"Recovery of sparse translation-invariant signals with continuous basis pursuit"
We consider the problem of decomposing a signal into a linear combination of features, each a continuously
translated version of one of a small set of elementary features. Although these constituents are drawn from a
continuous family, most current signal decomposition methods rely on a finite dictionary of discrete examples
selected from this family (e.g.,a set of shifted copies of a set of basic waveforms), and apply sparse optimization
methods to select and solve for the relevant coefficients. Here, we generate a dictionary that includes auxilliary
interpolation functions that approximate translates of features via adjustment of their coefficients. We formulate a
constrained convex optimization problem, in which the full set of dictionary coefficients represent a linear
approximation of the signal, the auxiliary coefficients are constrained so as to onlyrepresent translated features, and
sparsity is imposed on the non-auxiliary coefficients using an L1 penalty. The well-known basis pursuit denoising
(BP) method may be seen as a special case, in which the auxiliary interpolation functions are omitted, and we thus
refer to our methodology ascontinuous basis pursuit (CBP). We develop two implementations of CBP for a one-
dimensional translationinvariant source, one using a first-order Taylor approximation, and another using a form of
trigonometric spline. We examine the tradeoff between sparsity and signal reconstruction accuracy in these methods,
demonstrating empirically that trigonometric CBP substantially outperforms Taylor CBP, which in turn offers
substantial gains over ordinary BP. In addition, the CBP bases can generally achieve equally good or better
approximations with much coarser sampling than BP, leading to a reduction in dictionary dimensionality.
Dr. Simoncelli is a Professor of Neural Science, Mathematics, and Psychology at New York University. He began
his higher education as a physics major at Harvard, studied mathematics at Cambridge University for a year and a
half on a Knox Fellowship , and earned a doctorate in electrical engineering and computer science at the
Massachusetts Institute of Technology. He then joined the faculty of the Computer and Information Science
Department at the University of Pennsylvania. In 1996, he moved to NYU as part of the Sloan Center for
Theoretical Visual Neuroscience. In August 2000, he became an Investigator of the Howard Hughes Medical
Institute, under their new program in Computational Biology. Dr. Simoncelli became an Associate member of
CIFAR's Neural Computation & Adaptive Perception in 2010.
His research interests span a wide range of topics in the representation and analysis of visual images, in both
machine and biological systems. Since 2000, he's been an Investigator of the Howard Hughes Medical Institute,
under their program in computational biology.