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Kolmogorov Entropy

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

Figure 1: Coverings of KK by balls of radius ϵϵ.
Figure 1 (figure4.png)

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,

K⊆U i=1 N b(f i ,ϵ).K⊆U 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 K⊂XK⊂X, 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 f∈Kf∈K. 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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