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NuMax: A Convex Approach for Learning Near-Isometric Linear Embeddings

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journal contribution
posted on 01.07.2015, 00:00 authored by Chinmay Hegde, Aswin C. Sankaranarayanan, Wotao Yin, Richard G. Baraniuk

We propose a novel framework for the deterministic construction of linear, near-isometric embeddingsof a finite set of data points. Given a set of training points X RN, we consider the secant set S(X) that consists of all pairwise difference vectors of X, normalized to lie on the unit sphere. We formulate an affine rank minimization problem to construct a matrix that preserves the norms of all the vectors in S(X) up to a distortion parameter . While affine rank minimization is NP-hard, we show that this problem can be relaxed to a convex formulation that can be solved using a tractable semidefinite program (SDP). In order to enable scalability of our proposed SDP to very large-scale problems, we adopt a twostage approach. First, in order to reduce compute time, we develop a novel algorithm based on the Alternating Direction Method of Multipliers (ADMM) that we call Nuclear norm minimization with Max-norm constraints (NuMax) to solve the SDP. Second, we develop a greedy, approximate version ofNuMax based on the column generation method commonly used to solve large-scale linear programs. We demonstrate that our framework is useful for a number of signal processing applications via a range of experiments on large-scale synthetic and real datasets.


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