In the modern world, systems are routinely monitored by multiple sensors. This generates "Big Data" in the form of large collections of time series. On the other hand, scaleinvariant (fractal) systems are of great interest in signal processing since they naturally emerge in several fields of application. In this work, we put forward an algorithm for the statistical identification of high-dimensional self-similarity. In the threefold limit as dimension, sample size and scale go to infinity, the method builds upon spectral clustering applied to large wavelet random matrices to consistently estimate the Hurst modes (exponents) and, hence, characterize high-dimensional self-similarity. Monte Carlo simulations show that the proposed methodology is efficient for realistic sample sizes.