What does a subscript F represent?

Question

On page 11 of the slide,

Sum-of-least-square loss:

$$ \ell\left(\mathbf{\tilde W}\right) = \sum_{n=1}^N \left\| \mathbf{\tilde W}^T\mathbf{\tilde x^{(n)}} -\mathbf{t}^{(n}) \right\|^2 = \left\|\mathbf{\tilde X\tilde W-T}\right\|^2_F $$

• the $n$-th row of $\mathbf{\tilde X}$ is $\left[\mathbf{\tilde x}^{(n)}\right]^T$
• the $n$-th row of $\mathbf{T}$ is $\left[\mathbf{t}^{(n)}\right]^T$

I don't get what the subscript $F$ means in the equation.

I don't even know what tags I should put. Any modification would be appreciated!

Answer 1

The subscript $F$ denotes the Frobenius norm: if $A=[a_{ij}]$ in $n\times m$ matrix then $$\|A\|_F=\sqrt{\text{trace}(A^*A)}=\sqrt{\sum_{i=1}^n\sum_{j=1}^m|a_{ij}|^2}.$$