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An Optimal Diffusion Approach to Quadratic Rate-Distortion Problems: New Solution and Approximation Methods

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
information theoryrate-distortiondiffusion processesstochastic control
Abstract:

When compressing continuous data, some loss of information is inevitable, and this incurred a distortion when reconstruction the data. The Rate–Distortion (RD) function characterizes the minimum achievable rate for a code whose decoding permits a specified amount of distortion. We exploit the connection between rate-distortion theory and entropic optimal transport to propose a novel stochastic-control formulation for the former, and use a classic result dating back to Schrodinger to show that the tradeoff between rate and mean squared error distortion is equivalent to a tradeoff between control energy and the differential entropy of the terminal state, whose probability law defines the reconstruction distribution. For a special class of sources, we show that the optimal control law and the corresponding trajectory in the space of probability measures are obtained by solving a backward heat equation. In more general settings, our approach yields a numerical method that estimates the RD function using diffusion processes with a constant diffusion coefficient. We demonstrate the effectiveness of our method through several examples.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper proposes a stochastic-control formulation for computing rate-distortion functions by connecting rate-distortion theory to entropic optimal transport. It resides in the 'Diffusion-Based and Optimal Transport Methods' leaf under 'Practical Coding Schemes and Approximations'. Notably, this leaf contains only the original paper itself—no sibling papers are present. This isolation suggests the approach represents a relatively unexplored direction within the broader taxonomy of 50 papers spanning approximately 36 topics, indicating a sparse research area for diffusion-based computational methods in classical rate-distortion theory.

The taxonomy tree reveals that neighboring leaves focus on traditional quantization techniques: 'Vector Quantization and Lattice Codebooks' (2 papers), 'Finite Reproduction Alphabet Encoding' (2 papers), and 'Overcomplete Representations and Consistency' (1 paper). These directions emphasize discrete codebook design and high-resolution approximations, contrasting sharply with the continuous diffusion-process framework proposed here. The parent category 'Practical Coding Schemes and Approximations' excludes pure theoretical characterizations, positioning this work as a computational tool rather than an analytical derivation. The broader field shows substantial activity in Gaussian source theory (9 papers across multiple leaves) and perception-augmented frameworks (5 papers), but minimal exploration of optimal-transport-based computation.

Among 22 candidates examined across three contributions, no refutable prior work was identified. The terminal-entropy control formulation examined 2 candidates with 0 refutations; the backward heat equation characterization examined 10 candidates with 0 refutations; and the R2D2 neural estimation method examined 10 candidates with 0 refutations. This limited search scope—22 papers from semantic search and citation expansion—suggests the contributions appear novel within the examined literature. However, the absence of refutations reflects the search scale rather than exhaustive coverage. The backward heat equation and neural estimation components, each scrutinized against 10 candidates, show no substantial overlapping prior work in the retrieved set.

Given the sparse taxonomy position and zero refutations across 22 examined candidates, the work appears to introduce a genuinely distinct computational perspective. The diffusion-based approach diverges from the field's dominant quantization and analytical traditions. However, the limited search scope—particularly the absence of sibling papers and the modest candidate pool—means this assessment is provisional. A broader literature review might uncover related optimal-transport or stochastic-control formulations in adjacent communities not captured by the current semantic search.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
22
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Computing rate-distortion functions for continuous sources under mean squared error distortion. The field is organized around several complementary perspectives. Gaussian Source Rate-Distortion Theory establishes foundational results for Gaussian models, while Rate-Distortion with Side Information and Output Constraints explores scenarios where encoders or decoders have access to auxiliary information or must satisfy additional output requirements, as seen in works like WynerZiv Multivariate Gaussian[7] and OutputConstrained Lossy Coding[10]. Rate-Distortion-Perception Frameworks introduce perceptual quality metrics beyond classical distortion, exemplified by RateDistortionPerception Tradeoff[2] and Gaussian RateDistortionPerception Computation[6]. Specialized Source Models and Distortion Measures address non-Gaussian or structured sources such as AlphaStable RateDistortion[5] and Uniform Sphere RateDistortion[12], while Sampling and Quantization for Continuous Sources investigates the interplay between sampling rates and quantization, including SubNyquist Gaussian Distortion[26]. Practical Coding Schemes and Approximations develop implementable algorithms, and Application-Specific Rate-Distortion Problems tackle domain-driven challenges like Wavefield Coding Geophone[9]. Theoretical Foundations and Extensions provide broader mathematical underpinnings. Recent activity highlights a tension between classical analytical methods and modern computational approaches. Many studies continue refining Gaussian models and side-information settings, such as Indirect NRDF GaussMarkov[3] and Copula Constrained RateDistortion[1], while others explore perception-aware formulations like Gaussian Process RateDistortionPerception[15]. Within Practical Coding Schemes and Approximations, Optimal Diffusion Quadratic[0] leverages diffusion models and optimal transport to compute rate-distortion functions, representing a shift toward data-driven and generative techniques. This contrasts with earlier quantization-focused methods like Lattice Vector Quantization[20] and error-feedback schemes. Optimal Diffusion Quadratic[0] sits alongside emerging computational tools that bridge classical information theory with modern machine learning, offering a fresh lens on long-standing problems while maintaining rigorous connections to mean squared error criteria.

Claimed Contributions

Terminal-Entropy Control formulation equivalent to rate-distortion

The authors introduce a stochastic control problem called Terminal-Entropy Control (TEC) that trades off control energy against the differential entropy of the terminal state. They prove this formulation is equivalent to the rate-distortion problem under mean squared error distortion.

2 retrieved papers
Characterization of optimal solution via backward heat equation

Under suitable regularity conditions, the authors show that the optimal control law and trajectory in probability space are obtained by solving a backward heat equation, providing an analytical characterization of the solution.

10 retrieved papers
R2D2 neural estimation method using diffusion processes

The authors develop R2D2, a neural network-based algorithm that estimates the rate-distortion function and reconstruction distributions by modeling the control function with a deep neural network and using diffusion processes with constant diffusion coefficient.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Terminal-Entropy Control formulation equivalent to rate-distortion

The authors introduce a stochastic control problem called Terminal-Entropy Control (TEC) that trades off control energy against the differential entropy of the terminal state. They prove this formulation is equivalent to the rate-distortion problem under mean squared error distortion.

Contribution

Characterization of optimal solution via backward heat equation

Under suitable regularity conditions, the authors show that the optimal control law and trajectory in probability space are obtained by solving a backward heat equation, providing an analytical characterization of the solution.

Contribution

R2D2 neural estimation method using diffusion processes

The authors develop R2D2, a neural network-based algorithm that estimates the rate-distortion function and reconstruction distributions by modeling the control function with a deep neural network and using diffusion processes with constant diffusion coefficient.

Table of Contents

An Optimal Diffusion Approach to Quadratic Rate-Distortion Problems: New Solution and Approximation Methods | Novelty Validation