existing sphere-packing and single-letterization methods to auxiliary random variables under logarithmic loss. It additionally establishes a code-level equivalence to the WAK problem to re-derive the WAK sphere-packing exponent with an operational interpretation. These steps rely on non-trivial extensions of established information-theoretic tools rather than self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The central claims remain independent of the target results and are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard information-theoretic properties of mutual information and rate-distortion functions plus extensions of classical bounding techniques; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard properties of mutual information and conditional distributions hold for the auxiliary random variables in the exponent optimizations.
    Invoked when defining the exponents via optimizations over auxiliary variables.
  • domain assumption Sphere-packing and single-letterization arguments extend directly to the IB setting with logarithmic loss.
    Stated as the basis for the matching converse bounds.

pith-pipeline@v0.9.0 · 5804 in / 1301 out tokens · 34010 ms · 2026-05-18T08:18:20.049573+00:00 · methodology

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    We establish both the exact error exponent and the exact strong converse exponent for IB source coding by deriving matching upper and lower exponential bounds. The obtained exponents involve optimizations over auxiliary random variables.

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Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    Universal coding, information, prediction, and estimation,

    J. Rissanen, “Universal coding, information, prediction, and estimation,”IEEE Trans. Inf. Theory, vol. 30, no. 4, pp. 629–636, 1984

    work page 1984

  2. [2]

    Universal prediction,

    N. Merhav and M. Feder, “Universal prediction,”IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2124–2147, 1998

    work page 1998

  3. [3]

    Multiterminal source coding under logarithmic loss,

    T. A. Courtade and T. Weissman, “Multiterminal source coding under logarithmic loss,”IEEE Trans. Inf. Theory, vol. 60, no. 1, pp. 740–761, 2014

    work page 2014

  4. [4]

    The information bottleneck method,

    N. Tishby, F. C. Pereira, and W. Bialek, “The information bottleneck method,” inProc. 37th Annu. Allerton Conf. Commun. Control Comput., 1999, pp. 368—377

    work page 1999

  5. [5]

    An information theoretic tradeoff between complexity and accuracy,

    R. Gilad-Bachrach, A. Navot, and N. Tishby, “An information theoretic tradeoff between complexity and accuracy,” inProc. 16th Annu. Conf. Learn. Theory, 7th Kernel Workshop, Washington, DC, USA, 2003, pp. 595–609

    work page 2003

  6. [6]

    The information bottleneck revisited or how to choose a good distortion measure,

    P. Harremoes and N. Tishby, “The information bottleneck revisited or how to choose a good distortion measure,” inProc. IEEE Int. Symp. Inf. Theory, Nice, France, 2007, pp. 566–570

    work page 2007

  7. [7]

    A single-shot approach to lossy source coding under logarithmic loss,

    Y . Y . Shkel and S. Verdú, “A single-shot approach to lossy source coding under logarithmic loss,”IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 129–147, 2018

    work page 2018

  8. [8]

    Error exponents for source coding under logarithmic loss,

    H. Joudeh and H. Wu, “Error exponents for source coding under logarithmic loss,” inProc. Int. Semin. Zürich Inf. Commun., Zürich, Switzerland, 2024, pp. 99–103

    work page 2024

  9. [9]

    Tradeoffs between the excess-code-length exponent and the excess-distortion exponent in lossy source coding,

    T. Weissman and N. Merhav, “Tradeoffs between the excess-code-length exponent and the excess-distortion exponent in lossy source coding,”IEEE Trans. Inf. Theory, vol. 48, no. 2, pp. 396–415, 2002

    work page 2002

  10. [10]

    Strong converse exponent for remote lossy source coding,

    H. Wu and H. Joudeh, “Strong converse exponent for remote lossy source coding,” inProc. IEEE Int. Symp. Inf. Theory, Ann Arbor, MI, USA, 2025

    work page 2025

  11. [11]

    On source coding with side information at the decoder,

    A. Wyner, “On source coding with side information at the decoder,”IEEE Trans. Inf. Theory, vol. 21, no. 3, pp. 294–300, 1975

    work page 1975

  12. [12]

    Source coding with side information and a converse for degraded broadcast channels,

    R. Ahlswede and J. Körner, “Source coding with side information and a converse for degraded broadcast channels,”IEEE Trans. Inf. Theory, vol. 21, no. 6, pp. 629–637, 1975

    work page 1975

  13. [13]

    On the information bottleneck problems: Models, connections, applications and information theoretic views,

    A. Zaidi, I. Estella-Aguerri, and S. Shamai Shitz, “On the information bottleneck problems: Models, connections, applications and information theoretic views,”Entropy, vol. 22, no. 2, p. 151, 2020

    work page 2020

  14. [14]

    An upper bound on the error exponent in lossless source coding with a helper,

    W. Kang and N. Liu, “An upper bound on the error exponent in lossless source coding with a helper,” inProc. IEEE Inf. Theory Workshop, Guangzhou, China, 2018, pp. 1–5

    work page 2018

  15. [15]

    Berger,Rate Distortion Theory: A Mathematical Basis for Data Compression

    T. Berger,Rate Distortion Theory: A Mathematical Basis for Data Compression. Englewood Cliffs, NJ, USA: Prentice-Hall, 1971

    work page 1971

  16. [16]

    Csiszár and J

    I. Csiszár and J. Körner,Information Theory: Coding Theorems for Discrete Memoryless Systems, 2nd ed. Cambridge, UK: Cambridge Univ. Press, 2011

    work page 2011

  17. [17]

    Exponential strong converse for source coding with side information at the decoder,

    Y . Oohama, “Exponential strong converse for source coding with side information at the decoder,”Entropy, vol. 20, no. 5, p. 352, 2018

    work page 2018

  18. [18]

    Strong converse using change of measure arguments,

    H. Tyagi and S. Watanabe, “Strong converse using change of measure arguments,”IEEE Trans. Inf. Theory, vol. 66, no. 2, pp. 689–703, 2020

    work page 2020

  19. [19]

    Tight exponential strong converse for source coding problem with encoded side information,

    D. Takeuchi and S. Watanabe, “Tight exponential strong converse for source coding problem with encoded side information,”IEEE Trans. Inf. Theory, vol. 71, no. 3, pp. 1533–1545, 2025

    work page 2025

  20. [20]

    Bounds for the exponent of the probability of error for a semicontinuous memoryless channel,

    E. Haroutunian, “Bounds for the exponent of the probability of error for a semicontinuous memoryless channel,”Probl. Peredachi Inf., vol. 4, no. 4, pp. 37–48, 1968

    work page 1968

  21. [21]

    Error exponent for source coding with a fidelity criterion,

    K. Marton, “Error exponent for source coding with a fidelity criterion,”IEEE Trans. Inf. Theory, vol. 20, no. 2, pp. 197–199, 1974

    work page 1974

  22. [22]

    Polyanskiy and Y

    Y . Polyanskiy and Y . Wu,Information Theory: From Coding to Learning. Cambridge, UK: Cambridge Univ. Press, 2025

    work page 2025

  23. [23]

    Information transmission with additional noise,

    R. Dobrushin and B. Tsybakov, “Information transmission with additional noise,”IRE Trans. Inf. Theory, vol. 8, no. 5, pp. 293–304, 1962

    work page 1962

  24. [24]

    Transmission of noisy information to a noisy receiver with minimum distortion,

    J. Wolf and J. Ziv, “Transmission of noisy information to a noisy receiver with minimum distortion,”IEEE Trans. Inf. Theory, vol. 16, no. 4, pp. 406–411, 1970

    work page 1970

  25. [25]

    Indirect rate distortion problems,

    H. Witsenhausen, “Indirect rate distortion problems,”IEEE Trans. Inf. Theory, vol. 26, no. 5, pp. 518–521, 1980

    work page 1980

  26. [26]

    Source coding theory for multiterminal communication systems with a remote source,

    H. Yamamoto and K. Itoh, “Source coding theory for multiterminal communication systems with a remote source,”IEICE Trans., vol. 63, no. 10, pp. 700–706, 1980

    work page 1980

  27. [27]

    The CEO problem,

    T. Berger, Z. Zhang, and H. Viswanathan, “The CEO problem,”IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 887–902, 1996

    work page 1996

  28. [28]

    The rate-distortion function for the quadratic Gaussian CEO problem,

    Y . Oohama, “The rate-distortion function for the quadratic Gaussian CEO problem,”IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 1057–1070, 1998

    work page 1998

  29. [29]

    Indirect and direct Gaussian distributed source coding problems,

    ——, “Indirect and direct Gaussian distributed source coding problems,”IEEE Trans. Inf. Theory, vol. 60, no. 12, pp. 7506–7539, 2014

    work page 2014

  30. [30]

    Remote source coding under Gaussian noise: Dueling roles of power and entropy power,

    K. Eswaran and M. Gastpar, “Remote source coding under Gaussian noise: Dueling roles of power and entropy power,”IEEE Trans. Inf. Theory, vol. 65, no. 7, pp. 4486–4498, 2019

    work page 2019

  31. [31]

    Nonasymptotic noisy lossy source coding,

    V . Kostina and S. Verdú, “Nonasymptotic noisy lossy source coding,”IEEE Trans. Inf. Theory, vol. 62, no. 11, pp. 6111–6123, 2016. 42

    work page 2016

  32. [32]

    Nonasymptotic oblivious relaying and variable-length noisy lossy source coding,

    Y . Liu, S. H. Advary, and C. T. Li, “Nonasymptotic oblivious relaying and variable-length noisy lossy source coding,” inProc. IEEE Int. Symp. Inf. Theory, Ann Arbor, MI, USA, 2025

    work page 2025

  33. [33]

    Indirect rate distortion functions withf-separable distortion criterion,

    P. A. Stavrou, Y . Shkel, and M. Kountouris, “Indirect rate distortion functions withf-separable distortion criterion,” inProc. IEEE Int. Symp. Inf. Theory, Taipei, Taiwan, 2023, pp. 1050–1055

    work page 2023

  34. [34]

    Vector Gaussian CEO problem under logarithmic loss and applications,

    Y . Ugur, I. E. Aguerri, and A. Zaidi, “Vector Gaussian CEO problem under logarithmic loss and applications,”IEEE Trans. Inf. Theory, vol. 66, no. 7, pp. 4183–4202, 2020

    work page 2020

  35. [35]

    Universal lossy compression under logarithmic loss,

    Y . Shkel, M. Raginsky, and S. Verdú, “Universal lossy compression under logarithmic loss,” inProc. IEEE Int. Symp. Inf. Theory, Aachen, Germany, 2017, pp. 1157–1161

    work page 2017

  36. [36]

    Universal compression, list decoding, and logarithmic loss,

    ——, “Universal compression, list decoding, and logarithmic loss,” inProc. IEEE Int. Symp. Inf. Theory, Vail, CO, USA, 2018, pp. 206–210

    work page 2018

  37. [37]

    Soft guessing under logarithmic loss,

    H. Wu and H. Joudeh, “Soft guessing under logarithmic loss,” inProc. IEEE Int. Symp. Inf. Theory, Taipei, Taiwan, 2023, pp. 466–471

    work page 2023

  38. [38]

    On source coding for stationary sources under logarithmic loss,

    O. K. Ulger and E. Erkip, “On source coding for stationary sources under logarithmic loss,” inProc. IEEE Int. Symp. Inf. Theory, Ann Arbor, MI, USA, 2025

    work page 2025

  39. [39]

    On the converse to the coding theorem for discrete memoryless channels,

    S. Arimoto, “On the converse to the coding theorem for discrete memoryless channels,”IEEE Trans. Inf. Theory, vol. 19, no. 3, pp. 357–359, 1973

    work page 1973

  40. [40]

    Reliability function of a discrete memoryless channel at rates above capacity,

    G. Dueck and J. Körner, “Reliability function of a discrete memoryless channel at rates above capacity,”IEEE Trans. Inf. Theory, vol. 25, no. 1, pp. 82–85, 1979

    work page 1979

  41. [41]

    Universal coding for the Slepian-Wolf data compression system and the strong converse theorem,

    Y . Oohama and T. S. Han, “Universal coding for the Slepian-Wolf data compression system and the strong converse theorem,”IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1908–1919, 1994

    work page 1908

  42. [42]

    Exponential strong converse for one helper source coding problem,

    Y . Oohama, “Exponential strong converse for one helper source coding problem,”Entropy, vol. 21, no. 6, p. 567, 2019

    work page 2019

  43. [43]

    Tight exponential strong converses for lossy source coding with side-information and distributed function computation,

    S. Watanabe, “Tight exponential strong converses for lossy source coding with side-information and distributed function computation,”

    work page

  44. [44]

    Available: http://arxiv.org/abs/2504.16380

    [Online]. Available: http://arxiv.org/abs/2504.16380

    work page arXiv

  45. [45]

    Nonasymptotic and second-order achievability bounds for coding with side-information,

    S. Watanabe, S. Kuzuoka, and V . Y . F. Tan, “Nonasymptotic and second-order achievability bounds for coding with side-information,” IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 1574–1605, 2015

    work page 2015

  46. [46]

    Dispersion bound for the Wyner-Ahlswede-Körner network via a semigroup method on types,

    J. Liu, “Dispersion bound for the Wyner-Ahlswede-Körner network via a semigroup method on types,”IEEE Trans. Inf. Theory, vol. 67, no. 2, pp. 869–885, 2021

    work page 2021

  47. [47]

    Source coding for a simple network,

    R. M. Gray and A. D. Wyner, “Source coding for a simple network,”Bell Syst. Tech. J., vol. 53, no. 9, pp. 1681–1721, 1974

    work page 1974

  48. [48]

    A converse bound on Wyner-Ahlswede-Körner network via Gray-Wyner network,

    S. Watanabe, “A converse bound on Wyner-Ahlswede-Körner network via Gray-Wyner network,” inProc. IEEE Inf. Theory Workshop, Kaohsiung, Taiwan, 2017, pp. 81–85

    work page 2017

  49. [49]

    Reliability in source coding with side information,

    B. G. Kelly and A. B. Wagner, “Reliability in source coding with side information,”IEEE Trans. Inf. Theory, vol. 58, no. 8, pp. 5086–5111, 2012

    work page 2012

  50. [50]

    Error exponents for oblivious relaying and connections to source coding with a helper,

    H. Wu and H. Joudeh, “Error exponents for oblivious relaying and connections to source coding with a helper,”IEEE Trans. Inf. Theory, Accepted, 2025

    work page 2025

  51. [51]

    Communication via decentralized processing,

    A. Sanderovich, S. Shamai Shitz, Y . Steinberg, and G. Kramer, “Communication via decentralized processing,”IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 3008–3023, 2008

    work page 2008

  52. [52]

    Lower bounds to error probability for coding on discrete memoryless channels. I,

    C. Shannon, R. Gallager, and E. Berlekamp, “Lower bounds to error probability for coding on discrete memoryless channels. I,”Inf. Control, vol. 10, no. 1, pp. 65–103, 1967

    work page 1967

  53. [53]

    Hypothesis testing and information theory,

    R. Blahut, “Hypothesis testing and information theory,”IEEE Trans. Inf. Theory, vol. 20, no. 4, pp. 405–417, 1974

    work page 1974

  54. [54]

    Guessing based on compressed side information,

    R. Graczyk, A. Lapidoth, N. Merhav, and C. Pfister, “Guessing based on compressed side information,”IEEE Trans. Inf. Theory, vol. 68, no. 7, pp. 4244–4256, 2022

    work page 2022

  55. [55]

    A simple proof of the blowing-up lemma,

    K. Marton, “A simple proof of the blowing-up lemma,”IEEE Trans. Inf. Theory, vol. 32, no. 3, pp. 445–446, 1986

    work page 1986

  56. [56]

    A. E. Gamal and Y .-H. Kim,Network Information Theory. Cambridge, UK: Cambridge Univ. Press, 2011

    work page 2011

  57. [57]

    Identification via compressed data,

    R. Ahlswede, E.-H. Yang, and Z. Zhang, “Identification via compressed data,”IEEE Trans. Inf. Theory, vol. 43, no. 1, pp. 48–70, 1997

    work page 1997

  58. [58]

    S. M. Moser,Advanced Topics in Information Theory, 5th ed., ETH Zürich, Switzerland, 2022. [Online]. Available: https://moser-isi.ethz.ch/cgi-bin/request_script.cgi?script=atit

    work page 2022

  59. [59]

    The generalized stochastic likelihood decoder: Random coding and expurgated bounds,

    N. Merhav, “The generalized stochastic likelihood decoder: Random coding and expurgated bounds,”IEEE Trans. Inf. Theory, vol. 63, no. 8, pp. 5039–5051, 2017

    work page 2017

  60. [60]

    Optimal noiseless coding of random variables,

    J. Dunham, “Optimal noiseless coding of random variables,”IEEE Trans. Inf. Theory, vol. 26, no. 3, pp. 345–345, 1980

    work page 1980

  61. [61]

    An upper bound on the entropy series,

    A. Wyner, “An upper bound on the entropy series,”Inf. Control, vol. 20, no. 2, pp. 176–181, 1972

    work page 1972

  62. [62]

    Variable-length lossy compression and channel coding: Non-asymptotic converses via cumulant generating functions,

    T. A. Courtade and S. Verdú, “Variable-length lossy compression and channel coding: Non-asymptotic converses via cumulant generating functions,” inProc. IEEE Int. Symp. Inf. Theory, Honolulu, HI, USA, 2014, pp. 2499–2503

    work page 2014