The duality diagram in data analysis: Examples of modern applications

Today's data-heavy research environment requires the integration of different sources of information into structured data sets that can not be analyzed as simple matrices. We introduce an old technique, known in the European data analyses circles as the Duality Diagram Approach, put to new uses through the use of a variety of metrics and ways of combining different diagrams together. This issue of the Annals of Applied Statistics contains contemporary examples of how this approach provides solutions to hard problems in data integration. We present here the genesis of the technique and how it can be seen as a precursor of the modern kernel based approaches.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS408 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Perturbation splitting for more accurate eigenvalues

Let $T$ be a symmetric tridiagonal matrix with entries and eigenvalues of different magnitudes. For some $T$, small entrywise relative perturbations induce small errors in the eigenvalues, independently of the size of the entries of the matrix; this is certainly true when the perturbed matrix can be written as $\widetilde{T}=X^{T}TX$ with small $||X^{T}X-I||$. Even if it is not possible to express in this way the perturbations in every entry of $T$, much can be gained by doing so for as many as possible entries of larger magnitude. We propose a technique which consists of splitting multiplicative and additive perturbations to produce new error bounds which, for some matrices, are much sharper than the usual ones. Such bounds may be useful in the development of improved software for the tridiagonal eigenvalue problem, and we describe their role in the context of a mixed precision bisection-like procedure. Using the very same idea of splitting perturbations (multiplicative and additive), we show that when $T$ defines well its eigenvalues, the numerical values of the pivots in the usual decomposition $T-\lambda I=LDL^{T}$ may be used to compute approximations with high relative precision.Fundação para a Ciência e Tecnologia (FCT) - POCI 201

A new perturbation bound for the LDU factorization of diagonally dominant matrices

This work introduces a new perturbation bound for the L factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields L and U factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the D and U factors in [F. M. Dopico and P. Koev, Numer. Math., 119 (2011), pp. 337– 371] establish that if diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce tiny relative normwise variations of L and U and tiny relative entrywise variations of D when column diagonal dominance pivoting is used. These results will allow us to prove in a follow-up work that such perturbations also lead to strong perturbation bounds for many other problems involving diagonally dominant matrices.Research supported in part by Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542.Publicad

RELATIVE PERTURBATION THEORY FOR DIAGONALLY DOMINANT MATRICES

Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well conditioned L factor. We then establish relative perturbation bounds for the inverse that are entrywise and independent of the condition number. This allows us to also present relative perturbation bounds for the linear system Ax=b that are independent of the condition number. Lastly, we continue the work of Ye to provide relative perturbation bounds for the eigenvalues of symmetric indefinite matrices and non-symmetric matrices

Computing the singular value decomposition with high relative accuracy

AbstractWe analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which in general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as finite element problems and quantum mechanics, it is the smallest singular values that have physical meaning, and should be determined accurately by the data. Many recent papers have identified special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite different, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD of G to high relative accuracy provided we can accurately factor G=XDYT where D is diagonal and X and Y are any well-conditioned matrices; furthermore, the LDU factorization frequently does the job. We provide many examples of matrix classes permitting such an LDU decomposition

Fast Macroscopic Forcing Method

The macroscopic forcing method (MFM) of Mani and Park and similar methods for obtaining turbulence closure operators, such as the Green's function-based approach of Hamba, recover reduced solution operators from repeated direct numerical simulations (DNS). MFM has been used to quantify RANS-like operators for homogeneous isotropic turbulence and turbulent channel flows. Standard algorithms for MFM force each coarse-scale degree of freedom (i.e., degree of freedom in the RANS space) and conduct a corresponding fine-scale simulation (i.e., DNS), which is expensive. We combine this method with an approach recently proposed by Sch\"afer and Owhadi (2023) to recover elliptic integral operators from a polylogarithmic number of matrix-vector products. The resulting Fast MFM introduced in this work applies sparse reconstruction to expose local features in the closure operator and reconstructs this coarse-grained differential operator in only a few matrix-vector products and correspondingly, a few MFM simulations. For flows with significant nonlocality, the algorithm first "peels" long-range effects with dense matrix-vector products to expose a local operator. We demonstrate the algorithm's performance for scalar transport in a laminar channel flow and momentum transport in a turbulent one. For these, we recover eddy diffusivity operators at 1% of the cost of computing the exact operator via a brute-force approach for the laminar channel flow problem and 13% for the turbulent one. We observe that we can reconstruct these operators with an increase in accuracy by about a factor of 100 over randomized low-rank methods. We glean that for problems in which the RANS space is reducible to one dimension, eddy diffusivity and eddy viscosity operators can be reconstructed with reasonable accuracy using only a few simulations, regardless of simulation resolution or degrees of freedom.Comment: 16 pages, 10 figures. S. H. Bryngelson and F. Sch\"afer contributed equally to this wor