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
Academic Journal
DOC. TYPE
Article
ABSTRACT
This paper considers the approximate reconstruction of points ∈ ℝD which are close to a given compact d-dimensional submanifold ℳ of ℝD 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 ℳ 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 � 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. 59:1207�1223, ; 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, ...⊥ 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 “nice” 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 ≤ n ≤ 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 × f N2 of two Riemannian manifolds (N1,g1) and (N2,g2) is the product manifold N1× 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 Σn,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

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics