An Information Theoretic Condition for Perfect Reconstruction.

Area of Science:

• Information Theory
• Probability Theory
• Discrete Mathematics

Background:

• Reconstructing a discrete random variable (X) from functions of X is a fundamental problem.
• Existing methods lack a unified theoretical framework, often relying on dispersed knowledge.
• Shannon's 1953 lattice theory provides a foundational, yet underutilized, approach.

Purpose of the Study:

• To present a novel information theoretic condition for reconstructing a discrete random variable (X).
• To synthesize and clarify concepts from Shannon's lattice theory, including entropic metrics.
• To establish a geometric interpretation for a necessary (and sometimes sufficient) reconstruction condition.

Main Methods:

• Derivation of reconstruction conditions from Shannon's lattice theory and entropic metrics (Shannon, Rajski).
• Synthetic description and proof of concepts: total, common, and complementary information.
• Geometric interpretation of the lattice structure to define reconstruction criteria.

Main Results:

• A new information theoretic condition for reconstructing X from a set of its functions {X1,...,Xn}.
• Demonstration that X is reconstructable if its functions are sufficiently dependent on X (not too far in entropic distance).
• Illustration of the condition with five diverse examples: random variable reconstruction, word reconstruction, integer reconstruction (via prime signature and Chinese remainder theorem), and permutation reconstruction.

Conclusions:

• The derived condition provides a necessary, and sometimes sufficient, criterion for perfect reconstruction.
• Geometric insights into the lattice structure offer a foundation for a novel 'perfect reconstruction theory'.
• The study highlights the importance of sufficient dependence between the source variable and its observed functions for successful reconstruction.