Simon Foucart
· ProfessorVerifiedTexas A&M University · Mathematics
Active 2004–2026
About
Simon Foucart is a Professor of Mathematics at Texas A&M University and serves as the Associate Director of the Institute of Data Science. His current research activity centers on Mathematical Data Science, with a particular emphasis on Compressive Sensing and a strong influence from Approximation Theory. His research objective is to find, exploit, and create synergies between Classical Approximation Theory, Sparse and Structured Recovery, (Deep) Learning, and Scientific Computing, with applications in Engineering and Bioinformatics. Foucart has a distinguished academic background, having earned his PhD from the University of Cambridge in 2005, following a Masters of Engineering from Ecole Centrale Paris and the Part III of the Math Tripos with Distinction at Cambridge. He has held faculty positions at the University of Georgia and Drexel University before joining Texas A&M University, where he has been a professor since 2019. His professional experience also includes postdoctoral research at the University of Paris 6 and Vanderbilt University, as well as several visiting positions at prestigious institutions such as the University of Cambridge, Isaac Newton Institute, and Los Alamos National Laboratory. Foucart has been recognized with numerous honors and awards, including the Joseph F. Traub Prize for Achievement in Information-Based Complexity in 2026, the Frontiers of Science Award, and the Presidential Impact Fellow distinction at Texas A&M University. He is also actively involved in editorial work for several journals and book series in his field. His teaching portfolio includes courses on Topics in Mathematical Data Science, Optimization, Foundations and Methods of Approximation, and Matrix Analysis, among others. Foucart has authored and edited several books, including "Mathematical Pictures at a Data Science Exhibition" and "A Mathematical Introduction to Compressive Sensing," reflecting his expertise and contributions to the field of mathematical data science and approximation theory.
Research topics
- Artificial Intelligence
- Computer Science
- Machine Learning
- Mathematics
- Combinatorics
- Applied mathematics
- Mathematical optimization
- Mathematical analysis
- Discrete mathematics
- Arithmetic
- Algorithm
Selected publications
Learning the maximum of a Hölder function from inexact data
Proceedings of the American Mathematical Society · 2026-01-16
article1st authorCorrespondingWithin the theoretical framework of optimal recovery, one determines in this article the <italic>best</italic> procedures to learn a quantity of interest depending on a Hölder function acquired via inexact point evaluations at fixed datasites. <italic>Best</italic> here refers to procedures minimizing worst-case errors. The elementary arguments hint at the possibility of tackling nonlinear quantities of interest, with a particular focus on the function maximum. In a local setting, i.e., for a fixed data vector, the optimal procedure (outputting the so-called Chebyshev center) is precisely described relatively to a general model of inexact evaluations. Relatively to a slightly more restricted model and in a global setting, i.e., uniformly over all data vectors, another optimal procedure is put forward, showing how to correct the natural underestimate that simply returns the data vector maximum. Jitterred data are also briefly discussed as a side product of evaluating the minimal worst-case error optimized over the datasites.
Optimal Algorithms for Nonlinear Estimation with Convex Models
ArXiv.org · 2025-12-23
articleOpen access1st authorCorrespondingA linear functional of an object from a convex symmetric set can be optimally estimated, in a worst-case sense, by a linear functional of observations made on the object. This well-known fact is extended here to a nonlinear setting: other simple functionals of the object can be optimally estimated by functionals of the observations that share a similar simple structure. This is established for the maximum of several linear functionals and even for the $\ell$th largest among them. Proving the latter requires an unusual refinement of the analytical Hahn--Banach theorem. The existence results are accompanied by practical recipes relying on convex optimization to construct the desired functionals, thereby justifying the term of estimation algorithms.
Optimal prediction of vector-valued functions from point samples
Journal of Complexity · 2025-08-18
articleOpen access1st authorCorrespondingPredicting the value of a function f at a new point given its values at old points is an ubiquitous scientific endeavor, somewhat less developed when f produces several values depending on one another, e.g. when it outputs a probability vector. Considering the points as fixed (not random) entities and focusing on the worst-case, this article uncovers a prediction procedure that is optimal relatively to some model-set information about the vector-valued function f . When the model sets are convex, this procedure turns out to be an affine map constructed by solving a convex optimization program. The theoretical result is specified in the two practical frameworks of (reproducing kernel) Hilbert spaces and of spaces of continuous functions.
Linear Algebra and its Applications · 2025-12-18
articleOpen accessCorrespondingIt is shown in this note that one can decide whether an n -dimensional subspace of ℓ ∞ N is isometrically isomorphic to ℓ ∞ n by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an n -dimensional subspace of ℓ ∞ N with projection constant equal to one must be isometrically isomorphic to ℓ ∞ n .
When is a subspace of $\ell_\infty^N$ isometrically isomorphic to $\ell_\infty^n$?
ArXiv.org · 2025-09-20
preprintOpen accessIt is shown in this note that one can decide whether an $n$-dimensional subspace of $\ell_\infty^N$ is isometrically isomorphic to $\ell_\infty^n$ by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an $n$-dimensional subspace of $\ell_\infty^N$ with projection constant equal to one must be isometrically isomorphic to $\ell_\infty^n$.
Worst-Case Learning under a Multifidelity Model
SIAM/ASA Journal on Uncertainty Quantification · 2025-02-11
article1st authorCorrespondingOptimal Algorithms for Nonlinear Estimation with Convex Models
arXiv (Cornell University) · 2025-12-23
preprintOpen access1st authorCorrespondingA linear functional of an object from a convex symmetric set can be optimally estimated, in a worst-case sense, by a linear functional of observations made on the object. This well-known fact is extended here to a nonlinear setting: other simple functionals of the object can be optimally estimated by functionals of the observations that share a similar simple structure. This is established for the maximum of several linear functionals and even for the $\ell$th largest among them. Proving the latter requires an unusual refinement of the analytical Hahn--Banach theorem. The existence results are accompanied by practical recipes relying on convex optimization to construct the desired functionals, thereby justifying the term of estimation algorithms.
Least multivariate Chebyshev polynomials on diagonally determined sets
arXiv (Cornell University) · 2024-05-29
preprintOpen accessWe consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $Π^*_n$ be the subset of polynomials of degree at most $n$ in $d$ variables, whose homogeneous part of degree $n$ has coefficients summing up to $1$. The problem is determining a polynomial in $Π^*_n$ with the smallest uniform norm on a domain $Ω$, which we call a least Chebyshev polynomial (associated with $Ω$). Our main result solves the problem for $Ω$ belonging to a non-trivial class of sets that we call diagonally-determined, and establishes the remarkable result that a least Chebyshev polynomial can be given via the classical, univariate, Chebyshev polynomial. In particular, the solution can be independent of the dimension. Diagonally-determined domains include centered balls in $\mathbb{R}^d$ in any norm, but can be non-convex and even non-simply connected. We also introduce a computational procedure, based on semidefinite programming hierarchies, to detect if a given semi-algebraic set is diagonally-determined.
Journal of Complexity · 2024-03-08 · 2 citations
article1st authorCorrespondingarXiv (Cornell University) · 2024-01-20
preprintOpen access1st authorCorrespondingFor objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters, short of being explicit, can be precomputed by solving a semidefinite program. This general framework can be swiftly applied to several scenarios: the two-space problem, the problem of recovery from $\ell_2$-inaccurate data, and the problem of recovery from a mixture of accurate and $\ell_2$-inaccurate data. With more effort, it can also be applied to the problem of recovery from $\ell_1$-inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition. Experimentally, this condition seems to hold whenever the level of $\ell_1$-inaccuracy is small enough. We also point out that, independently of the inaccuracy level, the minimal worst-case error of a linear recovery method can be found by semidefinite programming.
Recent grants
NSF · $666k · 2011–2014
CDS&E-MSS: Optimal Recovery in the Age of Data Science
NSF · $150k · 2021–2024
CDS&E-MSS: Recovery of High-Dimensional Structured Functions
NSF · $100k · 2016–2020
Frequent coauthors
- 23 shared
Holger Rauhut
- 12 shared
Jean B. Lasserre
- 11 shared
Rémi Gribonval
- 10 shared
Mary Wootters
- 10 shared
Yaniv Plan
- 10 shared
Deanna Needell
- 10 shared
David Koslicki
Pennsylvania State University
- 8 shared
Chunyang Liao
University of California, Los Angeles
Labs
Mathematical Data Science (including Compressive Sensing) with a strong Approximation Theory influence
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