Low-rank approximation of images via singular value decomposition is well-received in the era of big data. However, singular value decomposition (SVD) is only for order-two data, i.e., matrices. It is necessary to flatten a higher order input into a matrix or break it into a series of order-two slices to tackle higher order data such as multispectral images and videos with the SVD. Higher order singular value decomposition (HOSVD) extends the SVD and can approximate higher order data using sums of a few rank-one components. We consider the problem of generalizing HOSVD over a finite dimensional commutative algebra. This algebra, referred to as a t-algebra, generalizes the field of complex numbers. The elements of the algebra, called t-scalars, are fix-sized arrays of complex numbers. One can generalize matrices and tensors over t-scalars and then extend many canonical matrix and tensor algorithms, including HOSVD, to obtain higher-performance versions. The generalization of HOSVD is called THOSVD. Its performance of approximating multi-way data can be further improved by an alternating algorithm. THOSVD also unifies a wide range of principal component analysis algorithms.To exploit the potential of generalized algorithms using tscalars for approximating images, we use a pixel neighborhood strategy to convert each pixel to a "deeper-order" tscalar. Experiments on publicly available images show that the generalized algorithm over t-scalars, namely THOSVD, compares favorably with its canonical counterparts.

Precipitation type is a key parameter used for better retrieval of precipitation characteristics as well as to understand the cloud-convection-precipitation coupling processes. Ice crystals and water droplets inherently exhibit different characteristics in different precipitation regimes (e.g., convection, stratiform), which reflect on satellite remote sensing measurements that help us distinguish them. The Global Precipitation Measurement (GPM) Core Observatory’s Microwave Imager (GMI) and Dual-Frequency Precipitation Radar (DPR) together provide ample information on global precipitation characteristics. As an active sensor, DPR provides an accurate precipitation type assignment, while passive sensors like GMI are traditionally only used for empirical understanding of precipitation regimes. Using collocated precipitation type flags from DPR as the “truth”, this paper employs machine learning (ML) models to train and test the predictability and accuracy of using passive GMI-only observations together with ancillary information from reanalysis and GMI surface emissivity retrieval products. Out of six ML models, four simple ones (Support Vector Machine, Neural Network, Random Forest, and Gradient Boosting) and the 1-D convolutional neural network (CNN) model are identified to produce 90% - 94% prediction accuracy globally for 5 types of precipitation (convective, stratiform, mixture, no precipitation, and other precipitation), which is much more robust than previous similar effort. One novelty of this work is to introduce data augmentation (subsampling and bootstrapping) to handle extremely unbalanced samples in each category. Careful evaluation of Impact matrices demonstrate that polarization difference (PD) and surface emissivity at high-frequency channels dominate the decision process, which are consistent with the physical understanding of polarized microwave radiative transfer over different surface types, as well as in snow and liquid clouds with different microphysical properties. Furthermore, the view-angle dependency artifact that DPR precipitation flag bears with does not propagate into the conical-viewing GMI retrievals. This work provides a new and promising way for future physics-based ML retrieval algorithm development.