  In this paper, we construct the wavelet eigenvalue regression methodology in
high dimensions. We assume that possibly non-Gaussian, finite-variance
$p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$)
fractional stochastic process with non-canonical scaling coordinates and in the
presence of additive high-dimensional noise. The measurements are correlated
both time-wise and between rows. Building upon the asymptotic and large scale
properties of wavelet random matrices in high dimensions, the wavelet
eigenvalue regression is shown to be consistent and, under additional
assumptions, asymptotically Gaussian in the estimation of the fractal structure
of the system. We further construct a consistent estimator of the effective
dimension $r$ of the system that significantly increases the robustness of the
methodology. The estimation performance over finite samples is studied by means
of simulations.
