Skip to content Skip to navigation

Connexions

You are here: Home » Content » Kolmogorov Entropy

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the authors

Recently Viewed

This feature requires Javascript to be enabled.

Kolmogorov Entropy

Module by: Ronald DeVore, Shriram Sarvotham

Figure 1: Coverings of KK by balls of radius ϵϵ.
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.

Comments, questions, feedback, criticisms?

Send feedback