A fast non-negative latent factor (FNLF) model adopts a single latent factor-dependent, non-negative, multiplicative and momentum-incorporated update (SLF-NM2U) algorithm, which can ensure fast convergence on a high-dimensional and sparse (HiDS) matrix according to empirical studies in prior researches. However, it is crucial to investigate the theoretical proof regarding the reason why incorporation of a generalized momentum method into an SLF-NM2U algorithm can ensure the fast convergence of an FNLF model, which has not been addressed in previous work. Therefore, this paper aims to unveil how a generalized momentum method improves the convergence rate of an FNLF model in the discrete time case by combining physical analysis. The FNLF model is superior to the NLF model in terms of the convergence rate and the prediction accuracy of missing data. This conclusion is obtained by empirically research on the HiDS matrix in industrial applications, which also provides empirical basis for theoretical proof.

Our first goal in this paper is to investigate stationarization and asymptotic decorrelation for fractional Brownian motion (fBm) using wavelet packet transform. The wavelet packets are generated by the Nth-order Daubechies scaling function and wavelet. To decorrelate the wavelet packet coefficients asymptotically, we take two strategies: fix N and let absolute scale difference or absolute difference between time shifts get large; and fix scale level and time shift and let N get large. Our second goal is to present the asymptotic properties of the entropy-like cost functional and denoising cost functional and their impact on the selection of best wavelet packet bases when used for fBm plus or not plus independent Gaussian white noise.

The asymptotic decorrelation of the discrete wavelet transform for long-memory processes (such as the fractionally differenced (FD) process, the autoregressive fractionally integrated moving average (ARFIMA) process, and the Gegenbauer autoregressive moving average (GARMA) process) as well as the statistical inference techniques based on this property, have received much attention nowadays. In this paper, we investigate the asymptotic decorrelation property of the discrete wavelet packet transform (DWPT) for two classes of discrete-time long-memory processes containing the ARFIMA and the GARMA processes. Especially, we prove theoretically that the covariance across between-packet DWPT coefficients decays hyperbolically or exponentially fast as the width of the underlying Daubechies scaling and wavelet filters to generate the DWPT gets large. Meanwhile, we show that the covariance between within-packet DWPT coefficients converges hyperbolically fast to its corresponding counterpart when the underlying scaling and wavelet filters to generate the DWPT are the Shannon's ideal low- and high-pass filters.