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A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces U = M i=1 S i . The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method.Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.

Abstract. This paper studies the cardinal interpolation operators associated with the general multiquadrics, φα,c(x) = ( x 2 + c 2 ) α , x ∈ R d . These operators take the formwhere Lα,c is a fundamental function formed by integer translates of φα,c which satisfies the interpolatory condition Lα,c(kWe consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter c → ∞. In the univariate case, we consider the norm of the operator Iα,c acting on p spaces as well as prove decay rates for Lα,c using a detailed analysis of the derivatives of its Fourier transform, Lα,c.

A Riesz-basis sequence for L 2 [−π, π] is a strictly increasing sequence X := (x j ) j∈Z in R such that the set of functions  e −i x j (·)  j∈Z is a Riesz basis for L 2 [−π, π]. Given such a sequence and a parameter 0 < h ≤ 1, we consider interpolation of functions g ∈ W k 2 (R) at the set (hx j ) j∈Z via translates of the Gaussian kernel. Existence is shown of an interpolant of the formwhich is continuous and square-integrable on R, and satisfies the interpolatory condition I h X (g)(hx j ) = g(hx j ), j ∈ Z. Moreover, the use of the parameter h gives approximation rates of order h k . Namely, there is a constant independent of g such that ∥I h X (g) − g∥ L 2 (R) ≤ Ch k | g | W k 2 (R) . Interpolation using translates of certain functions other than the Gaussian, so-called regular interpolators, is also considered and shown to exhibit the same approximation rates.

This article pertains to interpolation of Sobolev functions at shrinking lattices hZ d from Lp shift-invariant spaces associated with cardinal functions related to general multiquadrics, φα,c(x) := (|x| 2 + c 2 ) α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, Lp error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h k ) for functions with derivatives of order up to k in Lp, 1 < p < ∞). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.