Rate-distortion, Quantization, and Decoding Unite under a Variational Framework with Legendre Duality

The fundamental limits of data compression, the efficient representation of continuous information, and reliable communication are long-studied problems in information theory, yet a unifying mathematical framework has remained elusive. Bruno Macchiavello from University of Brasília, alongside colleagues, now demonstrates striking connections between these traditionally separate fields, revealing a shared underlying structure. The team establishes a novel variational approach to rate-distortion theory, showing that optimal compression strategies align with well-known principles from statistical physics, and extends this to distributed data compression using lattice quantization. Crucially, this work formalises decoding processes, such as those used in modern error-correcting codes, as forms of statistical inference, thereby unifying compression, quantization, and decoding as convex projections, and opening avenues for advancements in neural compression and beyond.

Algorithms for Information, Game, and Optimisation Theory

This collection of code implements algorithms and simulations exploring information theory, game theory, and optimisation, designed for educational and research purposes. The code relies on numpy for numerical calculations and matplotlib for data visualisation, and is structured with modularity in mind, making it relatively easy to understand and modify. Each file focuses on a specific algorithm, and the code is well-commented, explaining the purpose of each section and the logic behind the algorithms. The nash_equilibrium_check. py file finds pure-strategy Nash equilibria in a two-player game by checking for mutual best responses, demonstrating the concept of Nash equilibrium in game theory.

The bp_ldpc. py file implements the Belief Propagation algorithm for decoding Low-Density Parity-Check codes, a technique widely used in modern communication systems, plotting message changes to show the algorithm’s progress. The reverse_water_filling. py file implements the reverse water-filling algorithm, which allocates power or rates to different components of a Gaussian source under a total rate constraint, finding applications in power allocation in wireless communication and rate allocation in data compression.

Lattice Quantization, Rate-Distortion, and Legendre Duality

This research presents a unified mathematical framework connecting rate-distortion theory, lattice quantization, and modern coding techniques, emphasising their shared variational and convex-analytic structure. Scientists established a Gibbs-type variational formulation of the rate-distortion function, demonstrating that optimal test channels form an exponential family where the Kullback-Leibler divergence functions as a Bregman divergence, yielding a generalized Pythagorean theorem for projections and a Legendre duality that couples distortion constraints with inverse temperature parameters. The team extended the reverse water-filling metaphor to distributed lattice quantization, deriving distortion allocation bounds across the eigenmodes of conditional covariance matrices, allowing for precise control over the distribution of distortion. Experiments confirm that these bounds provide a robust framework for allocating resources in distributed coding scenarios, maximising compression efficiency while maintaining acceptable fidelity.

Researchers formalised inference as decoding, demonstrating that belief propagation in LDPC ensembles and polarization in polar codes can be interpreted as recursive variational inference procedures. Data shows these techniques, commonly used in error correction, are fundamentally linked to the optimisation principles governing information compression. The team achieved this by demonstrating that both methods represent convex projections of continuous information onto discrete manifolds, unifying compression, quantization, and decoding within a single mathematical framework. This research delivers a powerful synthesis of classical results, supported by reproducible numerical scripts, ensuring transparency and accessibility for verification and extension to modern contexts like neural compression and quantum information theory.

Information Theory, Physics, and Variational Principles

This work synthesises classical results from information theory, including Shannon’s rate-distortion formulation, Berger’s Gibbs characterisation, and the Nyquist-Shannon sampling theorem, into a unified framework grounded in variational optimisation and resource allocation. By reinterpreting established concepts, such as the Gibbs test channel as free energy minimisation and reverse water-filling as equilibrium across eigenmodes, the researchers highlight deep structural analogies connecting information theory to physics, economics, and engineering. Beyond clarifying historical foundations, the framework demonstrates how these classical results can be understood through modern interdisciplinary lenses, offering a cohesive perspective on compression, quantization, and sampling. The provision of reproducible numerical scripts ensures transparency and accessibility, facilitating verification, extension, and adaptation of these ideas to contemporary contexts, including neural compression, adversarial learning, and quantum information. The impact of this work lies in both consolidating the theoretical basis of rate-distortion, quantization, and sampling, and in opening new pathways for interdisciplinary exploration by revealing common principles across diverse domains. The authors acknowledge that future research could extend this unified perspective to emerging areas of information theory and computation, particularly by formalising connections between imitation, compression, and cognition, and incorporating richer models of context and interaction.