Practical Algorithms for Learning Near-Isometric Linear Embeddings

Abstract: We propose two practical non-convex approaches for learning near-isometric, linear embeddings of finite sets of data points. Given a set of training points $\mathcal{X}$, we consider the secant set $S(\mathcal{X})$ that consists of all pairwise difference vectors of $\mathcal{X}$, normalized to lie on the unit sphere. The problem can be formulated as finding a symmetric and positive semi-definite matrix $\boldsymbol{\Psi}$ that preserves the norms of all the vectors in $S(\mathcal{X})$ up to a distortion parameter $\delta$. Motivated by non-negative matrix factorization, we reformulate our problem into a Frobenius norm minimization problem, which is solved by the Alternating Direction Method of Multipliers (ADMM) and develop an algorithm, FroMax. Another method solves for a projection matrix $\boldsymbol{\Psi}$ by minimizing the restricted isometry property (RIP) directly over the set of symmetric, postive semi-definite matrices. Applying ADMM and a Moreau decomposition on a proximal mapping, we develop another algorithm, NILE-Pro, for dimensionality reduction. FroMax is shown to converge faster for smaller $\delta$ while NILE-Pro converges faster for larger $\delta$. Both non-convex approaches are then empirically demonstrated to be more computationally efficient than prior convex approaches for a number of applications in machine learning and signal processing.