Restricted Isometric Projections for Differentiable Manifolds and Applications

The restricted isometry property (RIP) is at the center of important developments in compressive sensing. In RN, RIP establishes the success of sparse recovery via basis pursuit for measurement matrices with small restricted isometry constants δ2s \u3c 1=3. A weaker condition, δ2s \u3c 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 Rm ,m \u3c\u3c 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 \u3e\u3e N)