Let V be an r-dimensional $${\mathbb{F}_{{q^n}}}$$ -vector space. We call $${\mathbb{F}_{q}}$$ -subspace U of h-scattered if meets the h-dimensional -subspaces in dimension at most h. In 2000 Blokhuis and Lavrauw proved that $${\dim_{\mathbb{F}_{q}}}$$ ? rn/2 when is 1-scattered. Sub-spaces attaining this bound have been investigated intensively because their relations with projective two-weigh...

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if E0 does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace ...

Results are proved indicating that the Veronese map vd often increases independence of both sets of points and sets of subspaces. For example, any d + 1 Veronesean points of degree d are independent. Similarly, the dth power map on the space of linear forms of a polynomial algebra also often increases independence of both sets of points and sets of subspaces. These ideas produce d+ 1-independen...

The notion of the central projection in spaces of pencils is generalized and new concepts of projections are introduced. The category of projectivities with segment subspaces as objects arises. These general projectivities are collineations given by linear maps. Properties of pencils of segment subspaces and projections between segments are investigated. Classical projections of lines onto penc...

This paper is a survey about recent results on sparse representations and optimal models in different settings. Given a set of functions, we show that there exists an optimal collection of subspaces minimizing the sum of the square of the distances between each function and its closest subspace in the collection. Further, this collection of subspaces gives the best sparse representation for the...

Subspace-based signal processing traditionally focuses on problems involving a few subspaces. Recently, a number of problems in different application areas have emerged that involve a significantly larger number of subspaces relative to the ambient dimension. It becomes imperative in such settings to first identify a smaller set of active subspaces that contribute to the observation before furt...

High-dimensional data often lie in low-dimensional subspaces corresponding to different classes they belong to. Finding sparse representations of data points in a dictionary built from the collection of data helps to uncover the low-dimensional subspaces and, as a result, address important problems such as compression, clustering, classification, subset selection and more. However, an important...