Optimal sampling in general domains

When approximating a multivariate function defined on an irregular domain, a good choice of sampling points is critical. In this paper, my PhD student Juan and I develop new, practical sampling strategies for which the sample complexity is near-optimal: specifically, it is linear (up to a log factor) in the degree of the approximation. This improves previous approaches which were at best quadratic in the degree. Here’s the paper:

Optimal sampling strategies for multivariate function approximation on general domains

New group members

I am pleased to announce the arrival of three new members to my group this fall:

  • Nick Dexter is a PIMS Postdoctoral Fellow, joining from the University of Tennessee in the USA.
  • Juan Manuel Cárdenas is a PhD student, joining from the University of Concepcion in Chile.  He previously visited my group in Spring 2017.
  • Matthew King-Roskamp is an NSERC MSc student, and a former undergraduate honours student at SFU.  He was previously an undergraduate researcher in my group.


Approximating high-dimensional functions on irregular domains

Multivariate polynomials are excellent means of approximating high-dimensional functions on tensor-product domains.  But what about approximations on irregular domains, as is quite common in applications?  In our new paper, Daan Huybrechs and I tackle this question using tools from frame theory and approximation theory:

Approximating smooth, multivariate functions on irregular domains

We establish a series of results on approximation rates and sample complexity, deriving bounds that scale well with dimension in a variety of cases.

Congratulations Casie!

My former undergraduate student Anyi (Casie) Bao was a winner of the SFU Department of Mathematics Undergraduate Research Prize. Her work involved the development and analysis of compressed sensing-based strategies for correcting for corrupted measurements in Uncertainty Quantification.  A draft version of the resulting paper can be found here:

Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations

Congratulations (again) Matt!

Matt King-Roskamp – an undergraduate student in my group co-supervised with Simone Brugiapaglia – was awarded runner-up in two poster competitions this summer:

His work, entitled Optimal Sampling Strategies for Compressive Imaging, presents new, theoretically optimal sampling techniques for imaging using compressed sensing.

Robustness to unknown error in sparse regularization

Simone Brugiapaglia, Rick Archibald and I have been investigating the robustness of constrained l1-minimization regularization to unknown measurement errors.  The vast majority of existing theory for such problems requires an a priori bound on the noise level.  Yet in many, if not most, real-world applications such a bound is unknown.  Our work provides the first recovery guarantees for a large class of practical measurement matrices, thereby extending existing results for subgaussian random matrices.  Besides this, our work sheds new light on sparse regularization in practice, addressing questions about the relationship between model fidelity, noise parameter estimation and reconstruction error.

Simone will be presenting this work at SPARS2017 and SampTA 2017.  In the meantime our conference paper is available here:

Recovery guarantees for compressed sensing with unknown errors