We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction of Le et al. (2013) as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept ...

The 2-D discrete phase retrieval problem is to reconstruct an image defined at integer coordinates and having known finite spatial extent from the magnitude of its discrete Fourier transform. Most methods for solving this problem are iterative but not POCS, and they tend to stagnate. Recently we developed a new approach that unwrapped the 2-D problem into a 1-D problem with bands of zeros in it...

We derive an explicit count for the number of singular n × n Hankel (Toeplitz) matrices whose entries range over a finite field with q elements by observing the execution of the Berlekamp/ Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n × n Toeplitz matrices with ...

Abstract. We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1+i2+· · ·+ik)). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg’s original formula amounts to the evaluation of all Ha...

Given a linear system Ax = b and some vector x̃, the backward error characterizes the smallest relative perturbation of the input data such that x̃ is a solution of the perturbed system. If the input matrix has some structure such as being symmetric or Toeplitz, perturbations of the input matrix may be restricted to perturbations within the same class of structured matrices. For normwise perturba...

The authors of the title proved in [2] an elegant identity expressing a Toeplitz determinant in terms of the Fredholm determinant of an infinite matrix which (although not described as such) is the product of two Hankel matrices. The proof used combinatorial theory, in particular a theorem of Gessel expressing a Toeplitz determinant as a sum over partitions of products of Schur functions. The p...