https://r2.erweima.ai/mathbot/image/1718192604805p74wjozu.png

The equations in the image can be broken down as follows:

1. **Equation for \([U \; V]^T\):**

   \[
   \begin{bmatrix}
   U \\
   V
   \end{bmatrix} = f(X, M_1, \ldots, M_R) = \sigma \left( \begin{bmatrix}
   H_u \\
   H_v
   \end{bmatrix} W^T \right)
   \]

2. **Definition of \([H_u \; H_v]^T\):**

   \[
   \begin{bmatrix}
   H_u \\
   H_v
   \end{bmatrix} = \sigma \left( \left( \sum_{r=1}^{R} D^{-1} \begin{pmatrix}
   0 & M_r \\
   M_r^T & 0
   \end{pmatrix} \right) X W_r^T \right)
   \]

### Explanation:

1. **Equation for \([U \; V]^T\):**
   - The matrix \([U \; V]^T\) is computed using a function \(f\) that takes inputs \(X\) and a series of matrices \(M_1\) to \(M_R\).
   - This function \(f\) involves applying a non-linear activation function \(\sigma\) to the result of a matrix multiplication.

2. **Definition of \([H_u \; H_v]^T\):**
   - The components \(H_u\) and \(H_v\) are defined using a summation involving the matrices \(M_r\), a diagonal matrix \(D\), and the input \(X\).
   - The matrix inside the summation is a block matrix with \(M_r\) and its transpose \(M_r^T\).
   - The summation term is multiplied by \(X\) and a weight matrix \(W_r^T\).
   - Finally, the result is passed through the activation function \(\sigma\).

If you have specific questions about any part of these equations or need further elaboration, please let me know.

Would you like more details or have any questions? Here are some questions you might consider:

1. Can you explain the function \(f\) in more detail?
2. What does the activation function \(\sigma\) represent in this context?
3. How does the summation over \(r\) affect the calculation of \(H_u\) and \(H_v\)?
4. What is the role of the diagonal matrix \(D^{-1}\)?
5. How do the weight matrices \(W\) and \(W_r\) influence the output?

**Tip:** When dealing with complex equations, breaking them down into smaller parts can help in understanding each component's role.

Explore the breakdown of equations and definitions in neural network architecture, including concepts such as linear algebra, activation functions, and neural networks. Dive deep into understanding how matrices and summations contribute to the computation of network outputs. Suitable for advanced learners and researchers.

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