Minimally Redundant Laplacian Eigenmaps

Spectral algorithms for learning low-dimensional data manifolds have largely been supplanted by deep learning methods in recent years. One reason is that classic spectral manifold learning methods often learn collapsed embeddings that do not fill the embedding space. We show that this is a natural consequence of data where different latent dimensions have dramatically different scaling in observation space. We present a simple extension of Laplacian Eigenmaps to fix this problem based on choosing embedding vectors which are both orthogonal and \textit{minimally redundant} to other dimensions of the embedding. In experiments on NORB and similarity-transformed faces we show that Minimally Redundant Laplacian Eigenmap (MR-LEM) significantly improves the quality of embedding vectors over Laplacian Eigenmaps, accurately recovers the latent topology of the data, and discovers many disentangled factors of variation of comparable quality to state-of-the-art deep learning methods.