Capacity-Distortion Function Overview

• Capacity-distortion function is defined as the maximum reliable communication rate achievable under a prescribed distortion constraint for state estimation.
• It extends classical rate-distortion and channel capacity formulations by integrating state sensing, with applications in ISAC, broadcast channels, and feedback systems.
• Methodologies such as nested coding, proximal optimization, and neural network estimators are used to evaluate and optimize capacity-distortion tradeoffs.

The capacity-distortion function, usually denoted $C(D)$, is the supremum of all reliable communication rates achievable while satisfying a prescribed distortion constraint on estimating or reconstructing a state process. In contemporary information theory it appears most prominently in state-dependent channels and integrated sensing and communication (ISAC), where the same waveform must support both message transmission and state inference; closely related variants include the capacity-distortion-cost function $C(D,B)$, which adds an input-cost constraint, and the classical rate-distortion and distortion-rate functions $R(D)$ and $D(R)$, which provide the source-coding prototype for these formulations (Ahmadipour et al., 2021, Li et al., 28 Apr 2025, Arumugam et al., 2022).

1. Formal definitions and conceptual lineage

The classical source-coding prototype is the rate-distortion function

$$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$

together with its inverse distortion-rate form

$$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$

In the formulation surveyed for capacity-limited cognition and reinforcement learning, $R(D)$ is interpreted as the minimum number of bits that must be retained on average from a source in order to achieve a target fidelity level, while $D(R)$ is the minimum distortion achievable under a rate budget $R$ (Arumugam et al., 2022).

The communication-side analogue replaces “minimum bits for a given fidelity” with “maximum reliable rate for a given sensing fidelity.” In the memoryless single-receiver ISAC model, $C(D)$ is defined as the largest reliable rate below which a message can be conveyed while satisfying a distortion constraint on state sensing. This generalizes ordinary channel capacity: when sensing is ignored, the formulation reduces to standard communication capacity (Ahmadipour et al., 2021). The same reduction appears in the continuous-memoryless capacity-distortion-cost framework: if the input-cost constraint is removed, or if it is inactive, then $C(D,B)$0 reduces to the standard capacity-distortion function $C(D,B)$1 (Li et al., 28 Apr 2025).

The conceptual lineage is also visible at the coding-theoretic level. For arbitrary discrete memoryless sources, nested polar codes achieve the Shannon rate-distortion function

$C(D,B)$2

and for arbitrary discrete memoryless channels they achieve the Shannon capacity

$C(D,B)$3

The nested construction uses a pair $C(D,B)$4 and auxiliary channels whose symmetric capacities differ by the target mutual information quantity, yielding $C(D,B)$5 in lossy compression and $C(D,B)$6 in channel coding (Sahebi et al., 2014). This establishes the rate-distortion and capacity endpoints that capacity-distortion problems interpolate between.

2. Single-user capacity-distortion functions in state-dependent channels

A canonical single-user formulation considers a memoryless channel with i.i.d. time-varying state sequence $C(D,B)$7. The transmitter sends a message $C(D,B)$8, the receiver decodes from $C(D,B)$9, and the state sequence is estimated under a single-letter distortion measure $R(D)$0. The average distortion constraint is

$R(D)$1

and the operational question is the maximum reliable rate compatible with that bound (Ahmadipour et al., 2021).

For memoryless single-receiver channels with i.i.d. state, the capacity-distortion tradeoff is characterized by a single-letter optimization. In one common form,

$R(D)$2

More generally, when strictly causal generalized feedback is used at the transmitter, the coding theorem introduces an auxiliary random variable $R(D)$3,

$R(D)$4

subject to a distortion constraint induced by the estimator based on the available sensing variables. The interpretation is that $R(D)$5 carries the message-bearing structure, $R(D)$6 is the channel input, $R(D)$7 is used for both decoding and sensing, and the optimal tradeoff is obtained by jointly choosing the input law and the estimator (Ahmadipour et al., 2021).

A broader SD-DMC framework incorporates encoder side information $R(D)$8, receiver side information $R(D)$9, and optional feedback $D(R)$0, with $D(R)$1. There a rate-distortion pair $D(R)$2 is achievable if

$D(R)$3

where

$D(R)$4

The corresponding capacity-distortion function is

$D(R)$5

Its single-letter rate functional is

$D(R)$6

with admissible sets specialized to strictly causal, causal, and noncausal encoder side information. The theorem is tight for strictly causal and causal SI-T,

$D(R)$7

while for noncausal SI-T it yields an achievable inner bound,

$D(R)$8

Within this formulation, $D(R)$9 is the communication term and $$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$0 is the penalty for conveying the information needed for state estimation. The resulting $$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$1 is non-decreasing and concave in $$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$2 (Li et al., 2024).

Two operational extremes are explicit in this framework. In communication-only mode, $$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$3 and the state-estimation constraint disappears. In sensing-only mode, the rate is forced to zero and the problem becomes one of minimizing distortion subject to a nonnegative residual rate condition. The paper uses these extreme points to show that simple time-sharing is generally suboptimal because the capacity-distortion function is concave (Li et al., 2024).

3. Broadcast channels, degradedness, and rate-limited feedback

For broadcast settings, the scalar function $$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$4 is replaced by a capacity-distortion region over rate-distortion tuples. In the state-dependent broadcast extension of joint sensing and communication, the region consists of all tuples

$$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$5

that are simultaneously achievable. For physically degraded broadcast channels, the region admits a full characterization based on the familiar superposition structure, with rate constraints such as

$$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$6

together with distortion constraints of the form

$$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$7

For general broadcast channels, inner and outer bounds are available, and the paper identifies a sufficient condition under which the capacity-distortion region factors as

$$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$8

so that communication and sensing decouple (Ahmadipour et al., 2021).

A more general degraded SD-DMBC formulation with common and private messages introduces auxiliaries $$R(D)=\inf I(X;Z)\quad \text{such that}\quad \mathbb{E}[d(X,Z)]\le D,$$9 and an achievable region whose rate constraints include

$$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$0

$$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$1

with additional conditions on $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$2 and $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$3 that quantify the rate needed for state-description information. The achievable region is monotone in $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$4, convex, and can be traced through weighted-sum-rate optimization. Tightness is established for several special cases, including a classical ISAC configuration in which one decoder performs sensing and the other performs communication only (Li et al., 2024).

A distinct capacity-distortion viewpoint arises in the two-user broadcast erasure channel with one-sided, rate-limited feedback. There the forward channel carries independent messages $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$5 and $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$6, while only $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$7 provides causal rate-limited feedback. The feedback link is treated as conveying a lossy reconstruction of the erasure-state sequence $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$8 under Hamming distortion. The minimum achievable distortion $$D(R)=\inf \mathbb{E}[d(X,Z)]\quad \text{such that}\quad I(X;Z)\le R.$$9 is defined by the paper’s rate-distortion relation

$R(D)$0

with notation as in the paper, and the resulting outer bound on the forward capacity region is the symmetric pair of weighted sum-rate inequalities

$R(D)$1

where the weight $R(D)$2 is determined by the minimum feedback distortion. A key converse step is

$R(D)$3

The proof proceeds by showing that rate-limited feedback imposes a minimum reconstruction distortion, that this distortion constrains the set of indistinguishable channel-state sequences, and that a worst-case modified state sequence can then be chosen to maximize correlation with the silent receiver’s state. The central claim is therefore not merely that low-rate feedback is scarce, but that its scarcity is operationally expressed through distortion in reconstructed CSI, which in turn induces the outer bound on the forward capacity region (Vahid, 2021).

4. Information-spectrum generalizations and distortion criteria

The most general formulas in the supplied literature are information-spectrum capacity-distortion formulas for action-dependent ISAC with arbitrary alphabets, nonstationary or nonergodic states, and general state-dependent channels

$R(D)$4

An action sequence $R(D)$5 generates the state $R(D)$6, imperfect encoder side information $R(D)$7, and imperfect decoder side information $R(D)$8. The estimator reconstructs the state from the message-dependent action, encoder-side information, and feedback $R(D)$9 (Chen et al., 2023).

Two distortion criteria are distinguished. Under average distortion, the relevant asymptotic quantity is

$D(R)$0

leading to the average-distortion capacity

$D(R)$1

Under maximal distortion, the criterion is

$D(R)$2

with corresponding capacity

$D(R)$3

The formulas are generalized Gel'fand-Pinsker expressions: $D(R)$4 shapes the state, $D(R)$5 is an auxiliary coding process, the communication term is the information delivered through the channel and decoder side information, and the penalty term accounts for dependence on encoder-side state information (Chen et al., 2023).

This framework subsumes several standard models by specialization. Setting $D(R)$6 and $D(R)$7 recovers a general Gel'fand-Pinsker capacity expression. Setting $D(R)$8 yields the general point-to-point capacity formula. The same paper also treats memoryless and mixed channels, and extends the formulation to rate-limited CSI at one side. For rate-limited CSI at the encoder, the region is characterized by

$D(R)$9

whereas for rate-limited CSI at the decoder the region becomes

$R$0

$R$1

These forms make explicit that the distortion-constrained sensing component can be combined with action-dependent state generation and compressed side information (Chen et al., 2023).

5. Capacity-distortion-cost functions and numerical computation

For continuous memoryless channels, the capacity-distortion-cost function is defined as

$R$2

where the optimization is over Borel probability measures $R$3 subject to the input-cost constraint

$R$4

and the expected state-estimation distortion constraint

$R$5

The optimal estimator is

$R$6

with posterior

$R$7

The central difficulty is that the optimization is infinite-dimensional and, in general, $R$8 has no closed-form expression (Li et al., 28 Apr 2025).

The proposed computational method alternates between updating the input distribution and the estimator. The input update is posed in Wasserstein space through a proximal-point iteration,

$R$9

and approximated by a particle pushforward

$C(D)$0

using the particle representation

$C(D)$1

Importance sampling is used for the mutual-information and distortion integrals, and the estimator is parameterized as $C(D)$2, with the numerical section employing a small fully connected neural network with ReLU activations. Dual ascent updates the Lagrange multipliers associated with cost and distortion. The paper guarantees local convergence of the Wasserstein $C(D)$3-update under suitable conditions when the estimator is fixed, but does not establish convergence of the full alternating scheme including neural-network updates (Li et al., 28 Apr 2025).

Complementary computational results appear in the discrete SD-DMC setting, where a proximal block coordinate descent method is proposed for evaluating point-to-point and broadcast capacity-distortion formulas. For variables $C(D)$4, the generic update is

$C(D)$5

and any limit point is a stationary point. The stopping rule is based on a certificate $C(D)$6 satisfying

$C(D)$7

with termination when

$C(D)$8

Earlier single-user work also proposed a Blahut-Arimoto-type numerical method that iteratively updates the input law or auxiliary distribution under a distortion-constrained Lagrangian (Li et al., 2024, Ahmadipour et al., 2021).

The continuous CDC computations also expose operational structure. In the ISAC example, when $C(D)$9 the problem reduces to pure communication under power constraint, yielding capacity $C(D,B)$00 and Gaussian input with zero mean and variance $C(D,B)$01. As $C(D,B)$02, the sensing term dominates and the input distribution concentrates on a small number of points, revealing the reported random-deterministic trade-off (Li et al., 28 Apr 2025).

6. Related rate-distortion viewpoints, task-oriented distortion, and coding implications

A related but distinct use of rate-distortion ideas appears in capacity-limited cognition and reinforcement learning. There a policy is treated as a communication channel subject to a mutual-information budget,

$C(D,B)$03

and the more explicit learning-target formulation introduces an episode-indexed rate-distortion function

$C(D,B)$04

The distortion used in the principal RL construction is value-based,

$C(D,B)$05

and the resulting regret bounds are written both in $C(D,B)$06 form and in inverse $C(D,B)$07 form. The operational point is that the learner need not identify the exact environment if a compressed surrogate preserves the optimal $C(D,B)$08-values relevant to decision quality (Arumugam et al., 2022).

Across the communication settings in this literature, the distortion metric is explicitly task dependent. In the rate-limited-feedback broadcast erasure channel it is Hamming distortion on the reconstructed erasure-state sequence. In continuous ISAC it can be squared error on a channel state such as angle of arrival. In the generalized SD-DMC and ISAC models it is an arbitrary single-letter or block distortion, under either average or maximal criteria. In the RL extension it is value distortion or expected squared regret rather than model mismatch (Vahid, 2021, Li et al., 28 Apr 2025, Li et al., 2024, Arumugam et al., 2022).

Several recurring structural themes follow directly from the cited work. First, the distortion constraint is often task-oriented rather than model-oriented: what matters is preserving the state information relevant to decoding, sensing, or control. Second, joint design can dominate resource splitting. In the single-user and broadcast ISAC formulations, a single carefully chosen input distribution can simultaneously support communication and sensing, and the examples show better rate-distortion points than naive splitting of time, power, or bandwidth (Ahmadipour et al., 2021). Third, rate limits can act through induced distortion rather than only through explicit side-channel budgets, as in the outer-bound technique for one-sided rate-limited feedback (Vahid, 2021).

Coding-theoretically, nested polar codes do not by themselves solve a general capacity-distortion problem, but they do achieve the two limiting Shannon objects—rate-distortion and capacity—through a common nested architecture whose net rate is a difference of symmetric-capacity terms (Sahebi et al., 2014). This suggests that nested constructions are naturally aligned with problems in which reliable communication and controlled distortion must be balanced within a single coding framework.