Invited Session Mon.1.H 1028

Monday, 10:30 - 12:00 h, Room: H 1028

Cluster 21: Sparse optimization & compressed sensing [...]

New models and algorithms in sparse optimization

 

Chair: Benjamin Recht

 

 

Monday, 10:30 - 10:55 h, Room: H 1028, Talk 1

Nicolas Boumal
Riemannian algorithms and estimation bounds for synchronization of rotations

Coauthors: Pierre-Antoine Absil, Amit Singer

 

Abstract:
We estimate unknown rotation matrices Ri in SO(n) from a set of measurements of relative rotations Ri^{}RjT. Each measurement is either slightly noisy, or an outlier bearing no information. We study the case where most measurements are outliers. We propose a Maximum Likelihood Estimator (MLE) approach, explicitly acknowledging outliers in the noise model.
The MLE maximizes the log-likelihood function over the parameter space. That space is a product of rotation groups, possibly quotiented to account for invariance under a common rotation of the estimators.
To compute the MLE, we use Riemannian trust-region methods to maximize the log-likelihood function over the parameter space. That space is a matrix manifold, hence tools and analyses from (Absil et al., Optimization Algorithms on Matrix Manifolds, Princeton Univ. Press, 2008) apply gracefully.
We derive Riemannian Cramer-Rao bounds for synchronization, valid for a broad class of problem dimensions and noise distributions. These bounds admit a simple expression in terms of an information-weighted Laplacian of the measurement graph. Numerical tests suggest the MLE is asymptotically efficient in many cases.

 

 

Monday, 11:00 - 11:25 h, Room: H 1028, Talk 2

Mark A. Davenport
A simple framework for analog compressive sensing

Coauthor: Michael B. Wakin

 

Abstract:
Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this talk, I will describe a new approach to bridging this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. I will provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements.

 

 

Monday, 11:30 - 11:55 h, Room: H 1028, Talk 3

Benjamin Recht
Atomic norm denoising with applications to spectrum estimation and system identification

Coauthors: Badri N. Bhaskar, Parikshit Shah, Gonnguo Tang

 

Abstract:
One of the most common goals of data analysis is to reject noise by leveraging the latent structure present in the true signal. This talk will propose a general approach to such denoising problems by regularizing data fidelity with a penalty called the atomic norm. Atomic norm denoising is posed as a convex optimization problem and has generic, mean-squared-error guarantees. For sparse signals, atomic norm denoising is equivalent to soft-thresholding, but our techniques can be applied to estimate a variety of other objects and structures beyond sparse signals and images.
To demonstrate the wide applicability of atomic norm denoising, I will specialize the abstract formulation to two applications of practical interest. First, I will present a convex approach to superresolution, estimating the frequencies and phases of a mixture of complex exponentials. I will then apply the framework to identify linear dynamical systems from noisy, incomplete measurements. In both cases, atomic norm denoising efficiently achieves comparable or better error to existing heuristics without a priori knowledge of system parameters such as model order.

 

  Most online loan lenders allow getting New Jersey Payday Loans without visiting a bank, straight to your bank account. On the global pharmaceutical market this medicine was issued in 2003 by two companies - Eli Lilly and ICOS. Initially, permission to sell Cialis was obtained in Europe, Australia, New Zealand.