Holdout cross-validation for large non-Gaussian covariance matrix estimation using Weingarten calculus

Cross-validation is one of the most widely used methods for model selection and evaluation; its efficiency for large covariance matrix estimation appears robust in practice, but little is known about the theoretical behavior of its error. In this paper, we derive the expected Frobenius error of the holdout method, a particular cross-validation procedure that involves a single train and test split, for a generic rotationally invariant multiplicative noise model, therefore extending previous results to non-Gaussian data distributions. Our approach involves using the Weingarten calculus and the Ledoit-P\'ech\'e formula to derive the oracle eigenvalues in the high-dimensional limit. When the population covariance matrix follows an inverse Wishart distribution, we approximate the expected holdout error, first with a linear shrinkage, then with a quadratic shrinkage to approximate the oracle eigenvalues. Under the linear approximation, we find that the optimal train-test split ratio is proportional to the square root of the matrix dimension. Then we compute Monte Carlo simulations of the holdout error for different distributions of the norm of the noise, such as the Gaussian, Student, and Laplace distributions and observe that the quadratic approximation yields a substantial improvement, especially around the optimal train-test split ratio. We also observe that a higher fourth-order moment of the Euclidean norm of the noise vector sharpens the holdout error curve near the optimal split and lowers the ideal train-test ratio, making the choice of the train-test ratio more important when performing the holdout method.

Scale Dependencies and Self-Similarity Through Wavelet Scattering Covariance

We introduce a scattering covariance matrix which provides non-Gaussian models of time-series having stationary increments. A complex wavelet transform computes signal variations at each scale. Dependencies across scales are captured by the joint covariance across time and scales of complex wavelet coefficients and their modulus. This covariance is nearly diagonalized by a second wavelet transform, which defines the scattering covariance. We show that this set of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. This is analyzed for a variety of processes, including fractional Brownian motions, Poisson, multifractal random walks and Hawkes processes. We prove that self-similar processes have a scattering covariance matrix which is scale invariant. This property can be estimated numerically and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering covariance coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.

Explore or exploit? A generic model and an exactly solvable case

Finding a good compromise between the exploitation of known resources and the exploration of unknown, but potentially more profitable choices, is a general problem, which arises in many different scientific disciplines. We propose a stylized model for these exploration-exploitation situations, including population or economic growth, portfolio optimisation, evolutionary dynamics, or the problem of optimal pinning of vortices or dislocations in disordered materials. We find the exact growth rate of this model for tree-like geometries and prove the existence of an optimal migration rate in this case. Numerical simulations in the one-dimensional case confirm the generic existence of an optimum.