Rate-Distortion-Complexity Analysis

• RDC analysis is defined as a framework that quantifies the three-way tradeoff among communication rate, signal fidelity, and computational complexity.
• It extends classical rate-distortion theory by adding explicit constraints on model complexity and semantic distance, influencing modern neural codecs and semantic communication design.
• Applications span neural image/video compression, Reed-Solomon decoding, and broadcast joint source-channel coding, providing actionable insights for system optimization.

Rate-distortion-complexity (RDC) analysis studies the joint behavior of three coupled quantities: the communication or coding rate, the fidelity of reconstruction or task performance, and the computational or representational complexity required to attain that fidelity. In contemporary semantic communication, RDC analysis is formulated as an extension of classical rate-distortion theory that adds explicit constraints on semantic distance and model complexity, thereby yielding a fundamental three-way tradeoff among achievable rate, semantic distance, and model complexity (Chai et al., 16 Feb 2026). Closely related formulations now appear in neural image and video compression, broadcast joint source-channel coding, Reed-Solomon multiple decoding, and control-communication settings, although the precise meanings of rate, distortion, and complexity vary with the application (Gao et al., 2023, Chen et al., 2024, Qin et al., 23 Mar 2026, 0908.2828, Wendel et al., 2023).

1. Conceptual foundations

The modern RDC viewpoint begins from the classical rate-distortion problem and augments it with a third constraint or penalty representing either model complexity, decoding complexity, algorithmic cost, or computational budget. In the semantic communication formulation, the extension is explicit: a source $S$ is inferred from indirect observations $X$, a representation $U$ is transmitted or processed, and the decoder produces $\hat S$. The optimization minimizes an achievable rate under simultaneous constraints on distortion, semantic distance, and complexity (Chai et al., 16 Feb 2026).

A broader historical context shows that the relationship between distortion and complexity has been treated in several non-identical ways. One line of work argues that complexity-distortion is only a special case of rate tolerance when constraint sets are clear sets that look like balls with the same radius, and that complexity distortion is not generally equivalent to rate distortion (Lu, 2012). Another line defines the complexity of a continuous-time linear system as the minimum number of bits required to describe its forward increments to a prescribed level of fidelity, identifying complexity with a rate-distortion function for the increment distribution (Wendel et al., 2023). These formulations do not collapse into a single universal definition, but they share the same information-theoretic pattern: complexity is treated as an explicit design variable rather than as an incidental implementation detail.

In current machine learning systems, this shift is especially consequential because rate-distortion optimality alone often pushes architectures toward high model size, high memory footprint, or long sequential decoding time. RDC analysis therefore serves both as a theoretical generalization and as a design methodology for systems operating under bandwidth and compute constraints (Gao et al., 2023, Chen et al., 2024).

2. Formal problem statements and metrics

A canonical RDC formulation for semantic communication is

$\min_{p_{XU\hat S}} I(U;\hat S)$

subject to

$$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$

Here, $I(U;\hat S)$ is the achievable communication rate, $d(S,\hat S)$ is a bit-level distortion metric such as MSE or Hamming distance, $d_p(p_S,p_{\hat S})$ is a semantic distance or divergence between source and reconstruction distributions, and $I(X;U)$ is the complexity measure (Chai et al., 16 Feb 2026).

Term	Formula / metric	Role
Rate	$X$0	Minimum achievable communication rate
Distortion	$X$1	Symbol-level fidelity
Semantic distance	$X$2	Divergence between source and recovery
Complexity	$X$3	Model complexity

Within this framework, semantic distance is any statistical divergence between the distributions of $X$4 and $X$5. The examples explicitly named are Wasserstein distance, KL divergence, and total variation. The rationale is that low symbol-wise distortion does not by itself guarantee preservation of meaning for downstream tasks (Chai et al., 16 Feb 2026).

The complexity term $X$6 is motivated by the Minimum Description Length principle and the Information Bottleneck. It quantifies the number of bits that the encoder output carries about its input, and thus its representational capacity. In that sense, it is an information-theoretic complexity measure rather than a raw architectural count (Chai et al., 16 Feb 2026).

Outside semantic communication, the same three-axis structure is instantiated differently. In neural image compression, decoding complexity is modeled as the proportion of spatial positions decoded sequentially,

$X$7

and the training objective becomes

$X$8

(Gao et al., 2023). In neural video coding, complexity is measured by encoding and decoding kMACs per pixel, model size, and the channel size $X$9 of the full-resolution conditional signal (Chen et al., 2024). In broadcast rateless DeepJSCC, receiver-side complexity depends on the number of rateless symbols $U$0 and the number of belief propagation iterations $U$1, with

$U$2

(Qin et al., 23 Mar 2026). In application-level codec selection, the three variables are unified through a Lagrangian cost,

$U$3

where $U$4 encode application priorities (Queiroz et al., 7 Sep 2025).

3. Theoretical characterizations and asymptotics

The semantic communication RDC framework includes closed-form theoretical results for both Gaussian and binary semantic sources. For the Gaussian semantic source, with $U$5, MSE distortion, and Wasserstein semantic distance, the minimum achievable rate is given by a piecewise function $U$6, with parameters

$U$7

For the binary semantic source, the RDC function is reported as

$U$8

where $U$9 is the binary entropy function and $\hat S$0 is determined from the constraints using mutual information and crossover probabilities. The interpretation stated for the Gaussian case is that increasing model complexity allows lower communication rates at the same distortion and divergence level (Chai et al., 16 Feb 2026).

A related but distinct asymptotic notion is the rate-distortion dimension (RDD) of an analog stationary process,

$\hat S$1

which is shown to equal the upper information dimension under stated regularity conditions. For a piecewise constant process that jumps with probability $\hat S$2, the RDD equals $\hat S$3. This makes the small-distortion asymptotics of $\hat S$4 an operational measure of source complexity (Rezagah et al., 2016).

The same interplay between performance and algorithmic burden appears in successive refinement. A source is strongly successively refinable if successive refinement coding can achieve the second-order optimum rate, including the dispersion terms, at both decoders. The reported positive cases are any discrete source under Hamming distortion and the Gaussian source under quadratic distortion. The same work shows that layered code constructions can reduce encoding complexity while retaining asymptotically optimal rate-distortion performance; when the number of layers grows with block length $\hat S$5, an $\hat S$6 algorithm can asymptotically achieve the rate-distortion bound (No et al., 2015).

For continuous-time linear systems, complexity is defined as the minimum number of bits required to describe forward increments to a desired level of fidelity. For Gaussian increments with covariance $\hat S$7, the rate-distortion function takes the standard Gaussian form, and when specialized to system increments it yields an explicit complexity measure in bits per increment. This ties dynamical uncertainty accumulation directly to an information-theoretic complexity quantity (Wendel et al., 2023).

4. Communication-theoretic instantiations

In semantic communication, RDC analysis is used to formalize a resource-allocation problem between communication rate and local computation. The framework explicitly permits one to shift resource burden between rate and model complexity while holding distortion and semantic relevance fixed. The reported experiments on classification, image generation, and video compression validate the three-way tradeoff and show that $\hat S$8 effectively correlates with practical computational costs such as FLOPs, especially while the representational bottleneck is active (Chai et al., 16 Feb 2026).

In Reed-Solomon decoding, RDC analysis appears in a different operational guise. An error pattern $\hat S$9 and an erasure pattern $\min_{p_{XU\hat S}} I(U;\hat S)$0 are related by a letter-by-letter distortion matrix

$\min_{p_{XU\hat S}} I(U;\hat S)$1

and total distortion

$\min_{p_{XU\hat S}} I(U;\hat S)$2

The classical decoding condition becomes a distortion threshold condition,

$\min_{p_{XU\hat S}} I(U;\hat S)$3

Multiple decoding attempts are then interpreted as a covering problem in which the set of erasure patterns must cover likely error patterns under the distortion threshold. The rate $\min_{p_{XU\hat S}} I(U;\hat S)$4 is the log-number of decoding attempts, and thus directly measures complexity (0908.2828).

A refinement of that approach uses the rate-distortion exponent (RDE) to analyze finite-length reliability. The central bound is

$\min_{p_{XU\hat S}} I(U;\hat S)$5

where $\min_{p_{XU\hat S}} I(U;\hat S)$6. This gives exponentially tight error-probability bounds and a direct performance-complexity tradeoff for multiple decoding attempts of Reed-Solomon codes (Nguyen et al., 2010).

In rateless DeepJSCC for broadcast channels, the tradeoff is receiver-adaptive. Learned nonlinear source transforms are combined with physical-layer LT codes, and each receiver can vary both the number of downloaded rateless symbols $\min_{p_{XU\hat S}} I(U;\hat S)$7 and the number of belief propagation iterations $\min_{p_{XU\hat S}} I(U;\hat S)$8. The design objective is

$\min_{p_{XU\hat S}} I(U;\hat S)$9

The training loss is summarized as

$$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$0

and the reported operational effect is a controllable tradeoff between distortion, rate, and decoding complexity for heterogeneous receivers (Qin et al., 23 Mar 2026).

5. Neural compression, codecs, and application space

In neural image compression, RDC analysis is used to convert decoding latency into an optimization variable. The variable-complexity codec employs a spatial binary mask $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$1 to decide which latent positions are decoded sequentially and which are decoded in parallel. Training proceeds with the loss

$$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$2

and the same model can be conditioned on a user-specified complexity indicator $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$3 at decode time. Reported Kodak decoding times range from 114 ms for a hyperprior-only model with $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$4 to 7283 ms for the variable-complexity model at $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$5, close to 7227 ms for a full context plus hyperprior model. The empirical result is a continuous, monotonic RDC curve: higher decoding time yields better rate-distortion performance (Gao et al., 2023).

In neural video coding, the central comparison is among conditional coding (CC), conditional residual coding (CR), and masked conditional residual coding (MCR). The reported conclusion is that CR and MCR achieve similar or better rate-distortion performance than CC at lower complexity, particularly when channel count $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$6 is reduced. One concrete comparison states that CC with $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$7 has BD-rate $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$8, 1153 encoding kMACs/pixel, 762 decoding kMACs/pixel, and model size 7.94 MB, whereas MCR with $$\mathbb{E}[d(S,\hat S)] \leq \theta_d, \qquad d_p(p_S,p_{\hat S}) \leq \theta_p, \qquad I(X;U) \leq \theta_c .$$9 has BD-rate $I(U;\hat S)$0, 970 encoding kMACs/pixel, 598 decoding kMACs/pixel, and model size 7.81 MB (Chen et al., 2024).

Dynamic point cloud compression introduces an explicitly named rate-distortion-complexity optimization (RDCO) framework. AdaDPCC uses a slimmable model with multiple coding routes, route-specific multipliers $I(U;\hat S)$1, and the per-route loss

$I(U;\hat S)$2

At inference time, a rate control module predicts bitrate for each route and selects a route to satisfy the remaining bit budget. The reported experimental results are an average BD-Rate reduction of 5.81%, a BD-PSNR improvement of 0.42 dB, mean bitrate error of 0.40%, and coding time reduction of up to 44.6% compared to D-DPCC (Zhang et al., 28 Aug 2025).

At the decision-theoretic level, RDC analysis has also been used to compare entire codec families. The proposed framework characterizes each codec by a cloud of points in RDC space and evaluates it using

$I(U;\hat S)$3

An application is then associated with a point $I(U;\hat S)$4 in an application space. The reported result is that, within the stated RDC computation constraints, only four neural video codecs came out as the best suited for any application, depending on where its desirable $I(U;\hat S)$5 lies (Queiroz et al., 7 Sep 2025).

6. Related notions, misconceptions, and terminological boundaries

A persistent misconception is that complexity distortion and rate distortion are generally equivalent. The contrary position stated in the literature is explicit: complexity-distortion is only a special case of rate tolerance when constraint sets are clear sets that look like balls with the same radius, and rate distortion can only be equivalent to rate tolerance under those conditions (Lu, 2012). This matters because modern RDC analyses frequently rely on soft, learned, or distributional constraints rather than on fixed-radius hard distortion balls.

Another boundary concerns what complexity measures capture. In semantic communication, $I(U;\hat S)$6 is presented as a theoretically grounded measure of effective complexity rather than a mere proxy for network size or depth. The reported experiments further state that once the representation saturates, increasing network size can increase FLOPs without increasing $I(U;\hat S)$7, indicating diminishing returns (Chai et al., 16 Feb 2026). By contrast, in neural codecs complexity may be measured directly as kMACs/pixel, decoding time, model size, or the number of sequential decoding operations (Gao et al., 2023, Chen et al., 2024). RDC analysis is therefore not tied to a single canonical complexity metric; it is a framework whose third axis must be interpreted relative to the system model.

The acronym “RDC” is also not semantically unique across the literature. In a separate task-oriented lossy compression line, “RDC” denotes rate-distortion-classification rather than rate-distortion-complexity. For Bernoulli sources under Hamming distortion, that formulation derives closed-form one-shot RDC and dual DRC tradeoffs, an achievable distortion-classification region induced by a fixed representation, and lower and upper bounds on the minimum asymptotic rate required for universal encoders (Nguyen et al., 17 Jan 2026). The overlap in acronym underscores a broader methodological point: current research increasingly treats rate-distortion analysis as a scaffold to which additional operational constraints—complexity, semantics, perception, or classification—can be attached.

Taken together, these works establish RDC analysis as an information-theoretic and systems-level methodology for quantifying how much bitrate, distortion, and computational effort must be exchanged against one another. Its modern importance lies not in a single closed-form theorem, but in the systematic conversion of computational resources into first-class variables of communication and compression theory (Chai et al., 16 Feb 2026, Queiroz et al., 7 Sep 2025).