Spatially resolved transcriptomics involves a set of emerging technologies that enable the transcriptomic profiling of tissues with the physical location of expressions. Although a variety of methods have been developed for data integration, most of them are for single-cell RNA-seq datasets without consideration of spatial information. Thus, methods that can integrate spatial transcriptomics data from multiple tissue slides, possibly from multiple individuals, are needed. Here, we present PRECAST, a data integration method for multiple spatial transcriptomics datasets with complex batch effects and/or biological effects between slides. PRECAST unifies spatial factor analysis simultaneously with spatial clustering and embedding alignment, while requiring only partially shared cell/domain clusters across datasets. Using both simulated and four real datasets, we show improved cell/domain detection with outstanding visualization, and the estimated aligned embeddings and cell/domain labels facilitate many downstream analyses. We demonstrate that PRECAST is computationally scalable and applicable to spatial transcriptomics datasets from different platforms.

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bound on the approximation error of deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are of independent interest. Deep Ritz method, deep neural networks, convergence rate analysis. 62G05, 65N12, 65N15, 68T07 COPYRIGHT: © Global Science Press

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU$^3$ network, which is of immense independent interest. PINNs, ReLU$^3$ neural network, B-splines, Rademacher complexity 62G05, 65N12, 65N15, 68T07 COPYRIGHT: © Global Science Press

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the preasymptotic convergence behavior of the randomized Kaczmarz method, and show that the low-frequency error (with respect to the right singular vectors) decays faster during first iterations than the high-frequency error. Under the assumption that the initial error is smooth (e.g. sourcewise representation), the result explains the fast empirical convergence behavior, thereby shedding new insights into the excellent performance of the randomized Kaczmarz method in practice. Further, we propose a simple strategy to stabilize the asymptotic convergence of the iteration by means of variance reduction. We provide extensive numerical experiments to confirm the analysis and to elucidate the behavior of the algorithms.