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Lie Operators For Compressive Sensing, Chinmay Hegde, Aswin C. Sankaranarayanan, Richard G. Baraniuk
Lie Operators For Compressive Sensing, Chinmay Hegde, Aswin C. Sankaranarayanan, Richard G. Baraniuk
Chinmay Hegde
We consider the efficient acquisition, parameter estimation, and recovery of signal ensembles that lie on a low-dimensional manifold in a high-dimensional ambient signal space. Our particular focus is on randomized, compressive acquisition of signals from the manifold generated by the transformation of a base signal by operators from a Lie group. Such manifolds factor prominently in a number of applications, including radar and sonar array processing, camera arrays, and video processing. Leveraging the fact that Lie group manifolds admit a convenient analytical characterization, we develop new theory and algorithms for: (1) estimating the Lie operator parameters from compressivemeasurements, and (2) …
Near-Isometric Linear Embeddings Of Manifolds, Chinmay Hegde, Aswin C. Sankaranarayanan, Richard G. Baraniuk
Near-Isometric Linear Embeddings Of Manifolds, Chinmay Hegde, Aswin C. Sankaranarayanan, Richard G. Baraniuk
Chinmay Hegde
We propose a new method for linear dimensionality reduction of manifold-modeled data. Given a training set X of Q points belonging to a manifold M ⊂ ℝN, we construct a linear operator P : ℝN → ℝM that approximately preserves the norms of all (2Q) pairwise difference vectors (or secants) of X. We design the matrix P via a trace-norm minimization that can be efficiently solved as a semi-definite program (SDP). When X comprises a sufficiently dense sampling of M, we prove that the optimal matrix P preserves all pairs of secants over M. …