COMP9444/9844 Neural Networks

Feedback Quiz on Hopfield Nets and BSB - Solutions

It is best not read the answers until you've tried to answer the questions yourself.

1. Name three types of attractor in a dynamical system.

   Answer: point attractor, limit cycle, chaotic attractor

2. What is a separatrix?

   Answer: the boundary separating one basis of attraction from another.

3. What is the relationship between the sigmoid function 1/(1+exp(–x)) and the sigmoid function tanh(x/2)?

   Answer: 1/(1 + exp(–x) = (tanh(x/2) + 1)/2

4. What is the energy function of the discrete Hopfield model?

   Answer: E = ½ Σ_i,jw_jix_jx_i

5. How do we know that the discrete Hopfield model will always settle into a fixed point from any initial condition?

   Answer: The discrete Hopfield model is the limiting case of the continuous Hopfield model, for which this property can be proven mathematically.

6. How do you store a set of fundamental memories ξ_μ in a Hopfield net?

   Answer: Form a weight matrix by summing the outer products of each ξ_μ with itself, then zero the diagonal of the weight matrix, and divide by the dimension of the vectors.

7. If y is a state in a Hopfield network, what does y = sgn(Wy + b) signify?

   Answer: that the state y is stable.

8. What is a spurious stable state of a Hopfield net?

   Answer: a stable state which is not designed into the system.

9. What is the squashing function φ for the BSB model?

   Answer:

   φ(z) = 1 if z > 1
   φ(z) = z if –1 ≤ z ≤ 1
   φ(z) = –1 if z < –1

10. What are the two equations that define the state udpate for the BSB model?

    Answer:

    y(n) = x(n) + βWx(n)
    x(n+1) = &phi(y(n))

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