Backpropagation for autoencoders. In an autoencoder, we seek to recon$-$ struct the original data after some operation that reduces the data$$s dimensionality. We may be interested in reducing the data$$s dimensionality to gain a more compact representation of the data.

For example, consider x in Rn$.$ Further, consider W in Rm$\backslash$times n where m $<$ n$.$ Then Wx is of lower dimensionality than x$.$ One way to design W so that Wx still contains key features of x is to minimize the following expression