  High-dimensional data sets are often analyzed and explored via the
construction of a latent low-dimensional space which enables convenient
visualization and efficient predictive modeling or clustering. For complex data
structures, linear dimensionality reduction techniques like PCA may not be
sufficiently flexible to enable low-dimensional representation. Non-linear
dimension reduction techniques, like kernel PCA and autoencoders, suffer from
loss of interpretability since each latent variable is dependent of all input
dimensions. To address this limitation, we here present path lasso penalized
autoencoders. This structured regularization enhances interpretability by
penalizing each path through the encoder from an input to a latent variable,
thus restricting how many input variables are represented in each latent
dimension. Our algorithm uses a group lasso penalty and non-negative matrix
factorization to construct a sparse, non-linear latent representation. We
compare the path lasso regularized autoencoder to PCA, sparse PCA, autoencoders
and sparse autoencoders on real and simulated data sets. We show that the
algorithm exhibits much lower reconstruction errors than sparse PCA and
parameter-wise lasso regularized autoencoders for low-dimensional
representations. Moreover, path lasso representations provide a more accurate
reconstruction match, i.e. preserved relative distance between objects in the
original and reconstructed spaces.
