TITLE

# Approximation of points on low-dimensional manifolds via random linear projections

AUTHOR(S)
Iwen, Mark A.; Maggioni, Mauro
PUB. DATE
June 2013
SOURCE
Information & Inference: A Journal of the IMA;Jun2013, Vol. 2 Issue 1, p1
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
This paper considers the approximate reconstruction of points âˆˆ â„D which are close to a given compact d-dimensional submanifold â„³ of â„D using a small number of linear measurements of . In particular, it is shown that a number of measurements of , which is independent of the extrinsic dimension D suffices for a highly accurate reconstruction of . Furthermore, it is also proved that all vectors which are sufficiently close to â„³ can be reconstructed with uniform approximation guarantees when the number of linear measurements of depends logarithmically on D. Finally, the proofs of these facts are constructive: a practical algorithm for manifold-based signal recovery is presented in the process of proving the two main results mentioned above.
ACCESSION #
88429485

## Related Articles

• SUBMANIFOLDS WITH POINTWISE 1-TYPE GAUSS MAP. Kim, Young Ho // Bulletin of the Transilvania University of Brasov, Series III: M;2008, Vol. 1 Issue 50, p201

We introduce the background of the notion of pointwise 1-type Gauss map defined on the submanifolds of a Euclidean space or a pseudo-Euclidean space and the recent results related to it.

• Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing. Wojtaszczyk, P. // Foundations of Computational Mathematics;Feb2010, Vol. 10 Issue 1, p1

In compressed sensing, we seek to gain information about a vector x?R N from d ï¿½ N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, ; Candes et al., Commun. Pure Appl. Math. 59:1207ï¿½1223, ; Donoho, IEEE Trans. Inf....

• Spherical Submanifolds of a Euclidean Space. AL-ODAN, HAILA; DESHMUKH, SHARIEF // Quarterly Journal of Mathematics;Sep2002, Vol. 53 Issue 3, p249

The article discusses the spherical submanifolds of a Euclidean space. It presents the ...T, ...âŠ¥ as tangential and normal components of the position vector ... in Rn+p. It analyses the codimension-2 isometric immersions ... Mn ... Rn+p. The integral formulae utilized to prove the spherical...

• DIFFEOMORPHIC MATCHING AND DYNAMIC DEFORMADLE SURFACES IN 3D MEDICAL IMAGING. Azencott, R.; Glowinski, R.; He, J.; Jajoo, A.; Li, Y.; Martynenko, A.; Hoppe, R. H. W.; Benzekry, S.; Little, S. H.; Zoghbi, W. A. // Computational Methods in Applied Mathematics;2010, Vol. 10 Issue 3, p235

We consider optimal matching of submanifolds such as curves and surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. In an abstract setting, the optimal matching is formulated as a minimization problem involving actions of diffeomorphisms on regular Borel...

• Universal Models For Real Submanifolds. Beloshapka, V. K. // Mathematical Notes;Mar/Apr2004, Vol. 75 Issue 3/4, p475

In previous papers by the present author, a machinery for calculating automorphisms, constructing invariants, and classifying real submanifolds of a complex manifold was developed. The main step in this machinery is the construction of a â€œniceâ€ model surface. The nice model surface...

• Curvature estimates for submanifolds with prescribed Gauss image and mean curvature. Xin, Y. L. // Calculus of Variations & Partial Differential Equations;Mar2010, Vol. 37 Issue 3/4, p385

We study that the n-graph defined by a smooth map $${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$$ in $${\mathbb R^{m+n}}$$ of the prescribed mean curvature and the Gauss image. Under the condition we derive the interior curvature estimates when 2 â‰¤ n â‰¤ 5 with constant C...

• GEOMETRY OF WARPED PRODUCT SUBMANIFOLDS: A SURVEY. BANG-YEN CHEN // Journal of Advanced Mathematical Studies;Aug2013, Vol. 6 Issue 2, p1

The warped product N1 Ã— f N2 of two Riemannian manifolds (N1,g1) and (N2,g2) is the product manifold N1Ã— N2 equipped with the warped product metric g = g1 + f1g2, where f is a positive function on N1. The notion of warped product manifolds is one of the most fruitful generalizations of...

• Exact moduli space metrics for hyperbolic vortex polygons. Krusch, S.; Speight, J. M. // Journal of Mathematical Physics;Feb2010, Vol. 51 Issue 2, p022304

Exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic N-vortices are derived. These submanifolds, denoted as Î£n,m, are spaces of Cn-invariant vortex configurations with n single vortices at the vertices of a regular polygon and m=N-n coincident vortices...

• Invariant submanifolds of Sasakian space forms. Yıldız, Ahmet; Murathan, Cengizhan // Journal of Geometry;2009, Vol. 95 Issue 1/2, p135

In the present study, we consider isometric immersions $${f : M \rightarrow \tilde{M}(c)}$$ of (2 n + 1)-dimensional invariant submanifold M2 n+1 of (2 m + 1) dimensional Sasakian space form $${\tilde{M}^{2m+1}}$$ of constant $${ \varphi}$$-sectional curvature c. We have shown that if f...

Share