$[3$ pts$]$ In Skip$-$Gram, show that $F=$exp$(x)$ satisfies the following:

$F((w_i-w_i)widetilde(w{)}_k)=\frac{F(w_i\text{w}\text{i}\text{d}\text{e}\text{t}\text{i}\text{l}\text{d}\text{e}(w{)}_k)}{(vec())(w_j\text{w}\text{i}\text{d}\text{e}\text{t}\text{i}\text{l}\text{d}\text{e}(w{)}_k)}$

$[3$ pts$]$ If we remove $f(x_{ij})$ from the following formula, what problems do we have?

${\displaystyle \underset{i}{\overset{?}{}}},$iinDf$(x_{ij})(w_i^T$widetilde$(w{)}_j-log{x}_{ij}+(widetilde(b_j))+b_i{)}^2$

Autoencoder can reduce input dimensions.

$[3$ pts$]$ In Sparse autoencoder, the latent space dimension is larger than the input dimension. Explain

how a sparse autoencoder can reduce the input dimension.

$[5$ pts$]$ What happens if a sparse autoencoder uses L$2($Ridge$)$ regularization? Explain it$.$