Channel Capacity Constrained Estimation

• C3E is a framework that formalizes estimation under finite information flow, linking the achievable estimation accuracy directly to the channel's capacity.
• It unifies principles from information theory, communications, control, and machine learning by rigorously quantifying the tradeoffs between communication rate, distortion, and uncertainty.
• Practical schemes using random coding and ML decoding demonstrate near-optimal performance, with extensions addressing robustness, multi-constraint scenarios, and over-squashing in graph neural networks.

Channel Capacity Constrained Estimation (C3E) encompasses a family of methodologies and theoretical frameworks in which estimation or learning tasks are fundamentally limited by the information-theoretic capacity of an underlying channel. These frameworks unite concepts from information theory, communications, estimation, and learning, rigorously quantifying the tradeoffs and achievable performance when inference must proceed under explicit or implicit channel capacity constraints.

1. Fundamental Concepts and Problem Formulation

Channel Capacity Constrained Estimation formalizes estimation tasks under fundamental information-theoretic limitations imposed by a communication channel's capacity. Consider a system with input $X$, channel state $S$ (possibly random or unknown), channel output $Y$, and a decoder or estimator seeking to reconstruct a desired quantity given $Y$ (and sometimes $X$). The core question is: how accurately can estimation or inference proceed, and at what communication rate, when the information flow $I(X;Y)$ cannot exceed the channel capacity $C$?

A representative canonical C3E problem is the joint communication-and-estimation scenario for a memoryless channel $P_{Y|X,S}$ with unknown i.i.d. state $S$ and an average distortion measure $d(S,\hat S)$. The capacity-distortion function is defined as: $$C(D) = \sup_{P_X: \mathbb{E}[d^*(X)] \leq D} I(X;Y),$$ where $$d^*(x) = \inf_{h_0:\mathcal{X}\times\mathcal{Y}\to\mathcal{S}} \mathbb{E}[d(S, h_0(x,Y))|X=x]$$ quantifies the "estimation cost" associated with each channel input symbol (0801.1136, Zhang et al., 2011).

This constrained optimization sets a rigorous link between communication rate, estimation accuracy, and the stochastic or adversarial uncertainty about the channel state.

2. Theoretical Characterizations and Tradeoffs

All C3E formulations rest on classical information-theoretic bounds and coding theorems. Central results include:

• Distortion-constraint $\rightarrow$ Input-cost equivalence: The estimation constraint (on expected distortion of $S$) is equivalent to an average input-cost constraint where the "cost" is the minimum achievable distortion when using symbol $x$.
• Single-letter Capacity-Distortion Function: For discrete memoryless channels (DMCs) or Gaussian channels (with appropriate regularity conditions), the optimal communication-estimation tradeoff is characterized by a single-letter maximization:

$C(D) = \max_{P_X: \mathbb{E}[d^*(X)]\le D} I(X;Y).$

This result is applicable under fairly general conditions, including multiple access channels (MAC) via suitable extensions (0801.1136, Zhang et al., 2011).

• Optimal Schemes: Random coding with decoding-first, estimate-per-symbol (using the one-shot optimal estimator $h^*(x,y)$ for each coordinate, once the transmitted $x^n$ has been recovered) is provably asymptotically optimal in the blocklength sense.
• Extensions and Special Cases: The framework accommodates multiple constraints (e.g., simultaneous energy and estimation costs), compound/uncertain channels (pessimization over channel parameters), and measures such as capacity per unit distortion.
• Examples: On memoryless Rayleigh channels, the per-input estimation cost is $d^*(x) = 1/(|x|^2+1)$ for MMSE distortion, yielding the constraint:

$$C(D) = \max_{P_X: \mathbb{E}[1/(|X|^2+1)] \leq D,\, \mathbb{E}[|X|^2]\leq \rho} I(X;Y).$$

At high SNR, the difference between achievable rates and unconstrained capacity is tightly characterized up to 1.443 bits (Zhang et al., 2011).

3. Practical Achievability Schemes

The C3E achievability approach proceeds via random coding and typical set decoding, followed by estimator application:

1. Codebook Generation: Draw codewords $X^n(m)$ i.i.d. under $P_X$ constrained by the average estimation cost $d^*(x)$.
2. Decoding: Use standard noncoherent maximum-likelihood (ML) or typicality decoding to recover the message.
3. State Estimation: Upon decoding, treat the recovered $X^n$ as known, and run $h^*(x_i, y_i)$ (the single-symbol estimator which achieves $d^*(x_i)$) for each symbol $i$.

This "decode-then-estimate-per-symbol" scheme not only matches performance to within $o(1)$ of the optimal $C(D)$ as $n \to \infty$ but is also constructive for implementation.

4. Achievable Rates, Outage Capacity, and Robustness

C3E generalizes to the practical setting of channel uncertainty and imperfect channel estimation. The resulting notions are:

• Composite Channel Model: For unknown parameter $\theta \in \Theta$ with a (possibly conditionally) known distribution, decoder operations are based on the "composite" channel,

$$\widetilde W(y|x, \hat\theta) = \int_\Theta W(y|x, \theta) \psi(\theta|\hat\theta)\,d\theta.$$

Optimizing using this metric yields a practically implementable nearest-neighbor decoder whose performance achieves the capacity of the composite channel (0706.2809).

• Estimation-Induced Outage (EIO) Capacity: Even with the optimal decoder, achievable rates in the presence of estimation errors are random (depend on $\theta$). The EIO capacity for outage probability $\gamma_{QoS}$ is:

$$C(\gamma_{QoS}, \hat\theta) = \max_{P_X} \sup_{\Lambda: \Pr(\Lambda|\hat\theta)\ge 1-\gamma_{QoS}} \inf_{\theta\in\Lambda} I(P_X, W(\cdot|\cdot,\theta)).$$

This captures the maximal rate sustainable with error probability not exceeding $\gamma_{QoS}$ over the uncertainty induced by channel estimation.

• Performance Benchmarks: In MIMO-BICM, the optimal C3E-metric decoder can yield up to 2 dB improvement at BER $10^{-3}$ over mismatched ML decoding when the training is short (as low as $N=2$ symbols), with complexity only marginally higher than standard ML decoding (0706.2809).

5. Extensions to Control, Learning, and Graph Neural Networks

C3E unifies a wide range of estimation settings where channel capacity is a hard bottleneck:

• Networked Control and Nonlinear Estimation: In the context of estimation over communication-constrained networks, the minimum channel capacity required for arbitrarily accurate state estimation is given by the restoration entropy $h_{res}$ (for regular observation), or topological entropy $h_{top}$ (for observation with exactness) of the underlying system, aligning the estimation limit with a dynamical systems invariant (Hafstein et al., 2018).
• Graph Representation Learning: In spectral graph neural networks (GNNs), the C3E framework recasts the width and depth selection problem as maximizing mutual information under structural and architectural constraints, using a nonlinear programming approach. Here, the "channel" is the propagation of node signals through layers; "over-squashing" describes catastrophic information loss when the network's effective channel capacity is less than the entropy of the relevant graph region. C3E methods yield architectures that provably avoid over-squashing and maximize learned representation quality across various benchmarks (You et al., 9 Nov 2025).

6. Numerical and Algorithmic Solutions

Multiple algorithmic strategies have been developed for computing C3E tradeoffs:

• Linear and Semidefinite Programming: For control scenarios (e.g., Lorenz attractor), certified upper bounds are produced via finite-dimensional LMIs or LPs on triangulations of the state space, ensuring convergence to the restoration-entropy limit as the mesh is refined (Hafstein et al., 2018).
• Monte Carlo and Sampling Methods: In settings where the channel is not analytically tractable, data-driven estimation of capacity-distortion functions employs alternating optimization, mutual information estimation by neural network discriminators, and Wasserstein-proximal updating of input distributions, with integrals estimated by importance sampling and particles (Li et al., 28 Apr 2025).
• Channel Decoding under Imperfect Estimation: For memoryless channels with imperfect channel estimation, the C3E-optimal decoding metric is

$$D_M(x, y|\hat\theta) = -\log \Bigg( \int W(y|x, \theta) \, d\psi(\theta|\hat\theta) \Bigg)$$

and can be implemented with minimal additional complexity over ML decoding, e.g., by adjusted Euclidean distances in MIMO systems (0706.2809).

7. Applications and Impact

C3E frameworks have wide-ranging implications:

• Reliability and Security in Communications: C3E quantifies the ultimate rate-distortion tradeoff and the effect of channel uncertainty, enabling robust communication system design.
• Joint Sensing and Communication: In integrated systems where both message detection and state estimation (e.g., for radar or environmental sensing) are simultaneous objectives, the C3E rate-distortion-cost framework precisely quantifies achievable tradeoffs.
• Machine Learning Architectures: C3E optimizing mutual information under width/depth constraints guides principled neural network design, particularly in situations with severe information bottlenecks (e.g., over-squashing in GNNs).
• Control and Estimation under Bit-rate Constraints: C3E provides data-rate theorems for real-time state estimation in networked systems, relating the required channel capacity to dynamical system invariants.
• Robustness to Model Uncertainty: By formulating worst-case (compound channel) or outage capacities under model uncertainty, C3E delivers designs that remain reliable across a range of operating regimes.

In summary, Channel Capacity Constrained Estimation furnishes a rigorous, broadly applicable theoretical and practical foundation for any system—communication, control, or learning—where estimation must proceed under the inescapable limits of finite information flow. Its single-letter capacity-distortion formulas, robust achievability constructions, extensions to uncertainty and nonlinearity, and algorithmic tractability form a cornerstone for the rational design of estimation under constraints.