Singular Values of Large Matrices Subject to Gaussian Perturbation,
Abstract
Extending the work of Wachter (1978, 1980) and many others, we study the configuration of the singular values (s.v.'s) of an a by b matrix of the form X = M + sigma Z where M is a constant matrix, and the elements of Z are i.i.d., standard Gaussian, in the limit as a and b increase in constant ratio. We put N = a + b and suppose a = alpha N, b = Beta N, with (sigma of order 1 square root of N. Let the empirical distribution of the s.v.'s of X be GN, and let the corresponding moment-generating-function (m.g.f) be gN(t). These are random quantities; their distributions depend only on sigma and the empirical distribution Fn of the s.v.'s of M. We derive a differential equation that governs the evolution of E(gN) as sigma increases. In the limit as N yields infinity we can solve this equation and hence exhibit the limiting (non-random) g itself. This study was motivated by some blood-pressure data collected by a new type of transducer. It suggests a novel way of adjusting large matrices to reduce the effect of additive contamination.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1992
- Accession Number
- ADP007105
Entities
People
- Colin Mallows
- Lorraine Denby