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Theoretical Limits of Language Model Alignment $f$-Divergence Regularized RLHF: Two Tales of Sampling and Unified Analyses A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation When Semantic Communication Meets Queueing: Cross-Layer Latency and Task Fidelity Optimization Convexity in Disguise: A Theoretical Framework for Nonconvex Low-Rank Matrix Estimation Conditional Diffusion Under Linear Constraints: Langevin Mixing and Information-Theoretic Guarantees Sharp Capacity Thresholds in Linear Associative Memory: From Winner-Take-All to Listwise Retrieval Expert Routing for Communication-Efficient MoE via Finite Expert Banks Contextual Memory-Enhanced Source Coding for Low-SNR Communications Realizable Bayes-Consistency for General Metric Losses Leveraging Code Automorphisms for Improved Syndrome-Based Neural Decoding A Hierarchical Sampling Framework for bounding the Generalization Error of Federated Learning Dueling DDQN-Based Adaptive Multi-Objective Handover Optimization for LEO Satellite Networks The Causal Description Gap: Information-Theoretic Separations Across Pearl's Hierarchy Optimization of CV-QKD Under Practical Constraints Benchmarking Wireless Representations: High-Dimensional vs. Compressed Embeddings for Efficiency and Robustness Real-Time Text Transmission via LLM-Based Entropy Coding over Fixed-Rate Channels SwiftChannel: Algorithm-Hardware Co-Design for Deep Learning-Based 5G Channel Estimation Evolving Token Communication with Parametric Memory Network Remote Action Generation: Remote Control with Minimal Communication The (Marginal) Value of a Search Ad: An Online Causal Framework for Repeated Second-price Auctions Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation Linear-Readout Floors and Threshold Recovery in Computation in Superposition Soft Graph Diffusion Transformer for MIMO Detection Hierarchical Federated Learning for Networked AI: From Communication Saving to Architecture-Aware Design Exponential families from a single KL identity MIFair: A Mutual-Information Framework for Intersectionality and Multiclass Fairness Diffusion-OAMP for Joint Image Compression and Wireless Transmission Decoupled Descent: Exact Test Error Tracking Via Approximate Message Passing
Set Shaping Theory and the Foundations of Redundancy-Free...
Aida Koch, Alix Petit · 2025-07-04 · via cs.IT updates on arXiv.org

To render a sequence testable, namely capable of identifying and detecting errors, it is necessary to apply a transformation that increases its length by introducing statistical dependence among symbols, as commonly exemplified by the addition of parity bits. However, since the decoder does not have prior knowledge of the original symbols, it must treat the artificially introduced symbols as if they were independent. Consequently, these additional symbols must be transmitted, even though their conditional probability, under ideal and error free conditions, would be zero. This sequence extension implies that not all symbol combinations of the new length are practically realizable: if an error modifies a sequence, making it inadmissible such an error becomes detectable. Recent developments in Set Shaping Theory have revealed a surprising result: it is always possible to transform a sequence into a longer version by carefully selecting which longer sequences are allowed, in such a way that the overall set of sequences becomes more structured and less complex than the original. This means that even though the sequence is extended and dependencies are introduced between symbols, the total amount of information contained in the new set does not increase proportionally on the contrary, it can be slightly reduced. In other words, one can construct a new set of longer sequences where each one corresponds uniquely to an original sequence, but the entire set is designed in such a way that it can be treated as if the symbols were independent, making encoding simpler. This allows sequence to become testable capable of detecting errors without adding visible redundancy or increasing the informational content.