If linear activations are used, or only a single sigmoid hidden layer, then the optimal solution to an autoencoder is strongly related to principal component analysis (PCA). The weights of an autoencoder with a single hidden layer of size formula_79 (where formula_79 is less than the size of the input) span the same vector subspace as the one spanned by the first formula_79 principal components, and the output of the autoencoder is an orthogonal projection onto this subspace. The autoencoder weights are not equal to the principal components, and are generally not orthogonal, yet the principal components may be recovered from them using the singular value decomposition.