Convex Combination of Constraint Vectors for Set-membership Affine Projection Algorithms

Set-membership affine projection (SM-AP) adaptive filters have been increasingly employed in the context of online data-selective learning. A key aspect for their good performance in terms of both convergence speed and steady-state mean-squared error is the choice of the so-called constraint vector. Optimal constraint vectors were recently proposed relying on convex optimization tools, which might some- times lead to prohibitive computational burden. This paper proposes a convex combination of simpler constraint vectors whose performance approaches the optimal solution closely, utilizing much fewer computations. Some illustrative examples confirm that the sub-optimal solution follows the accomplishments of the optimal one

Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensin

Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.Comment: Full version of L. Rademacher, S. Vempala: Dispersion of Mass and the Complexity of Randomized Geometric Algorithms. Proc. 47th IEEE Annual Symp. on Found. of Comp. Sci. (2006). A version of it to appear in Advances in Mathematic

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding