Exact and Soft Successive Refinement of the Information Bottleneck

Area of Science:

• Information Theory
• Machine Learning
• Computational Neuroscience

Background:

• The Information Bottleneck (IB) framework balances representation complexity with information extraction.
• Real-world systems require adaptable representations with varying complexity over time.
• The cost of updating representations is crucial for efficient information processing.

Purpose of the Study:

• To investigate how adaptive systems can reuse existing IB-optimal representations for new ones at different granularities.
• To explore the information-theoretic limits of adapting IB-optimal representations.
• To extend the concept of successive refinement within the IB framework.

Main Methods:

• Studied and extended the notion of successive refinement within the IB framework.
• Developed a new geometric characterization for analytical derivations.
• Provided a linear-programming-based tool for numerical investigation in discrete cases.
• Quantified information optimality loss using a measure of unique information.

Main Results:

• Analytically derived successive refinability for specific IB problems (binary, jointly Gaussian, deterministic functions).
• Developed a numerical tool for investigating successive refinement of IB in discrete settings.
• Quantified information optimality loss in multi-stage processing, finding it typically low but non-negligible.

Conclusions:

• Adaptive systems can effectively leverage existing IB-optimal representations for new ones.
• Successive refinement offers a theoretical limit for adapting representation granularity with minimal information loss.
• Results have implications for incremental learning, statistical decision problems, and deep neural network theory.