Feedback Nerual Network(二）：Stability and Attractor

Definition

 Attractor: If the state x of NN satisfies          then x is called as the Stable Point or Attractor of the NN

Definition of Energy Function: Energy Function is a function of neuron’s Status & Link-Weight,which as last should be stable.

Synchronous mode and Asynchronous mode is different.

 

Theorem3.1: Adjust the state of Hopfield Network by using Asynchronous mode and assume the Link-
Weight Matrix W is Symmetric, then for any Initial State, the NN will eventually converge to an Attractor.

 

Proof:

 

Theorem3.2: Adjust the state of DHNN by using Synchronous mode and assume the Link-weight
Matrix W is non-negative defined Symmetric, then for any Initial State, the NN will eventually converge to an Attractor.

 

Proof:

 

 

Compare Asynchronous Mode & Synchronous Mode
1 For Synchronous Mode, from Theorem3.2 one can see, if Link-Weight Matrix W is not non-negative defined
Symmetric matrix, then Hopfield could enter the self-oscillation, i.g the ultimate-loop极限环 could occurr.

2 Synchronous mode is faster than Asynchronous Mode.
3 For Asynchronous Mode, from Theorem3.1 one can see, requirement of W is not so strict, it has better stability than Synchronous.
4 Practically using Asynchronous Mode is more often. His main disadvantage is that lost the advantages of Concurrent Treatment of NN.

 

Characteristics of Attractor:

 

 

The Region of Attractor:
Definition of Absorbed Field: In order to implement the correct Associative Memorization, there must be
an Attractive Region. This Attractive Region is so called Absorbed Field Definition3.2 Weak/Strong Attract:
Suppose x(a) is an Attractor, for Asynchronous mode:
If there is an adjustment sequence so that from x varies to x(a), then call this as: x weakly attract to x(a).
If for any adjustment sequence so that from x varies to x(a), then call this as: x strongly attract to x(a).