Several statistics-based detectors, based on unimodal matrix models, for determining the number of sources in a field are designed. A new variance-ratio statistic is proposed, and its asymptotic distribution is analyzed. The variance-ratio detector is shown to outperform the alternatives. It is shown that further improvements are achievable via optimally selected rotations. Numerical experiments demonstrate the performance gains of our detection methods over the baseline approach.

We develop a general theory for the goodness-of-fit test to non-linear models. In particular, we assume that the observations are noisy samples of a submanifold defined by a sufficiently smooth non-linear map. The observation noise is additive Gaussian. Our main result shows that the “residual” of the model fit, by solving a non-linear least-square problem, follows a (possibly noncentral) χ2distribution. The parameters of the χ2distribution are related to the model order and dimension of the problem. We further present a method to select the model orders sequentially. We demonstrate the broad application of the general theory in machine learning and signal processing, including determining the rank of low-rank (possibly complex-valued) matrices and tensors from noisy, partial, or indirect observations, determining the number of sources in signal demixing, and potential applications in determining the number of hidden nodes in neural networks.

We consider the problem of determining the rank in the low-rank matrix completion. We propose a statistical model for noisy observation. It is important for many existing algorithms and sometimes has practical meanings. We construct a test statistics for the low rank approximation problem. Under this model, we derive the distribution of the test statistics. By applying the test statistics, we propose a sequential rank test procedure to determine the rank with statistical inference. In the numerical section, we illustrate our theoretical results and give examples of our proposed rank test procedure.