Trace Quotient Meets Sparsity: A Method for Learning Low Dimensional Image Representations

This paper presents an algorithm that allows to learn low dimensional representations of images in an unsupervised manner. The core idea is to combine two criteria that play important roles in unsupervised representation learning, namely sparsity and trace quotient. The former is known to be a convenient tool to identify underlying factors, and the latter is known as a disentanglement of underlying discriminative factors. In this work, we develop a generic cost function for learning jointly a sparsifying dictionary and a dimensionality reduction transformation. It leads to several counterparts of classic low dimensional representation methods, such as Principal Component Analysis, Local Linear Embedding, and Laplacian Eigenmap. Our proposed optimisation algorithm leverages the efficiency of geometric optimisation on Riemannian manifolds and a closed form solution to the elastic net problem.