Restricted Isometric Projections for Differentiable Manifolds and Applications

Author

Vasile Pop, University of South Florida

Graduation Year

2020

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Razvan Teodorescu, Ph.D.

Committee Member

Iuliana Teodorescu, Ph.D.

Committee Member

Sherwin Kouchekian, Ph.D.

Committee Member

Mohamed Elhamdadi, Ph.D.

Keywords

Sparsity, Varieties, Low Dimensional Models, Random Matrices, Covariant Exterior Differential

Abstract

The restricted isometry property (RIP) is at the center of important developments in compressive sensing. In R^N, RIP establishes the success of sparse recovery via basis pursuit for measurement matrices with small restricted isometry constants δ_2s < 1=3. A weaker condition, δ_2s < 0:6246, is actually sufficient to guarantee stable and robust recovery of all s-sparse vectors via l₁-minimization. In infinite Hilbert spaces, a random linear map satisfies a general RIP with high probability and allow recovering and extending many known compressive sampling results. This thesis extends the known restricted isometric projection of sparse datasets of vectors embedded in the Euclidean spaces RN down into low-dimensional subspaces R^m ,m << N; to Riemannian manifolds (M, g), of manifold dimension m, with Riemannian metric g equivalent to the induced metric from the embedding space RN. This will establish a higher-dimensional version of the Fisher-Kolmogorov test for comparing populations in usual statistical analysis, allowing to develop an inference procedure analogous to Generalized Linear Models in the usual case (#Σ >> N).

Scholar Commons Citation

Pop, Vasile, "Restricted Isometric Projections for Differentiable Manifolds and Applications" (2020). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/8278