Compressive Sensing (CS) theory states that real-world signals can often be recovered from much fewer measurements than those suggested by the Shannon sampling theorem. Nevertheless, recoverability does not only depend on the signal, but also on the measurement scheme. The measurement matrix should behave as close as possible to an isometry for the signals of interest. Therefore the search for optimal CS measurement matrices of size @math translates into the search for a set of @math @math -dimensional vectors with minimal coherence. Best Complex Antipodal Spherical Codes (BCASCs) are known to be optimal in terms of coherence. An iterative algorithm for BCASC generation has been recently proposed that tightly approaches the theoretical lower bound on coherence. Unfortunately, the complexity of each iteration depends quadratically on @math and @math . In this work we propose a modification of the algorithm that allows reducing the quadratic complexity to linear on both @math and @math . Numerical evaluation showed that the proposed approach does not worsen the coherence of the resulting BCASCs. On the contrary, an improvement was observed for large @math . The reduction of the computational complexity paves the way for using the BCASCs as CS measurement matrices in problems with large @math . We evaluate the CS performance of the BCASCs for recovering sparse signals. The BCASCs are shown to outperform both complex random matrices and Fourier ensembles as CS measurement matrices, both in terms of coherence and sparse recovery performance, especially for low @math , which is the case of interest in CS.
The construction method first introduced in @cite_81 and further developed and analyzed in @cite_68 exploits the fact that unitary transforms representing any point on the Grassmann manifold can be represented by the exponential of any element of the tangent space at the identity point. For clarity, in the rank one case we deal with in this paper this would mean an exponential of an @math -dimensional vector with only @math nonzero components. This is due to the fact that we pack subspaces of dimension one (lines) on the Grassmannian, which yields a tangent space of dimensionality @math . The core idea of the method is to design a coherent codebook in the tangent space, which yields an optimal non-coherent one on the Grassmannian. Further work exploiting the exponential parametrization of the Grassmannian can be found in @cite_36 , where different rotated lattices are used as initial codes in the tangent space.
@cite_36 @cite_68 @cite_81
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Geometric methods for construction of codes in the Grassmann manifolds are presented. The methods follow the geometric approach to space-time coding for the non-coherent MIMO channel where the code design is interpreted as a packing problem on Grassmann manifolds. The differential structure of the Grassmann manifold provides parametrization with the tangent space at the identity element. Grassmann codes for the non-coherent channel are constructed by mapping suitable subsets of lattices from the tangent space to the Grassmann manifold via the exponential map. As examples, constructions from the rotated Gosset, Barnes-Wall and Leech lattice are presented. Due to the specifics of the mapping, some of the structure is preserved after the mapping to the manifold. The method is further improved by modifying the mapping from the tangent space to the manifold. Ideas for other constructions of Grassmann codes are also presented and discussed. Abstract We describe a method for recovering the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based locally linear embedding, derives from a conceptual framework of local isometry in which the manifold M, viewed as a Riemannian submanifold of the ambient Euclidean space ℝn, is locally isometric to an open, connected subset Θ of Euclidean space ℝd. Because Θ does not have to be convex, this framework is able to handle a significantly wider class of situations than the original ISOMAP algorithm. The theoretical framework revolves around a quadratic form ℋ(f) = ∫M ∥Hf(m)∥dm defined on functions f : M ↦ ℝ. Here Hf denotes the Hessian of f, and ℋ(f) averages the Frobenius norm of the Hessian over M. To define the Hessian, we use orthogonal coordinates on the tangent planes of M. The key observation is that, if M truly is locally isometric to an open, connected subset of ℝd, then ℋ(f) has a (d + 1)-dimensional null space consisting of the constant functions and a d-dimensional space of functions spanned by the original isometric coordinates. Hence, the isometric coordinates can be recovered up to a linear isometry. Our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the Laplacian eigenmaps framework, where we substitute a quadratic form based on the Hessian in place of one based on the Laplacian. Motivated by image perturbation and the geometry of manifolds, we present a novel method combining these two elements. First, we form a tangent space from a set of perturbed images and observe that the tangent space admits a vector space structure. Second, we embed the approximated tangent spaces on a Grassmann manifold and employ a chordal distance as the means for comparing subspaces. The matching process is accelerated using a coarse to fine strategy. Experiments on the FERET database suggest that the proposed method yields excellent results using both holistic and local features. Specifically, on the FERET Dup2 data set, our proposed method achieves 83.8 rank 1 recognition: to our knowledge the currently the best result among all non-trained methods. Evidence is also presented that peak recognition performance is achieved using roughly 100 distinct perturbed images. As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.