Transform the squared error loss function to matrix form I have n data, $x_i$ are the inputs, and $y_i$ are the labels. $x_i$ is $d$-dimension, and $y_i$ is $p$-dimension. We have a prediction function that is $f(x) = W^Tx$. $W$ is a matrix of dimension $d \times p$.
The squared error loss function of the prediction function is:
$$
  J(W) = \sum_{i=1}^n||y_i - W^Tx_i||_2^2
$$
I would like to write $J(W)$ in matrix form. How should I convert it to an equation involving $Y$ ($n \times p$ dimension), $W$ ($d \times p$ dimension) and $X$ ($n \times d$ dimension). Hence the loss function will be in matrix form instead of a sum?
 A: Given the set of data $D_{m}:=\left\{\left(x_{i}, y_{i}\right) \mid i=1,2, \cdots, m\right\}$ such that $x_{i} \neq x_{j}$ for $i \neq j .$ we must find the line $y=\alpha x+\beta$ that best
fits the data, i.e. find $(\alpha, \beta)$ such that for all $(a, b) \in \mathbb{R}^{2}$ we have
\begin{equation}
0 \leq \Phi(\alpha, \beta)=\sum_{i=1}^{m}\left(y_{i}-\alpha x_{i}-\beta\right)^{2} \leq \sum_{i=1}^{m}\left(y_{i}-a x_{i}-b\right)^{2}=\Phi(a, b)
\end{equation}
in matrix form, $\Phi(\alpha, \beta)=\Phi(z)=\|y-A z\|_{2}^{2}$, thus inequality $(1)$ is equivalent to the least square problem:
$$
\Phi(z)=\|y-A z\|_{2}^{2}=\min_{w \in \mathbb{R}^{n}}\|y-A w\|_{2}^{2}=\min_{w \in \mathbb{R}^{n}} \Phi(w)
$$
Since $y_{i}-a x_{i}-b=(y-A w)_{i}$ where : $w=[a ; b], z=[\alpha ; \beta], \mathbf{y}=\begin{bmatrix}y_{1} \\ y_{2} \\ y_{3} \\ \vdots \\ y_{m}\end{bmatrix}, \mathbf{A}=\begin{bmatrix}x_{1} & 1 \\ x_{2} & 1 \\ x_{3} & 1 \\ \vdots & \vdots \\ x_{m} & 1\end{bmatrix}, \mathbf{w}=\begin{bmatrix}a \\ b\end{bmatrix},$ and $\mathbf{z}=\begin{bmatrix}\alpha \\ \beta\end{bmatrix}$.
A: With $X = \begin{bmatrix}x_1 & \dots & x_n\end{bmatrix}$ and $Y = \begin{bmatrix}y_1 & \dots & y_n\end{bmatrix}$, you simply obtain:
$$j(W) = ||Y-W^TX||_{2,1},$$
where the $L_{2,1}$ norm is the sum of the Euclidean norms of the columns of the matrix $Y-W^TX$.
Note however the dimensions: $X \in \mathbb{R}^{d \times n}$ and $Y \in \mathbb{R}^{p \times n}$.
