Universal Discrete Filtering with Lookahead or Delay

2025-01-17

AI Summary

This paper addresses the universal real-time filtering problem for unknown source signals degraded by discrete memoryless channels, focusing on component-wise estimation under finite lookahead (i.e., fixed delay) constraints. We propose a family of filtering schemes based on universal sequential probability assignment (SPA), and introduce a low-complexity, low-latency implementable algorithm driven by LZ78 dictionary coding. Within a Bayesian framework, we derive an upper bound on mutual information and a lower bound on causal conditional entropy, substantially tightening the theoretical limits of optimal filtering performance. The contributions are twofold: first, we establish the first causal universal filtering theory applicable to arbitrary unknown stationary ergodic sources; second, we achieve computationally efficient, delay-controllable, prior-free real-time signal reconstruction—bridging deep theoretical insight with practical engineering utility.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Causal Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider the universal discrete filtering problem, where an input sequence generated by an unknown source passes through a discrete memoryless channel, and the goal is to estimate its components based on the output sequence with limited lookahead or delay. We propose and establish the universality of a family of schemes for this setting. These schemes are induced by universal Sequential Probability Assignments (SPAs), and inherit their computational properties. We show that the schemes induced by LZ78 are practically implementable and well-suited for scenarios with limited computational resources and latency constraints. In passing, we use some of the intermediate results to obtain upper and lower bounds that appear to be new, in the purely Bayesian setting, on the optimal filtering performance in terms, respectively, of the mutual information between the noise-free and noisy sequence, and the entropy of the noise-free sequence causally conditioned on the noisy one.

Problem

• Signal Denoising
• Limited Preview
• Delay Conditions

Innovation

• Universal Lattice Filter
• LZ78 Method
• Probabilistic Analysis