NEAR-ISOMETRIC LINEAR EMBEDDINGS OF MANIFOLDS

We propose a new method for linear dimensionality reduction of manifold-modeled data. Given a training set chi of Q points belonging to a manifold M subset of R-N, we construct a linear operator P : R-N -> R-M that approximately preserves the norms of all ((Q)(2)) pairwise difference vectors (or secants) of chi. We design the matrix P via a trace-norm minimization that can be efficiently solved as a semi-definite program (SDP). When chi comprises a sufficiently dense sampling of M, we prove that the optimal matrix P preserves all pairs of secants over M. We numerically demonstrate the considerable gains using our SDP-based approach over existing linear dimensionality reduction methods, such as principal components analysis (PCA) and random projections.