Algorithmic information theory studies the complexity of data strings. Complex strings are difficult to compress. While compression depends on the codec, Turing-complete languages can translate between each other. For large strings, the translation overhead becomes negligible. Algorithmic complexity often labels random noise as complex, which contrasts with the understanding of complexity in complex systems. Information entropy also measures complexity, but it too assigns high complexity to randomness. Information fluctuation complexity, however, avoids this issue by focusing on entropy fluctuations, proving useful in applications. Machine learning (ML) research explores how data complexity impacts supervised classification. Ho and Basu proposed complexity measures for binary classification (Figure 4.1), focusing on:
Geometric, topological, and density properties of data manifolds.
Instance hardness, which generalizes to non-binary problems, assesses classification difficulty.
4.1 Algorithmic Information Theory
Algorithmic information theory focuses on several key aspects, including Kolmogorov complexity, algorithmic mutual information, their relationships to entropy and Shannon mutual information, the algorithmic minimal sufficient statistic, ...
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