Math Problem Statement
Find the weight matrix of the Hopfield network which stores two patterns, X1 = [1, 1, 1, -1, -1, -1] and X2 = [1, -1, 1, -1, 1, -1].
Solution
The task is to find the weight matrix of the Hopfield network that stores two patterns:
- X1=[1 1 1 −1 −1 −1]
- X2=[1 −1 1 −1 1 −1]
In a Hopfield network, the weight matrix W is determined by the Hebbian learning rule, which for two patterns is given by:
W=X1TX1+X2TX2
Where:
- X1T and X2T are the transposes of the vectors X1 and X2.
- The elements on the diagonal are set to zero, i.e., wii=0.
Step 1: Compute the weight matrix contributions from each pattern
For each pattern, the weight matrix is the outer product of the vector with itself.
- For X1=[1 1 1 −1 −1 −1]:
\begin{bmatrix}
1 \\
1 \\
1 \\
-1 \\
-1 \\
-1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 1 & -1 & -1 & -1
\end{bmatrix}
=
\begin{bmatrix}
1 & 1 & 1 & -1 & -1 & -1 \\
1 & 1 & 1 & -1 & -1 & -1 \\
1 & 1 & 1 & -1 & -1 & -1 \\
-1 & -1 & -1 & 1 & 1 & 1 \\
-1 & -1 & -1 & 1 & 1 & 1 \\
-1 & -1 & -1 & 1 & 1 & 1
\end{bmatrix}$$
2. **For $$X_2 = [1 \ -1 \ 1 \ -1 \ 1 \ -1]$$:**
$$X_2^T X_2 =
\begin{bmatrix}
1 \\
-1 \\
1 \\
-1 \\
1 \\
-1
\end{bmatrix}
\begin{bmatrix}
1 & -1 & 1 & -1 & 1 & -1
\end{bmatrix}
=
\begin{bmatrix}
1 & -1 & 1 & -1 & 1 & -1 \\
-1 & 1 & -1 & 1 & -1 & 1 \\
1 & -1 & 1 & -1 & 1 & -1 \\
-1 & 1 & -1 & 1 & -1 & 1 \\
1 & -1 & 1 & -1 & 1 & -1 \\
-1 & 1 & -1 & 1 & -1 & 1
\end{bmatrix}$$
### Step 2: Sum the two matrices
$$W = X_1^T X_1 + X_2^T X_2 =
\begin{bmatrix}
2 & 0 & 2 & -2 & 0 & -2 \\
0 & 2 & 0 & -2 & 0 & 2 \\
2 & 0 & 2 & -2 & 0 & -2 \\
-2 & -2 & -2 & 2 & 2 & 2 \\
0 & 0 & 0 & 2 & 2 & 0 \\
-2 & 2 & -2 & 2 & 0 & 2
\end{bmatrix}$$
### Step 3: Set diagonal elements to zero
We zero out the diagonal elements to get the final weight matrix.
$$W =
\begin{bmatrix}
0 & 0 & 2 & -2 & 0 & -2 \\
0 & 0 & 0 & -2 & 0 & 2 \\
2 & 0 & 0 & -2 & 0 & -2 \\
-2 & -2 & -2 & 0 & 2 & 2 \\
0 & 0 & 0 & 2 & 0 & 0 \\
-2 & 2 & -2 & 2 & 0 & 0
\end{bmatrix}$$
### Conclusion:
The correct matrix corresponds to option **(c)**.
Let me know if you need further details or have any questions!
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Here are 5 related questions to expand your understanding:
1. How does the Hebbian learning rule work in Hopfield networks?
2. Why are diagonal elements set to zero in a Hopfield network weight matrix?
3. How does the Hopfield network retrieve stored patterns?
4. Can a Hopfield network store more than two patterns effectively? What are the limitations?
5. How does the energy function of a Hopfield network relate to its convergence properties?
**Tip:** Hopfield networks are recurrent neural networks that operate by minimizing an energy function, similar to the way physical systems settle into states of minimal energy.