K-Deep Simplex: Deep Manifold Learning via Local Dictionaries

We propose K-Deep Simplex (KDS), a unified optimization framework for nonlinear dimensionality reduction that combines the strengths of manifold learning and sparse dictionary learning. Our approach learns local dictionaries that represent a data point with reconstruction coefficients supported on the probability simplex. The dictionaries are learned using algorithm unrolling, an increasingly popular technique for structured deep learning. KDS enjoys tremendous computational advantages over related approaches and is both interpretable and flexible. In particular, KDS is quasilinear in the number of data points with scaling that depends on intrinsic geometric properties of the data. We apply KDS to the unsupervised clustering problem and prove theoretical performance guarantees. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.