- #1
Squatchmichae
- 12
- 0
Hi All,
It's been years since I have re-visited PF.
I have an interesting problem today. I arises in a physical hypothesis testing problem:
Problem Statement: what's the density function for the sum of singular values (trace of the singular value matrix) for a square, Gaussian matrix?
My Approach thus far:
Suppose A is an NxN matrix whose elements are Gaussian R.V.s Under the assumption of mutual independence:
|| A ||[itex]^{2}_{F}[/itex] = || D ||[itex]^{2}_{F}[/itex]
where D is the eigenvalue value matrix of A. Because || A ||[itex]^{2}_{F}[/itex] is a sum of the IID Gaussian squares along the diagonal, it it Chi-square:
|| A ||[itex]^{2}_{F}[/itex] ~ [itex]\chi[/itex][itex]^{2}[/itex]
where the effective degrees of freedom are N[itex]^{2}[/itex], since the Frobenius norm-squared is tr( A[itex]^{T}[/itex]A ), and means we are summing inner products along a diagonal, making N[itex]^{2}[/itex] terms contributing to the sum in total.
I would like to find the seemingly related distribution for tr( S )--that is, the trace of the singular value matrix, S, which has square-rooted eigenvalues.
A naive approach would be to suggest that tr( S ) is a sum of Chi random variables, assuming each singular value is separately also a [itex]\chi[/itex][itex]^{2}[/itex] by Cochran's Theorem.
I am not a mathematician or EE, so if you have insight, I probably will get lost if said insight involves Lie algebras!
Thanks for reading this, and any insight you have is appreciated. This distribution is associated with a detection problem at Los Alamos, and is currently considered quite important for establishing false alarm and detection probabilities of the associated detection statistic.
It's been years since I have re-visited PF.
I have an interesting problem today. I arises in a physical hypothesis testing problem:
Problem Statement: what's the density function for the sum of singular values (trace of the singular value matrix) for a square, Gaussian matrix?
My Approach thus far:
Suppose A is an NxN matrix whose elements are Gaussian R.V.s Under the assumption of mutual independence:
|| A ||[itex]^{2}_{F}[/itex] = || D ||[itex]^{2}_{F}[/itex]
where D is the eigenvalue value matrix of A. Because || A ||[itex]^{2}_{F}[/itex] is a sum of the IID Gaussian squares along the diagonal, it it Chi-square:
|| A ||[itex]^{2}_{F}[/itex] ~ [itex]\chi[/itex][itex]^{2}[/itex]
where the effective degrees of freedom are N[itex]^{2}[/itex], since the Frobenius norm-squared is tr( A[itex]^{T}[/itex]A ), and means we are summing inner products along a diagonal, making N[itex]^{2}[/itex] terms contributing to the sum in total.
I would like to find the seemingly related distribution for tr( S )--that is, the trace of the singular value matrix, S, which has square-rooted eigenvalues.
A naive approach would be to suggest that tr( S ) is a sum of Chi random variables, assuming each singular value is separately also a [itex]\chi[/itex][itex]^{2}[/itex] by Cochran's Theorem.
I am not a mathematician or EE, so if you have insight, I probably will get lost if said insight involves Lie algebras!
Thanks for reading this, and any insight you have is appreciated. This distribution is associated with a detection problem at Los Alamos, and is currently considered quite important for establishing false alarm and detection probabilities of the associated detection statistic.