General-Purpose Bayesian Tensor Learning With Automatic Rank Determination and Uncertainty Quantification

A major challenge in many machine learning tasks is that the model expressive power depends on model size. Low-rank tensor methods are an efficient tool for handling the curse of dimensionality in many large-scale machine learning models. The major challenges in training a tensor learning model include how to process the high-volume data, how to determine the tensor rank automatically, and how to estimate the uncertainty of the results. While existing tensor learning focuses on a specific task, this paper proposes a generic Bayesian framework that can be employed to solve a broad class of tensor learning problems such as tensor completion, tensor regression, and tensorized neural networks. We develop a low-rank tensor prior for automatic rank determination in nonlinear problems. Our method is implemented with both stochastic gradient Hamiltonian Monte Carlo (SGHMC) and Stein Variational Gradient Descent (SVGD). We compare the automatic rank determination and uncertainty quantification of these two solvers. We demonstrate that our proposed method can determine the tensor rank automatically and can quantify the uncertainty of the obtained results. We validate our framework on tensor completion tasks and tensorized neural network training tasks.

ScLRTC: imputation for single-cell RNA-seq data via low-rank tensor completion

Background With single-cell RNA sequencing (scRNA-seq) methods, gene expression patterns at the single-cell resolution can be revealed. But as impacted by current technical defects, dropout events in scRNA-seq lead to missing data and noise in the gene-cell expression matrix and adversely affect downstream analyses. Accordingly, the true gene expression level should be recovered before the downstream analysis is carried out. Results In this paper, a novel low-rank tensor completion-based method, termed as scLRTC, is proposed to impute the dropout entries of a given scRNA-seq expression. It initially exploits the similarity of single cells to build a third-order low-rank tensor and employs the tensor decomposition to denoise the data. Subsequently, it reconstructs the cell expression by adopting the low-rank tensor completion algorithm, which can restore the gene-to-gene and cell-to-cell correlations. ScLRTC is compared with other state-of-the-art methods on simulated datasets and real scRNA-seq datasets with different data sizes. Specific to simulated datasets, scLRTC outperforms other methods in imputing the dropouts closest to the original expression values, which is assessed by both the sum of squared error (SSE) and Pearson correlation coefficient (PCC). In terms of real datasets, scLRTC achieves the most accurate cell classification results in spite of the choice of different clustering methods (e.g., SC3 or t-SNE followed by K-means), which is evaluated by using adjusted rand index (ARI) and normalized mutual information (NMI). Lastly, scLRTC is demonstrated to be also effective in cell visualization and in inferring cell lineage trajectories. Conclusions a novel low-rank tensor completion-based method scLRTC gave imputation results better than the state-of-the-art tools. Source code of scLRTC can be accessed at https://github.com/jianghuaijie/scLRTC.