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# Kolmogorov Entropy

Module by: Ronald DeVore, Shriram Sarvotham. E-mail the authors

Given ϵ>0ϵ>0, and the compact set KK, consider all coverings of KK by balls of radius ϵϵ, as shown in Figure 1. In other words,

KU i=1 N b(f i ,ϵ).KU i=1 N b(f i ,ϵ).
(1)
Let N ϵ :=inf{N:overallsuchcovers}N ϵ :=inf{N:overallsuchcovers}. N ϵ (K)N ϵ (K) is called the covering number of KK. Since it depends on XX and KK, we write it as N ϵ =N ϵ (K,X)N ϵ =N ϵ (K,X).

## definition 1: Kolmogorov entropy

The Kolmogorov entropy, denoted by H ϵ (K,X)H ϵ (K,X), of the compact set KK in XX is defined as the logarithm of the covering number:

H ϵ (K,X)=logN ϵ (K,X).H ϵ (K,X)=logN ϵ (K,X).
(2)

The Kolmogorov entropy solves our problem of optimal encoding in the sense of the following theorem.

## Theorem 1

For any compact set KXKX, we have n ϵ (K,X)=H ϵ (K,X)n ϵ (K,X)=H ϵ (K,X), where ·· is the ceiling function.

### Proof

Sketch: We can define an encoder-decoder as follows To encode: Say fKfK. Just specify which ball it is covered by. Because the number of balls is N ϵ (K,X ̲ ¯)N ϵ (K,X ̲ ¯), we need at most logN ϵ (K,X ̲ ¯)logN ϵ (K,X ̲ ¯) bits to specify any such ball ball.

To decode: Just take the center of the ball specified by the bitstream.

It is now easy to see that this encoder-decoder pair is optimal in either of the senses given above.

The above encoder is not practical. However, the Kolmogorov entropy tells us the best performance we can expect from any encoder-decoder pair. Kolmogorov entropy is defined in the deterministic setting. It is the analogue of the Shannon entropy which is defined in a stochastic setting.

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