  We present a comparison between various algorithms of inference of covariance
and precision matrices in small datasets of real vectors, of the typical length
and dimension of human brain activity time series retrieved by functional
Magnetic Resonance Imaging (fMRI). Assuming a Gaussian model underlying the
neural activity, the problem consists in denoising the empirically observed
matrices in order to obtain a better estimator of the true precision and
covariance matrices. We consider several standard noise-cleaning algorithms and
compare them on two types of datasets. The first type are time series of fMRI
brain activity of human subjects at rest. The second type are synthetic time
series sampled from a generative Gaussian model of which we can vary the
fraction of dimensions per sample q = N/T and the strength of off-diagonal
correlations. The reliability of each algorithm is assessed in terms of
test-set likelihood and, in the case of synthetic data, of the distance from
the true precision matrix. We observe that the so called Optimal Rotationally
Invariant Estimator, based on Random Matrix Theory, leads to a significantly
lower distance from the true precision matrix in synthetic data, and higher
test likelihood in natural fMRI data. We propose a variant of the Optimal
Rotationally Invariant Estimator in which one of its parameters is optimised by
cross-validation. In the severe undersampling regime (large q) typical of fMRI
series, it outperforms all the other estimators. We furthermore propose a
simple algorithm based on an iterative likelihood gradient ascent, providing an
accurate estimation for weakly correlated datasets.
