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Suppose we are given multi-class labels $\mathbf{y} \in \mathbb{R}^K$ where $K \in \mathbb{N^+}$ and data $\mathbf{X} \in \mathbb{R}^{P \times K}$ where $P \in \mathbb{N^+}$. We wish to minimise the squared error between the data and the labels to find the best linear relationship between the inputs and the labels. However, this is not the standard linear multi-class linear regression as we are restricted to a single coefficient per input dimension (i.e. $[w_1, \ldots, w_P ] = \mathbf{w}$). Specifically we wish to solve:

$$\hat{\mathbf{w}} = \mathop{\arg \min}\limits_\mathbf{w} \mathbb{E}[ \frac{1}{K}\sum_{k=1}^K ( y_k - \sum_{j=1}^{P} w_{j} x_{j,k} ) ^2]. $$

What is the solution to this problem?

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