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Complex dimension

About: Complex dimension is a research topic. Over the lifetime, 1168 publications have been published within this topic receiving 22217 citations.


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TL;DR: It is shown that by replacing the lscr1 norm with theLscrp norm, exact reconstruction is possible with substantially fewer measurements, and a theorem in this direction is given.
Abstract: Several authors have shown recently that It is possible to reconstruct exactly a sparse signal from fewer linear measurements than would be expected from traditional sampling theory. The methods used involve computing the signal of minimum lscr1 norm among those having the given measurements. We show that by replacing the lscr1 norm with the lscrp norm with p < 1, exact reconstruction is possible with substantially fewer measurements. We give a theorem in this direction, and many numerical examples, both in one complex dimension, and larger-scale examples in two real dimensions.

1,321 citations

Journal ArticleDOI
TL;DR: In this paper, Cartan gave a complete solution of the equivalence problem, which is, among other results, the problem of finding a complete system of analytic invariants for two real analytic real hypersurfaees in Cn+l to be locally equivalent under biholomorphic transformations.
Abstract: Whether one studies the geometry or analysis in the complex number space C a + l , or more generally, in a complex manifold, one will have to deal with domains. Their boundaries are real hypersurfaces of real codimension one. In 1907, Poincar4 showed by, a heuristic argument tha t a real hypersurface in (38 has local invariants unde r biholomorphie transformations [6]. He also recognized the importance of the special uni tary group which acts on the real hyperquadrics (cf. w Following a remark by B. ~Segre, Elie :Cartan took, up again the problem. In t w o profound papers [1], he gave, among other results, a complete solution of the equivalence problem, tha t is, the problem of finding a complete system of analytic invariants for two real analytic real hypersurfaees in C~ to be locally equivalent under biholomorphic transformations. Let z 1, ..., z n+l be the coordinates in Cn+r We s tudy a real hypersurface M at the origin 0 defined by the equation

1,037 citations

Journal ArticleDOI
TL;DR: In this paper, the authors discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors, and conclude that dimension of the natural measure is more important than the fractal dimension.

1,000 citations

Posted Content
TL;DR: A new algorithm for manifold learning and nonlinear dimension reduction is presented based on a set of unorganized data points sampled with noise from the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point.
Abstract: Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of the neighborhood connection matrix. We present a careful error analysis of our algorithm and show that the reconstruction errors are of second-order accuracy. We illustrate our algorithm using curves and surfaces both in 2D/3D and higher dimensional Euclidean spaces, and 64-by-64 pixel face images with various pose and lighting conditions. We also address several theoretical and algorithmic issues for further research and improvements.

654 citations

Journal ArticleDOI
TL;DR: In this paper, the authors give a complete topological characterization of Stein manifolds of complex dimension > 2 and present new topogical obstructions for the existence of a Stein complex structure on real manifold of dimension 4.
Abstract: In this paper I give a completed topological characterization of Stein manifolds of complex dimension >2. Another paper (see [E14]) is devoted to new topogical obstructions for the existence of a Stein complex structure on real manifolds of dimension 4. Main results of the paper have been announced in [E13].

496 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20238
202220
202142
202037
201941
201839