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

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

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$.$