The problem of single-tone frequency estimation for a discrete-time real sinusoid in white Gaussian noise is addressed. We first show that the frequency information is embedded in the principal singular vectors of a matrix which stores the observed data with no repeated entry. The technique of weighted least squares is then utilized for finding the frequency from the singular vectors. It is pro...

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras. This method, which proves to be very powerful, can be applied to most algebras and superalgebras, however. It allows: to determine maximal dimensions for a given type of singular vector space, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and...

In several dimension reduction techniques, the original variables are replaced by a smaller number of linear combinations. The coefficients of these linear combinations are typically the elements of the left singular vectors of a random matrix. We derive the asymptotic distribution of the left singular vectors of a random matrix that has a normal limit distribution. This result is then used to ...

Spectral methods, which use information relating to eigenvectors, singular vectors and generalized singular vectors, help us to visualize and summarize sets of pairwise interactions. In this work, we motivate and discuss the use of spectral methods by taking a matrix computation view and applying concepts from applied linear algebra. We show that this unified approach is sufficiently flexible t...

Much recent work in the theoretical computer science, linear algebra, and machine learning has considered matrix decompositions of the following form: given an m×n matrix A, decompose it as a product of three matrices, C, U , and R, where C consists of a small number of columns of A, R consists of a small number of rows of A, and U is a small carefully constructed matrix that guarantees that th...

The aim of this paper is to address computational aspects of the total energy singular vector analysis of atmospheric chemical transport models. We discuss the symmetry of the tangent-linear/adjoint operator for stiff systems. Results for a 3D simulation with real data reveal that the total energy singular vectors depend on the target domain, simulation window, chemical reactions, and meteorolo...

This paper presents a novel approach to compute the rank-deficiency locus of non-square Jacobian matrices. This algorithm is based on the computation of the singular vectors associated to zero singular values of the Jacobian. Examples are provided to illustrate the application of the algorithm. Results are shown for a four degree-of-freedom and a seven degree-of-freedom manipulator.

Here, we will consider one approach for extending the ideas underlying the least-squares algorithm we discussed in class to non-tall matrices. Let A ∈ Rn×d matrix, where both n and d are large, and where rank(A) = k exactly, and let B ∈ Rn×t. Consider the problem minX∈Rn×t‖AX −B‖, where ‖ · ‖ is a unitarily-invariant matrix norm. The solution to this problem is Xopt = AB, and here we consider a...

This paper discusses an efficient technique for computing filtered matrix-vector (mat-vec) products by exploiting the Lanczos algorithm. The goal of the proposed method, which is the same as that of the truncated singular value decomposition (SVD), is to preserve the quality of the resulting mat-vec product in major singular directions of the matrix. Unlike the SVD-based techniques, the propose...

We perform a non-asymptotic analysis on the singular vector distribution under Gaussian noise. In particular, we provide sufficient conditions on a matrix for its first few singular vectors to have near normal distribution. Our result can be used to facilitate the error analysis in linear dimension reduction.