# Eero Simoncelli

### Abstract

*"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 onedimensional 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

### Biography

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

# Video

## Eero Simoncelli

##### Abstract

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 onedimensional 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