Research
Brief Research Description:
My research lies in the broad area of applied harmonic analysis and is partially supported by the National Science Foundation under the award NSF DMS 1112612. More specifically, my research currently falls into two main categories:
- Composite Dilation Wavelets: Composite dilation wavelets, introduced by Guo, Labate, Lim, Weiss and Wilson, are affine systems generated by translations from a full rank lattice and dilations from two countable sets of matrices. These affine systems form an orthonormal basis or a Parseval frame for the space of square integrable functions in dimension n. In my research, I have focused on the situation where both sets of dilation matrices are subgroups of the invertible nxn matrices. Currently, I am working on finding smoother compactly supported composite wavelets than the known Haar-type wavelets (joint work with Ilya Krishtal) and classifying the composite dilation Haar-type wavelets (joint work with Kyle Steffen). I am also interested in developing implementation algorithms for composite wavelets and smoothing MSF composite dilation wavelets.
- Compressed Sensing and High Dimensional Geometry: Donoho and Tanner established necessary and sufficient conditions on a measurement matrix such that the l_1 minimization produces the sparsest solution: if the measurment matrix, A, preserves k-neighborliness when projecting the standard simplex or the l_1 ball, this measurement matrix is one such that linear programming will provide the sparesest solution to the incidence y=Ax. Simultaneously, Candes, Romberg, Tao, Vershynin and others introduced the restricted isometry property that also gaurantees that solving the l_1 minimization problem will give you the minimum l_0 (sparesest) solution. Certain random matrix ensembles have been shown to satisfy both conditions. I have been working on comparability of these and other compressed sensing results (joint work with Coralia Cartis, Jared Tanner, and Andrew Thompson).
Publications:
Journal Publications
- Performance Comparisons for Greedy Algorithms in Compressed Sensing (with Jared Tanner), submitted March 2013, (preprint version)
- Commentary: Toward Deterministic Compressed Sensing, Proc. National Academy of Sciences, 110(4), Early Edition, January 2013,
(preprint version)
- GPU Accelerate Greedy Algorithms for Compressed Sensing
(with Jared Tanner), submitted June 2012, (preprint version)
- Fast K-selection Algorithms for Graphics Processing Units
(with Tolu Alabi, Bradley Gordon, and Russel Steinbach), ACM Journal of Experimental Algorithmics, 17(2), Article 4.2, Pages 4.2:1-4.2:29, 2012 (preprint version)
- Recovery Guarantees for Rank Aware Pursuits (with Mike E. Davies), IEEE Signal Processing Letters, 19(7), 427-430, July 2012, (preprint version)
- Associated Video of Mike Davies' Presentation on these results from the SMALL Workshop, January 2011.
- Crystallographic Haar-type Composite Dilation Wavelets (with
Kyle R. Steffen), Wavelets and Multiscale Analysis: Theory and
Appilcations, Birkhauser, Cohen and Zayed eds., 2011 (preprint version)
- Matricial
Filters and
Crystallographic Composite
Dilation
Wavelets (with Ilya A. Krishtal), Mathematics of
Computation, 81, 905-922, 2012 (published online July 2011) (preprint version)
- Phase
Transitions for Greedy Sparse Approximation
Algorithms (with Coralia Cartis, Jared Tanner, and Andrew
Thompson)
Applied and Computational Harmonic Analysis, 30(2), 188-203, 2011 (preprint version)
- On
Support Sizes of Restricted Isometry Constants, (with Andrew
Thompson) Applied and Computational Harmonic Analysis, 29(3)
382-390, 2010 (preprint version)
- Compressed Sensing: How Sharp is the Restricted Isometry Property?, (with Coralia Cartis and Jared Tanner), SIAM Review, 53(1), 105-125, 2011 (preprint version)
- Extended technical report (formerly titled The restricted isometry property and l^q-regualrization: phase transitions for sparse approximation)
- Decay Properties of Restricted Isometry Constants, (with Coralia Cartis and Jared Tanner), IEEE Signal Processing Letters, 16(7), 572-575, 2009 (preprint version)
- Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets, Journal of Geometric Analysis, 19(1), 19-35, 2009 (preprint version)
- Minimally Supported Frequency Composite Dilation Wavelets, Journal of Fourier Analysis and Applications, 15(6), 796-815, 2009 (preprint version)
Proceedings
Thesis
Software
- GAGA: GPU Accelerated Greedy Algorithms for Compressed Sensing (with Jared Tanner). A software package for solving large compressed sensing problems with millions of unknowns in fractions of a second by exploiting the power of graphics processing units. This software is the focus of GPU Accelerated Greedy Algorithms for Compressed Sensing and was used to generate the data in Performance Comparisons of Greedy Algorithms in Compressed Sensing. (This link takes you to gaga4cs.org.)
- GGKS: Grinnell GPU k-Selection (with Tolu Alabi, Bradley Gordon, and Russel Steinbach). This code provides the source code for the k-selection algorithms discussed in Fast K-selection Algorithms for Graphics Processing Units. (This link takes you to the Google Code page.)
- MatricialFilterCode.zip (with Kyle Steffen). This zip file of Matlab code verifies the matricial filters equations in the papers Crystallographic Haar-type ... and one example in Matricial Filters ...
This code requires the symbolic toolbox. (To download: right-click, save-as.)
Copyrights for the preprint versions are identical to those of the published version. Submitted papers may change copyright holder without notice.
Selected Conferences:
- SIAM Southeastern Atlantic Sectional Conference: Minisymposium on Applied Harmonic Analysis, Huntsville, AL, March 2012
- PP12, SIAM Parallel Processing for High Performance Computing, Savannah, GA, February 2012
- MathFest Invited Paper Session: Polyhedra are Everywhere, Lexington, KY, August 2011
- 24th Biennial Conference on Numerical Analysis, Glasgow, Scotland, July 2011
- SPARS11, Workshop on Signal Processing with Adaptive Sparse Structured Representations, Edinburgh, Scotland, June 2011
- Scottish Parallel Computational Mathematics (SPARC-M), St. Andrews, Scotland, November 2010
- International Conference on Numerical Analysis and Applied Mathematics, Rodos, Greece, September 2010
- European Signal Processing Conference, Aalborg, Denmark, August 2010
- SIAM Annual Meeting, Symposium on Compressed Sensing, Pittsburgh, PA, July 2010
- PCMI Summer Program on Image Processing, Park City, Utah, June/July 2010
- 13th International Conference on Approximation Theory, San Antonio, TX, March 2010
- SMALL Meeting, Villars, Switzerland, February 2010
- MathFest and Project NExT Workshop, Portland, OR, August 2009
- 20 Years of Wavelets, Chicago, IL, May 2009
- SPARS'09, Workshop on Signal Processing with Adaptive Sparse Structured Representations, Saint-Malo, France, April, 2009
- AMS-MAA Joint Mathematics Meeting, Washington, D.C., January 2009
- Defense Threat Reduction Agency/National Science Foundation Algorithms Workshop, Baltimore, MD, November 2008
- MathFest and Project NExT Workshop, Madison, WI, July 2008
- Foundation of Computational Mathematics, Hong Kong, China, June 2008
- Classical and Modern Harmonic Analysis: from Theory to Numerical Computation, ICMS, Edinburgh, Scotland, May 2008
- AMS-MAA Joint Mathematics Meeting, Special Session on Wavelet Sets, San Diego, CA, January 2008
- Von Neumman Symposium: Sparse Approximation and High Dimensional Geometry, Snowbird, UT, July 2007
- STROBL07: Trends in Harmonic Analysis, Strobl, Austria, June 2007
- IPAM Short Course: Sparse Approximation and High Dimensional Geometry, Los Angeles, CA, May 2007
- AMS-MAA Joint Mathematics Meeting, New Orleans, LA, January 2007
- Washington University/University of Zagreb International Wavelet Workshop, St. Louis, MO, April 2006
- AMS Western Regional Meeting, Special Session on Wavelets, Eugene, OR, November 2005
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