## Presidential Impact Fellow## Texas A&M University## Professor## Department of Mathematics## Associate Director## TAMIDS## Director## CAMDA |
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Current Research Activity:

Mathematical Data Science (including Compressive Sensing)
with a strong Approximation Theory influence

- Classical Approximation Theory
- Sparse and Structured Recovery
- (Deep) Learning
- Scientific Computing
- Applications in Engineering and Bioinformatics

- Students:
Thomas Winckelman

Prospective - apply via the standard departmental process (link) - Postdocs:
none at the moment

Prospective - apply via mathjobs.org

S. F.,
Mathematical pictures at a data science exhibition.
Cambridge University Press. List of errata |
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S. F., H. Rauhut,
A mathematical introduction to compressive sensing.
Applied and Numerical Harmonic Analysis, Birkhäuser. List of errata |

Approximation Theory XV: San Antonio 2016, Springer Proceedings in Mathematics & Statistics, vol 201, 61--104.

- S. F., Linearly embedding sparse vectors from $\ell_2$ to $\ell_1$
via deterministic dimension-reducing maps.

(pdf) (reproducible) - S. F, C. Liao, S-procedure relaxation: a case of exactness involving Chebyshev centers.

(pdf)

- S. F., C. Liao,
Optimal recovery from inaccurate data in Hilbert spaces: regularize, but what of the parameter?

Constructive Approximation, 57, 489--520, 2023. (doi) (pdf) - I. Daubechies, R. DeVore, S. F., B. Hanin, G. Petrova,
Nonlinear approximation and (deep) ReLU networks.

Constructive Approximation, 55, 127--172, 2022. (doi) (pdf) - S. F., J. B. Lasserre,
Determining projection constants of univariate polynomial spaces.

Journal of Approximation Theory, 235, 74--91, 2018. (doi) (pdf) - R. Baraniuk, S. F., D. Needell, Y. Plan, M. Wootters,
Exponential decay of reconstruction error from binary measurements of sparse signals.

IEEE Transactions on Information Theory, 63/6, 3368--3385, 2017. (doi) (pdf) - S. F., L. Skrzypek,
On maximal relative projection constants.

Journal of Mathematical Analysis and Applications, 447/1, 309--328, 2017. (doi) (pdf) - S. F., Stability and robustness of $\ell_1$-minimizations with Weibull matrices and redundant dictionaries.

Linear Algebra and its Applications, 441, 4--21, 2014. (doi) (pdf) - D. Koslicki, S. F., G. Rosen,
Quikr: a method for rapid reconstruction of bacterial communities via compressive sensing.
(doi)
(pdf)

Bioinformatics, 29/17, 2096--2102, 2013. - S. F.,
Hard thresholding pursuit: an algorithm for compressive sensing.
(doi)
(pdf)

SIAM Journal on Numerical Analysis, 49/6, 2543--2563, 2011. - S. F., A. Pajor, H. Rauhut, T. Ullrich,
The Gelfand widths of $\ell_p$-balls for $0 < p \le 1$.
(doi)
(pdf)

Journal of Complexity, 26/6, 629--640, 2010.

**Mathematical Foundations for Data Science**(syllabus) (webpage)

**Foundations and Methods of Approximation**(updates forthcoming, restricted access) (notes) (codes)**Topics in Mathematical Data Science**(supplanted by my CUP book, restricted access) (pdf)**Matrix Analysis**(a few lectures missing) (pdf)**Mathematics of Genome Analysis**(very incomplete, restricted access) (pdf)**Problem Solving Competitions**(selected topics) (pdf)**Compressed Sensing**(supplanted by the book written with H. Rauhut) (pdf)**Numerical Mathematics**(updates required) (pdf)

Go to my Github page for download.

- MinProj
- This is a MATLAB package that computes exact projection constants and minimal projections in coordinate spaces and matrix spaces by solving linear programs, as well as approximate projection constants and minimal projections in polynomial spaces by solving linear or semidefinite programs. It relies on the external packages CVX and Chebfun.
- Basc
- This is a MATLAB package that computes Best Approximations by Splines under Constraints relative to various norms. Relying on the external packages CVX and Chebfun, it is based on a reformulation of constrained approximation problems as semidefinite programs. (demo)
- SplineDim
- This is a collection of SAGE routines designed to generate formulas for the dimension of multivariate spline spaces over specific partitions. It is based on Hilbert series computations. The core of the code was written by P. Clarke.
- Quikr and WGSQuikr
- These computational packages determine the composition of bacteria in an environmental sample analyzed by 16S rRNA amplicon and whole-genome shotgun sequencing technologies. The packages were assembled by D. Koslicki, who also set up this Galaxy server.
- HTP
- These are three MATLAB routines to be used when trying to recover a sparse vector x or a row-sparse matrix X from the incomplete linear measurements y=Ax or Y=AX. They are implementations of the HTP, FHTP, and SHTP algorithms.
- Allometry
- This is a collection of MATLAB routines to be used for the computation of exact constants in Banach space geometry.

- 2001-05: PhD, University of Cambridge.
- 2000-01: Part III of Math Tripos (Distinction), University of Cambridge.
- 1998-01: Masters of Engineering, Ecole Centrale Paris.

- 2019-now: Professor of Mathematics, Texas A&M University.
- 2015-19: Associate Professor of Mathematics, Texas A&M University.
- 2013-15: Assistant Professor of Mathematics, University of Georgia.
- 2010-13: Assistant Professor of Mathematics, Drexel University.
- 2009-10: Postdoctoral Researcher, University of Paris 6.
- 2006-09: Postdoctoral Researcher, Vanderbilt University.

- July 2023: Los Alamos National Laboratory.
- January-June 2019: University of Wisconsin-Madison.
- June 2018: LAAS-CNRS, Toulouse.
- December 2017: Hong Kong University of Science and Technology.
- May-June 2015: University of South Florida.
- July-August 2009: University of Bonn.

- 2019: Recipient of a Presidential Impact Fellowship, Texas A&M University.
- 2012: Recipient of the Antelo Devereux Award for Young Faculty, Drexel University.
- 2010: Journal of Complexity Best Paper Award.

- Srinivas Subramanian (grad student, Aug 2016-Oct 2023).
- Chunyang Liao (grad student, Aug 2019-May 2023, then postdoc at UCLA).
- Josiah Park (postdoc, Aug 2020-Dec 2022, now postdoc at Berkeley National Lab).
- Mahmood Ettehad (grad student, Aug 2016-Jul 2020, then postdoc at the IMA, University of Minnesota).
- Richard G. Lynch (postdoc, Aug 2016-Jun 2019, then Instructional Assistant Professor at Texas A&M University).
- Jean-Luc Bouchot (postdoc, Nov 2012-Aug 2014, then Assistant Professor at Beijing Institute of Technology).
- David Koslicki (postdoc, Jan-Sep 2012, now Associate Professor at Pennsylvania State University).
- Michael Minner (grad student, Sep 2012-Mar 2016, now at Sandia National Lab).

- Journal of Approximation Theory (journal, Aug 2017-now).
- Sampling Theory, Signal Processing, and Data Analysis (journal, Jun 2020-now).
- Journal of Numerical Mathematics (journal, May 2021-now).
- Numerical Mathematics and Scientific Computation (book series, Jul 2023-now).
- Journal of Complexity (journal, Sep 2023-now)
- Surveys in Approximation Theory (online journal, Sep 2023-now)

Consultation hours: Tu 9:15-9:45am, We 8-8:30am, Th 1:30-2pm, and by appointment.

Office location:
502D Blocker Building

Ireland Street

College Station

Texas

Mailing address:
Texas A&M University

Department of Mathematics

3368 TAMU

College Station, TX 77843-3368

E-mail: foucart@tamu.edu
or simon.foucart@centraliens.net

Zoom: tamu.zoom.us/my/foucart