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Sequential monitoring of high-dimensional nonlinear time series is studiedfor a projection of the second-moment matrix, a problem interesting in its ownright and specifically arising in finance and deep learning. Open-end as wellas closed-end monitoring is studied under mild assumptions on the trainingsample and the observations of the monitoring period. Asymptotics is based onGaussian approximations of projected partial sums allowing for an estimatedprojection vector. Estimation is studied both for classicalnon-$\ell_0$-sparsity as well as under sparsity. For the case that the optimalprojection depends on the unknown covariance matrix, hard- and soft-thresholdedestimators are studied. Applications in finance and training of deep neuralnetworks are discussed. The proposed detectors typically allow to reducedramatically the required computational costs as illustrated by monitoringsynthetic data.

Online Detection of Changes in Moment-Based Projections: When to Retrain Deep Learners or Update Portfolios?