21 research outputs found
Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models
This paper derives central limit theorems (CLTs) for general linear spectral
statistics (LSS) of three important multi-spiked Hermitian random matrix
ensembles. The first is the most common spiked scenario, proposed by Johnstone,
which is a central Wishart ensemble with fixed-rank perturbation of the
identity matrix, the second is a non-central Wishart ensemble with fixed-rank
noncentrality parameter, and the third is a similarly defined non-central
ensemble. These CLT results generalize our recent work to account for multiple
spikes, which is the most common scenario met in practice. The generalization
is non-trivial, as it now requires dealing with hypergeometric functions of
matrix arguments. To facilitate our analysis, for a broad class of such
functions, we first generalize a recent result of Onatski to present new
contour integral representations, which are particularly suitable for computing
large-dimensional properties of spiked matrix ensembles. Armed with such
representations, our CLT formulas are derived for each of the three spiked
models of interest by employing the Coulomb fluid method from random matrix
theory along with saddlepoint techniques. We find that for each matrix model,
and for general LSS, the individual spikes contribute additively to yield a
correction term to the asymptotic mean of the linear statistic, which we
specify explicitly, whilst having no effect on the leading order terms of the
mean or variance
Testing in high-dimensional spiked models
We consider the five classes of multivariate statistical problems identified
by James (1964), which together cover much of classical multivariate analysis,
plus a simpler limiting case, symmetric matrix denoising. Each of James'
problems involves the eigenvalues of where and are
proportional to high dimensional Wishart matrices. Under the null hypothesis,
both Wisharts are central with identity covariance. Under the alternative, the
non-centrality or the covariance parameter of has a single eigenvalue, a
spike, that stands alone. When the spike is smaller than a case-specific phase
transition threshold, none of the sample eigenvalues separate from the bulk,
making the testing problem challenging. Using a unified strategy for the six
cases, we show that the log likelihood ratio processes parameterized by the
value of the sub-critical spike converge to Gaussian processes with logarithmic
correlation. We then derive asymptotic power envelopes for tests for the
presence of a spike
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Detection of weak signals in high-dimensional complex-valued data
This paper considers the problem of detecting a few signals in
high-dimensional complex-valued Gaussian data satisfying Johnstone's (2001)
\textit{spiked covariance model}. We focus on the difficult case where signals
are weak in the sense that the sizes of the corresponding covariance spikes are
below the \textit{phase transition threshold} studied in Baik et al (2005). We
derive a simple analytical expression for the maximal possible asymptotic
probability of correct detection holding the asymptotic probability of false
detection fixed. To accomplish this derivation, we establish what we believe to
be a new formula for the \textit{% Harish-Chandra/Itzykson-Zuber (HCIZ)
integral} \int_{\mathcal{U}(p)}e^{\tr(AGBG^{-1})}dG , where has a
deficient rank . The formula links the HCIZ integral over to an HCIZ integral over a potentially much smaller unitary group
. We show that the formula generalizes to the integrals over
orthogonal and symplectic groups. In the most general form, it expresses the
hypergeometric function of two matrix
arguments as a repeated contour integral of the hypergeometric function
of two matrix arguments