Indirect Wyner–Ziv Rate–Distortion Function

• The indirect Wyner–Ziv rate–distortion function is a single-letter characterization that manages two fidelity constraints: one on the observed source and one on an unobserved latent semantic variable.
• The formulation uses an auxiliary random variable U and standard binning techniques to extend classical Wyner–Ziv coding, with explicit analyses in both asymptotic and finite-blocklength regimes.
• This framework underpins advances in distributed, goal-oriented, and semantic communications, leading to algorithmic solutions like the Blahut–Arimoto algorithm and neural estimation methods.

The indirect Wyner–Ziv rate–distortion function is the single-letter characterization of the minimum rate required when an encoder observes a source correlated with a decoder-side observation, while the fidelity criterion is imposed on a latent or remote variable that is not directly observed at either terminal. In the formulation developed in "Non-Asymptotic Achievable Rate-Distortion Region for Indirect Wyner-Ziv Source Coding" (Wei et al., 23 Jul 2025), the encoder observes $X$, the decoder has side information $Y$, and a latent source $S$ must be reconstructed together with $X$ under two simultaneous per-letter distortion constraints. This setup is increasingly relevant in goal-oriented communications, where $S$ can represent semantic information obtained from $X$ (Wei et al., 23 Jul 2025).

1. Canonical formulation

In the latent-source formulation, $S\in\mathcal S$ is a latent source, $X\in\mathcal X$ is the observed source at the encoder, and $Y\in\mathcal Y$ is side information available only at the decoder. The triple $(S,X,Y)$ is drawn i.i.d. according to

$Y$0

with the stated Markov chain

$Y$1

The decoder produces two reconstructions: $Y$2 for the observed source and $Y$3 for the latent source (Wei et al., 23 Jul 2025).

Two fidelity criteria are imposed. The first uses a per-letter distortion

$Y$4

with the requirement

$Y$5

The second uses a semantic distortion

$Y$6

with

$Y$7

Because $Y$8 is never observed by the encoder or the decoder, the semantic constraint is pulled back through the conditional law $Y$9 by defining

$S$0

so that

$S$1

This reduction makes explicit that the latent-source fidelity constraint is mediated by the observable pair $S$2 rather than by direct access to $S$3 (Wei et al., 23 Jul 2025).

A closely related single-distortion version uses a latent variable $S$4 to denote the remote source and reconstructs only $S$5 from an encoded observation $S$6 and decoder side information $S$7. In that setting, the problem is commonly termed indirect or remote Wyner–Ziv coding, and the operational objective is to minimize the rate required to meet a single average distortion constraint on $S$8 (Tang et al., 23 Jan 2025).

2. Asymptotic single-letter characterization

The asymptotic indirect Wyner–Ziv rate–distortion function is expressed through an auxiliary random variable $S$9 satisfying

$X$0

with deterministic reconstruction functions

$X$1

The rate–distortion function is

$X$2

subject to

$X$3

Equivalently,

$X$4

This is the natural extension of the Wyner–Ziv expression to a setting with two simultaneous distortion constraints, one on the observation and one on the latent semantic variable (Wei et al., 23 Jul 2025).

The auxiliary alphabet can be restricted by standard support-lemma or Carathéodory arguments. In the stated result, one may restrict

$X$5

or even

$X$6

Operationally, achievability follows by a standard Wyner–Ziv binning argument: the encoder finds a $X$7 typical with $X$8 and sends the bin index at rate $X$9; the decoder uses $S$0 to disambiguate $S$1 within the bin and then applies the reconstruction maps. The converse is obtained from

$S$2

followed by single-letterization with

$S$3

which recovers the lower bound $S$4 together with the distortion constraints (Wei et al., 23 Jul 2025).

3. Relation to classical Wyner–Ziv and remote source coding

In the ordinary Wyner–Ziv problem, only the reconstruction of $S$5 is relevant, and there is a single distortion constraint. The corresponding rate–distortion function is

$S$6

The indirect formulation differs in that fidelity is imposed on a latent variable, or on both the observed and latent variables simultaneously. In the latent-source setup, the extra semantic distortion couples through the modified distortion $S$7, and the encoder-decoder pair must choose $S$8, $S$9, and $X$0 to trade off two distortions rather than one (Wei et al., 23 Jul 2025).

When the formulation is specialized to a single latent reconstruction target $X$1 and a single distortion constraint, the indirect Wyner–Ziv theorem takes the form

$X$2

subject to the existence of a deterministic $X$3 such that

$X$4

Using

$X$5

this is equivalent to the usual Wyner–Ziv difference form. For $X$6, this formula is obtained directly as the single-encoder specialization of the distributed indirect source coding result under decoder side information (Tang et al., 23 Jan 2025). In the terminology used in the literature summarized in (Stylianou et al., 2021), this is the indirect or remote Wyner–Ziv problem associated with the line of results of Dobrushin–Tsybakov, Wyner–Ziv, and Draper–Wornell.

A common simplification is to treat indirect Wyner–Ziv coding as merely a notational variant of classical Wyner–Ziv coding. The latent-source formulation shows that this is not generally sufficient: the semantic constraint is not imposed directly on the encoder observation, and in the two-distortion setting the optimization is genuinely bivariate in $X$7 rather than scalar in $X$8 alone (Wei et al., 23 Jul 2025).

4. Non-asymptotic characterization and second-order behavior

Beyond the asymptotic regime, a non-asymptotic achievability bound can be derived via the Poisson matching lemma. In the stated result, the probability of exceeding either distortion is upper-bounded by

$X$9

where the information density is

$S\in\mathcal S$0

This form makes the finite-blocklength tradeoff explicit in terms of both information density and simultaneous distortion events (Wei et al., 23 Jul 2025).

A second-order approximation follows by invoking a multidimensional Berry–Esseen argument. For blocklength $S\in\mathcal S$1 and excess probability $S\in\mathcal S$2, the achievable rate satisfies

$S\in\mathcal S$3

where $S\in\mathcal S$4 is the covariance matrix of

$S\in\mathcal S$5

This identifies the dispersion-relevant quantities for the indirect Wyner–Ziv problem with two fidelity criteria and shows that the finite-blocklength penalty depends jointly on rate density and both distortion observables (Wei et al., 23 Jul 2025).

The non-asymptotic development is significant because it sharpens the asymptotic formula into a blocklength-sensitive achievable region. In the setting of semantic or goal-oriented communication, where short packets and task-level fidelity are often central, this finite-length perspective is not an ancillary refinement but part of the core operational description suggested by the problem statement in (Wei et al., 23 Jul 2025).

5. Structural properties and Gaussian specializations

For the decoder-only side-information version with a remote source $S\in\mathcal S$6, encoder observation $S\in\mathcal S$7, and decoder side information $S\in\mathcal S$8, the indirect Wyner–Ziv rate–distortion function is

$S\in\mathcal S$9

subject to the Markov chain

$X\in\mathcal X$0

In this general formulation, optimality requires the decoder to be the conditional mean,

$X\in\mathcal X$1

and the summary in (Stylianou et al., 2021) states that, in the jointly Gaussian–MSE case, side information at both the encoder and decoder does not reduce compression relative to decoder-only side information.

For multivariate jointly Gaussian random variables with squared-error fidelity, the problem admits a parallel-channel decomposition. With suitable whitening and singular-value decomposition, the rate becomes

$X\in\mathcal X$2

and the optimum is given by a water-filling solution

$X\in\mathcal X$3

This places the Gaussian indirect Wyner–Ziv problem within the usual water-filling paradigm after transformation to diagonal coordinates (Stylianou et al., 2021).

A vector Gaussian remote Wyner–Ziv problem under covariance distortion constraints admits a different closed form. If $X\in\mathcal X$4, then the rate–distortion function is

$X\in\mathcal X$5

The construction relies on a matrix “minimum” operator defined through joint diagonalization, and the formula shows how the distortion budget interacts with the portion of $X\in\mathcal X$6 observable only through $X\in\mathcal X$7 beyond what $X\in\mathcal X$8 already provides (Zahedi et al., 2015).

These Gaussian results clarify an important point. The indirect Wyner–Ziv rate–distortion function is not restricted to abstract single-letter existence statements; in several Gaussian regimes it admits explicit structure, optimal test channels, and water-filling or determinant expressions (Stylianou et al., 2021).

6. Distributed, algorithmic, and data-driven extensions

The indirect Wyner–Ziv paradigm extends naturally to distributed settings. In "Distributed Indirect Source Coding with Decoder Side Information" (Tang et al., 2024), $X\in\mathcal X$9 correlated sources $Y\in\mathcal Y$0 are independently encoded for a central decoder that also observes side information $Y\in\mathcal Y$1 and reconstructs a latent variable $Y\in\mathcal Y$2. An achievable region is given by the subset constraints

$Y\in\mathcal Y$3

and the same inequalities appear in the outer bound, with an exact characterization of the rate–distortion function when the sources are conditionally independent given $Y\in\mathcal Y$4. Under that conditional-independence assumption, the sum-rate reduces to

$Y\in\mathcal Y$5

For $Y\in\mathcal Y$6, the conditional-independence assumption is vacuous, and the result collapses to the standard indirect Wyner–Ziv formula (Tang et al., 2024).

Algorithmic computation has also received explicit treatment. The 2025 distributed follow-up develops a distributed Blahut–Arimoto algorithm and shows that, for $Y\in\mathcal Y$7, it reduces to the usual Wyner–Ziv BA: update a decoder-side distribution $Y\in\mathcal Y$8, then update the encoder-side test channel $Y\in\mathcal Y$9, and recompute the optimal reconstruction rule

$(S,X,Y)$0

Varying the Lagrange multiplier sweeps out the rate–distortion curve (Tang et al., 23 Jan 2025). In the latent-source two-distortion setting, (Wei et al., 23 Jul 2025) likewise proposes a Blahut–Arimoto algorithm tailored for the indirect Wyner–Ziv setup and uses it to numerically evaluate the achievable indirect rate–distortion region when $(S,X,Y)$1 is treated as a classification label.

When the joint law is unknown and only data are available, the indirect rate–distortion function can still be estimated. "Data-Driven Neural Estimation of Indirect Rate-Distortion Function" (Yu et al., 2024) emphasizes the reduction

$(S,X,Y)$2

and reformulates the indirect RDF as a variational problem in which the unknown reduced distortion is learned through an $(S,X,Y)$3 regression characterization of conditional expectation. The resulting neural approach alternates between a regression network that estimates the conditional distortion map and an outer variational optimization that learns the test channel, with strong consistency stated under sufficiently rich hypothesis classes, increasing sample sizes, and appropriate growth of inner and outer gradient steps (Yu et al., 2024).

Taken together, these extensions show that the indirect Wyner–Ziv rate–distortion function is simultaneously a single-encoder side-information problem, a distributed latent-source compression problem, a finite-blocklength achievability problem, and a computational object accessible through BA-type and data-driven estimation procedures. This suggests a unified role for indirect Wyner–Ziv theory in semantic communication, distributed learning, remote sensing, and related settings where the fidelity criterion is imposed on a task-relevant latent variable rather than on the raw observation itself (Tang et al., 2024).