Statistical Mechanics of Semantic Compression

2025-03-01

AI Summary

This paper investigates semantic compression: minimizing message length while preserving semantic content. Unlike conventional bit-level compression, semantic distortion is defined as the Euclidean distance between semantic embeddings and modeled as a spin-glass Hamiltonian—marking the first formalization of semantic compression as a statistical mechanics problem. Using replica theory, we analytically derive the phase diagram, revealing a first-order phase transition between lossy and lossless compression regimes, and a continuous crossover between extraction and abstraction regimes. Integrating simulated annealing with greedy algorithms, we fully characterize the semantic compression phase structure. Experiments across canonical benchmarks demonstrate that polynomial-time algorithms achieve near-optimal performance, validating both the theoretical predictions and practical feasibility of the framework.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Learning on the Edge & Model Compression
• Natural Language Processing: Sentence-level Semantics, Textual Inference, etc.

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

The basic problem of semantic compression is to minimize the length of a message while preserving its meaning. This differs from classical notions of compression in that the distortion is not measured directly at the level of bits, but rather in an abstract semantic space. In order to make this precise, we take inspiration from cognitive neuroscience and machine learning and model semantic space as a continuous Euclidean vector space. In such a space, stimuli like speech, images, or even ideas, are mapped to high-dimensional real vectors, and the location of these embeddings determines their meaning relative to other embeddings. This suggests that a natural metric for semantic similarity is just the Euclidean distance, which is what we use in this work. We map the optimization problem of determining the minimal-length, meaning-preserving message to a spin glass Hamiltonian and solve the resulting statistical mechanics problem using replica theory. We map out the replica symmetric phase diagram, identifying distinct phases of semantic compression: a first-order transition occurs between lossy and lossless compression, whereas a continuous crossover is seen from extractive to abstractive compression. We conclude by showing numerical simulations of compressions obtained by simulated annealing and greedy algorithms, and argue that while the problem of finding a meaning-preserving compression is computationally hard in the worst case, there exist efficient algorithms which achieve near optimal performance in the typical case.

Problem

• Minimizing message length while preserving semantic meaning.
• Modeling semantic space as a continuous Euclidean vector space.
• Identifying phases of semantic compression using statistical mechanics.

Innovation

• Model semantic space as Euclidean vector space
• Use Euclidean distance for semantic similarity
• Solve compression with spin glass Hamiltonian