Source-linked AI summary
Distortion-Rate Outage Probability for Non-ergodic Classes of Sources
Dongmin Park, Ibrahim Abou-Faycal, Razane Tajeddine
TL;DR
For non-ergodic source statistics, expected distortion may not capture reliability when one source state persists throughout a codeword. The paper defines DROP for slow source classes and studies its properties, including compact classes. DROP varies with rate and distortion, and the examples illustrate their tradeoff.
Problem
When a source parameter remains fixed over a codeword, averaging across source states loses operational meaning, leaving expected distortion unsuited to characterize reliability in this non-ergodic setting.
Source: §I · p. 1, §III · p. 3Method
The paper defines DROP for slow source classes, where one source parameter is held fixed during a codeword, and examines compact classes and examples.
Source: §I · p. 1, §III · p. 3, §IV · p. 4Results
For the compact-class example, Pout decreases in R and D and reaches zero once R ≥ 1/2 log2(σ_0^2/D) = supθ∈Λ Rθ(D).
Source: §IV · p. 5Takeaways & Limitations
The examples show how DROP characterizes the tradeoff among outage probability, rate, and distortion, and may support a joint outage notion for slow source classes transmitted over slow-fading channels.
Source: §V · p. 5
Abstract
from arXiv · showhide
The concept of outage has been commonly used in the context of transmission rates over non-ergodic channels. For those channels, one channel state or realization may be encountered for the whole transmission duration of a codeword. In the dual setting of source coding and whenever source statistics are not ergodic, we define in this work the notion of outage; we discuss related properties, consider the special case of compact source classes and provide examples.
I. INTRODUCTION
The paper extends outage from non-ergodic channels to source coding when source statistics vary across a class and are unknown to the encoder and decoder. It distinguishes fast-varying sources, which can be averaged, from slow sources, for which it defines Distortion-Rate Outage Probability (DROP).
I. INTRODUCTION
- For slow fading channels, averaging over channel states loses operational meaning, motivating outage when a prescribed probability of transmission failure is tolerated.The paper uses this channel-side concept as the analogy for source coding.Source: §I · p. 1
- The source-coding setting is blind: the encoder and decoder do not know which probability law pθ governs the source.Each θ in the class Λ is associated with its own probability law.Source: §I · p. 1
- Fast-varying source parameters can be averaged over Λ so the resulting symbol sequence is ergodic and standard rate-distortion tools apply.The parameter is redrawn on a short time scale according to an ergodic process over Λ.Source: §I · p. 1
- DROP treats as outage the event that a slow source’s required rate exceeds the available rate for meeting a target distortion.In the slow regime, one source parameter θ is drawn and held fixed over the codeword.Source: §I · p. 1
II. PRELIMINARIES
The preliminaries define source coding distortion and the information rate-distortion function for a class, then summarize Sakrison’s converse and compact-class achievability results. Compactness yields a computable worst-source rate characterization, while non-compact classes may require more rate than the converse bound.
II. PRELIMINARIES
- A code maps an IID source sequence to a reconstruction, with sequence distortion defined as the per-symbol average of symbol distortions.The encoder and decoder operate on the source sequence and its encoded message, respectively.Source: §II · p. 1
- The class information rate-distortion function minimizes worst-case mutual information using one conditional distribution that meets the distortion constraint for every source in Λ.The paper restricts attention to classes with finite expected distortion from a fixed reference point for every θ.Source: §II · p. 2
- Compactness requires most probability mass from every source in Λ to lie in a common totally bounded set; a finite class is a special case.The definition is attributed to Sakrison.Source: §II · p. 2
- For any class, Sakrison’s converse makes RI_Λ(D) a necessary rate for coding with distortion at most D uniformly across Λ.The achievability statement applies to compact classes: for positive D and δ, sufficiently long codes achieve distortion at most D + δ at rate no greater than supθ∈Λ Rθ(D) + δ.Source: §II · p. 2
- For compact Λ, the class rate-distortion function equals supθ∈Λ Rθ(D), which is often easier to compute than RI_Λ(D) directly.The converse and compact-class achievability together establish the equality.Source: §II · p. 2
- For non-compact Λ, only the necessary-rate direction is guaranteed, so RI_Λ(D) need not be achievable and the true minimum rate may be higher.The paper also states that the class information rate-distortion function is nonincreasing and convex in D.Source: §II · p. 2
III. DISTORTION-RATE OUTAGE PROBABILITY
The section defines outage for blind coding over a source class whose realization remains fixed across a codeword. It formalizes rate- and distortion-based versions, establishes their equivalence, and identifies a countable boundary exception.
III. DISTORTION-RATE OUTAGE PROBABILITY
- RΛ(D) is the infimum rate at which a single θ-blind code meets asymptotic distortion D for every source in Λ.The section uses this operational quantity as its benchmark, invoking supθ Rθ(D) for compact subclasses.Source: §III · p. 3
- For a slow source class, outage allows a nonzero probability of exceeding a distortion threshold or rate budget while the source parameter remains fixed over a codeword.The source symbols are IID under the selected θ, but averaging across Λ loses its operational meaning.Source: §III · p. 3
- The section defines rate- and distortion-outage probabilities by maximizing the probability of measurable subclasses subject to subclass rate–distortion or distortion–rate constraints.The encoders and decoders are blind to the chosen source θ.Source: §III · p. 3
- The rate- and distortion-outage definitions are equivalent because their dual subclass formulations optimize over the same feasible set.The paper thereafter uses Pout(R, D) for either quantity.Source: §III · p. 3
- For fixed D, at most countably many rate values have positive-probability boundary events, so the threshold convention is immaterial outside that exception.The paper leaves the exceptional boundary case unresolved as a definitional choice.Source: §III · p. 3
IV. COMPUTING THE OUTAGE PROBABILITY FOR COMPACT CLASSES
For compact source classes, the outage calculation reduces from optimizing over subclasses to checking individual sources’ rate–distortion functions. The section illustrates the computation for finite Bernoulli and compact Gaussian classes.
IV. COMPUTING THE OUTAGE PROBABILITY FOR COMPACT CLASSES
- For compact Λ, the outage-maximizing subclass is the set of sources whose individual rate–distortion functions are at most R, reducing the subclass optimization to pointwise checks.The outage probability is the prior mass of that maximizing subclass, provided it is measurable.Source: §IV · p. 4
- For a finite class, outage is computed by summing the prior probabilities of sources whose individual rate–distortion functions do not exceed R; no subclass search is needed.This follows from Corollary 1’s finite-class reduction.Source: §IV · p. 4
- At D = 0.05, Pout is 0.5 at R = 0.2, 0.2 at R = 0.65, and zero for R ≥ 0.714 in the Bernoulli example.The feasible source set grows at the stated rate breakpoints.Source: §IV · pp. 4–5
- For the finite Bernoulli class, tightening distortion from D = 0.05 to D = 0.01 shifts every outage breakpoint to a higher rate.Figure 2 plots outage probability against rate budget for both distortion levels.Source: §IV · p. 5, Figure 2 · p. 5
- For the compact Gaussian class, Pout decreases with R and D and reaches zero once R ≥ 1/2 log2(σ0^2/D), the worst-case source rate.With σ0^2 = 1, each plotted curve reaches zero at R = 1.Source: §IV · p. 5, Figure 3 · p. 5
V. CONCLUSION
The paper introduces DROP for slow source classes and illustrates its tradeoff among outage probability, rate, and distortion. It identifies a possible extension to joint outage for slow sources transmitted over slow-fading channels.
V. CONCLUSION
- DROP characterizes performance when the source remains fixed throughout a codeword, and the examples illustrate tradeoffs among outage probability, rate, and distortion.The formulation may serve as a foundation for joint outage analysis over slow-fading channels.Source: §V · p. 5
AI USAGE
The authors report using ChatGPT for English and LaTeX assistance and figure generation, while stating that the main ideas and technical content were produced independently.
AI USAGE
- The authors acknowledge ChatGPT assistance with English, LaTeX, and Figures 2–3, and Claude assistance reviewing and refining technical content.They state that the work’s main ideas and technical contents were produced independently.Source: AI USAGE · p. 5