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The Geometry Compression Discovers

What compression reveals about the shape of information

When a compression algorithm finds patterns, what has it discovered? LZ77, Huffman, arithmetic coding, and PPM each make implicit assumptions about where structure lives — in repetition, in frequency, in context. The compressibility of a sequence is not just about its entropy. It's about what geometry the algorithm can see.

The Geometry of Repetition

LZ77 is the simplest compression algorithm that actually works. It uses a sliding window to encode repeated substrings as back-references: "copy 12 bytes from 47 bytes back." This reveals a geometry: the algorithm assumes structure is local. If a pattern appeared recently, it might appear again. The sliding window enforces a distance horizon — structure beyond the window is invisible, even if it repeats perfectly.

This locality assumption encodes a belief about data: the near past predicts the near future. LZ77 discovers nothing that isn't within reach. But within that reach, it finds structure efficiently: a repeated phrase becomes a pointer, a copy command, a back-reference. The compression ratio is a measure of how much local repetition exists in the input.

The geometry here is spatial proximity in time. Patterns that cluster together compress well. Patterns that are scattered — appearing once, then again far later — don't compress at all under LZ77. The algorithm's window is its world.

Frequency and the Independence Assumption

Huffman coding encodes each symbol independently, using shorter codes for more frequent symbols. This assumes that symbol frequencies are stable and that symbols don't influence each other — that 'e' after 'th' is statistically the same 'e' as anywhere else.

This is almost never true. The frequency of 'e' after 'q' is near zero. The frequency of 'u' after 'q' is near one. But Huffman can't see these dependencies. It discovers a simpler geometry: the marginal distribution. Each symbol lives in its own frequency bin, and Huffman builds an optimal code for that distribution — optimal if symbols are independent.

What Huffman finds is real: frequent symbols really do benefit from short codes. What it misses is also real: context matters. The letter 'x' is rare overall, but 'x' after 'e' in English text is almost always followed by 't' or 'c' or the end of a word. Huffman can't capture this because its geometry has only one dimension: frequency.

Context and the Prediction Game

Arithmetic coding and PPM improve on Huffman by separating the modeling from the coding. The model predicts the next symbol's distribution; the coder encodes that symbol efficiently given the prediction. This separation reveals a deeper truth: compression is really about prediction.

PPM (Prediction by Partial Matching) goes further. It maintains context models of different depths: what follows 'a', what follows 'th', what follows 'the'. When predicting the next character, it tries the longest context first, falling back to shorter contexts if the longer one has insufficient data. This learns that 'the' is almost always followed by a space, that 'th' is usually followed by 'e', that 't' is followed by many possibilities.

The geometry PPM discovers is contextual. Symbols are not independent. They depend on what came before. PPM learns these dependencies from data, building up a hierarchy of contexts: order-0 (no context), order-1 (one previous symbol), order-2 (two previous), and so on. The model encodes the discovered structure: after this context, these symbols become more likely.

Arithmetic coding, combined with such a model, can encode each symbol using nearly the theoretical minimum bits: close to the entropy of the symbol given its context. This is the entropy that actually matters: conditional entropy, not marginal entropy. The gap between unconditional and conditional entropy measures how much structure exists in the dependencies between symbols.

Compressibility vs Entropy

Shannon defined entropy as the theoretical limit on compression: the minimum bits needed to encode a symbol from a distribution. But there's a catch. This bound assumes the distribution is known. In practice, you have to learn the distribution from data — or encode the model itself alongside the data.

A sequence of perfectly random bits has maximal entropy: no compression algorithm can shrink it. But a sequence that looks random but has hidden structure — a pseudo-random number generator's output, or a sequence generated by a simple cellular automaton — also has high entropy unless you know the generator.

This is where compression reveals something profound. The compressibility of a sequence is a measurement: not of the sequence itself, but of the match between the sequence's structure and the algorithm's assumptions. LZ77 compresses repetitive sequences. Hufman compresses skewed frequency distributions. PPM compresses context-dependent sequences. A sequence might compress well under one algorithm and poorly under another. Each algorithm measures a different kind of structure.

Kolmogorov Complexity and the Unknowable

There's a deeper limit. The Kolmogorov complexity of a sequence is the length of the shortest program that outputs it. This is compression's ideal: the most compact possible representation. But Kolmogorov complexity is uncomputable. There's no algorithm that can find the shortest program for an arbitrary sequence.

What compression algorithms do is approximate Kolmogorov complexity within a restricted model class. LZ77 assumes the shortest program uses back-references within a sliding window. Huffman assumes symbols are independent with stable frequencies. PPM assumes symbols depend on finite context histories. Each algorithm restricts the "programs" it considers, making compression tractable — but also making it blind to structure its model class cannot represent.

The geometry each algorithm discovers is not the true structure of the data. It's the structure the algorithm can see. A sequence generated by a depth-50 Markov process will compress poorly under a PPM model that only looks back 5 symbols. The structure is there, but the algorithm can't reach it.

What Compression Ratios Reveal

When you compress different kinds of data with the same algorithm, the ratios tell you something. Text compresses well because it has structure: local repetition, skewed frequencies, strong context dependencies. High-entropy random data compresses poorly because it lacks structure the algorithm can exploit. Executable binaries compress moderately: they have some repetition (common instruction sequences) and some high-entropy sections (embedded constants, jump tables).

The compression ratio is not just a number. It's a measurement of how much discoverable structure exists in the data, relative to the algorithm's model class. Switch algorithms and you measure different structure. Use a general-purpose compressor like gzip (LZ77 + Huffman) and you measure structure that locality and frequency can capture. Use a model-based compressor like PPM and you capture contextual structure. Use a neural compressor and you capture patterns in learned representations.

There's no neutral measurement. Every compression tool comes with baked-in assumptions about where structure lives. The geometry it discovers — repetition, frequency, context, representation — is the geometry it was designed to see.

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