In this paper we investigate least squares matching problems, extending the methods of our earlier paper 1] in such a way as to make them applicable to problems involving the sets of points that are so complex that approximate answers are of interest. These problems are formulated in terms of continuous descent equations, and lower bounds on the quality of the best match are obtained in terms o...

The notion of a low rank update arises in many important applications. This paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values. The setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated. Precise and easy-to-check necessary and sufficient ...

Upper and lower bounds on both the largest and smallest singular values of a tournament matrix M of order n are obtained. For most values of n, the matrices M for which equality holds are characterized.

The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on the absolute differences between approximate eigenvalues (singular values) and the true eigenvalues (singular values) of a matrix. These bounds may be bad news for small eigenvalues (singular values), which thereby suffer worse relative uncertainty than large ones. However, there are...

We consider recovery of low-rank matrices from noisy data by shrinkage of singular values, in which a single, univariate nonlinearity is applied to each of the empirical singular values. We adopt an asymptotic framework, in which the matrix size is much larger than the rank of the signal matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. For a variety of...

2 has nonzero entries only on its diagonal and first superdiagonal ) Compute orthogonal matrices P and Q such that Σ = P BQ is diagonal and nonnegat i 2 2 2 T 2 ive. The diagonal entries σ of Σ are the singular values of A . We will take them to be sorted in decreasing order: σ ≥ σ . The columns of Q= Q Q are the right singular vec i i + 1 1 2 t 1 2 ors, and the columns of P= P P are the left s...

The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect rega...

Let A and B be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of p does the majorisation σ(BpAp) ≺w σ((BA) p) hold, and for which values its reversed inequality σ(BpAp) ≻w σ((BA) p).

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on R that is supported in a relatively small ball and linear functionals are uniformly bounded in Lp for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1±c √ n/N ...

Every transcendental meromorphic function f in the plane which has only three critical values satisfies lim inf r→∞ T (r, f) log r ≥ √ 3