The increasing utilization of mouse models in human neuroscience research places higher demands on computational methods to translate findings from the mouse brain to the human one. In this study, we develop BrainAlign, a self-supervised learning approach, for the whole brain alignment of spatial transcriptomics (ST) between humans and mice. BrainAlign encodes spots and genes simultaneously in two separated shared embedding spaces by a heterogeneous graph neural network. We demonstrate that BrainAlign could integrate cross-species spots into the embedding space and reveal the conserved brain regions supported by ST information, which facilitates the detection of homologous regions between humans and mice. Genomic analysis further presents gene expression connections between humans and mice and reveals similar expression patterns for marker genes. Moreover, BrainAlign can accurately map spatially similar homologous regions or clusters onto a unified spatial structural domain while preserving their relative positions. Comparative transcriptomics of whole brains across species is vital in neuroscience. Here, authors develop a deep learning method, BrainAlign, to align spatial transcriptomics across human and mouse brains. BrainAlign identifies conserved brain regions and uncovers similar patterns for marker genes.Publication:Nature Communications volume 15, Article number: 6302 (30 July 2024)https://doi.org/10.1038/s41467-024-50608-2Author:Biao ZhangSchool of Mathematical Sciences, Fudan University, Shanghai, ChinaShuqin ZhangSchool of Mathematical Sciences, Fudan University, Shanghai, ChinaKey Laboratory of Mathematics for Nonlinear Science, Fudan University, Ministry of Education, Shanghai, ChinaShanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, ChinaEmail：zhangs@fudan.edu.cnShihua ZhangNCMIS, CEMS, RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, ChinaKey Laboratory of Systems Health Science of Zhejiang Province, School of Life Science, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Hangzhou, ChinaEmail：zsh@amss.ac.cn
In this paper we develop and analyze a variational data assimilation method with efficient decoupled iterative numerical algorithms for the Stokes--Darcy equations with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the variational data assimilation into an optimization problem, we establish the existence, uniqueness, and stability of the optimal solution. Based on the weak formulation of the Stokes--Darcy equations, the Lagrange multiplier rule is utilized to derive the first order optimality system for both the continuous and discrete variational data assimilation problems, where the discrete data assimilation is based on a finite element discretization in space and the backward Euler scheme in time. By rescaling the optimality system and then analyzing its corresponding bilinear forms, we prove the optimal finite element convergence rate with special attention paid to recovering uncertainties missed in the optimality system. To solve the discrete optimality system efficiently, three decoupled iterative algorithms are proposed to address the computational cost for both well-conditioned and ill-conditioned variational data assimilation problems, respectively. Finally, numerical results are provided to validate the proposed methods.Publication:SIAM Journal on Scientific Computing Volume 46, Issue 2, July 18, 2023https://doi.org/10.1137/22M1492994Author:XUEJIAN LIDepartment of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409 USAEmail: xlcdt@mst.eduWEI GONGLSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Haidian, Beijing 100190,People's Republic of ChinaEmail: wgong@lsec.cc.ac.cnXIAOMING HEDepartment of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409 USAEmail: hex@mst.eduTAO LINDepartment of Mathematics, Virginia Tech, Blacksburg, VA 24061 USAEmail:tlin@vt.edu
We introduce robust principal component analysis from a data matrix in which the entries of its columns have been corrupted by permutations, termed Unlabeled Principal Component Analysis (UPCA). Using algebraic geometry, we establish that UPCA is a well-defined algebraic problem since we prove that the only matrices of minimal rank that agree with the given data are row-permutations of the ground-truth matrix, arising as the unique solutions of a polynomial system of equations. Further, we propose an efficient two-stage algorithmic pipeline for UPCA suitable for the practically relevant case where only a fraction of the data have been permuted. Stage-I employs outlier-robust PCA methods to estimate the ground-truth column-space. Equipped with the column-space, Stage-II applies recent methods for unlabeled sensing to restore the permuted data. Allowing for missing entries on top of permutations in UPCA leads to the problem of unlabeled matrix completion, for which we derive theory and algorithms of similar flavor. Experiments on synthetic data, face images, educational and medical records reveal the potential of our algorithms for applications such as data privatization and record linkage.Publication:Journal of Machine Learning Research 25 (2. 2024)https://jmlr.org/papers/v25/22-0816.htmlAuthor:Yunzhen YaoSchool of Computer and Communication Sciences，EPFLCH-1015 Lausanne, SwitzerlandEmail：yunzhen.yao@epfl.chLiangzu PengEmail：lpenn@seas.upenn.eduManolis C. TsakirisKey Laboratory for Mathematics MechanizationAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing, 100190, ChinaEmail：manolis@amss.ac.cn
The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of one-dimensional compressible Euler equations is proved in the inviscid limit of compressible Navier-Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in [M. J. Kang and A. F. Vasseur, Invent. Math., 224 (2021), pp. 55--146]. The proof contains two new ingredients: (1) a cut-off technique and the expanding property of rarefaction are used to overcome the errors generated by the viscosity related to inviscid rarefaction; (2) the error terms concerning the interactions between shock and rarefaction are controlled by the compressibility of shock, the decay of derivative of rarefaction, and the separation of shock and rarefaction as time increases.Publication:SIAM Journal on Mathematical Analysis Volume 56, Issue 3,Jun 2024https://doi.org/10.1137/23M156584XAuthor:FEIMIN HUANGAcademy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, ChinaEmail: fhuang@amt.ac.cnWEIQIANG WANGAcademy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, ChinaEmail: wangweiqiang@amss.ac.cnYI WANGAcademy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing,100049, ChinaEmail: wangyi@amss.ac.cn
Learning operators mapping between infinite -dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, we propose an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features. To capture the discontinuities in both the input functions and the output solutions across the interface, IONet divides the entire domain into several separate subdomains according to the interface and uses multiple branch nets and trunk nets. Each branch net extracts latent representations of input functions at a fixed number of sensors on a specific subdomain, and each trunk net is responsible for output solutions on one subdomain. Additionally, tailored physics -informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the training dataset requirement and makes IONet effective without any paired input-output observations inside the computational domain. Extensive numerical studies demonstrate that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy and versatility.Publication:Journal of Computational Physics Volume 514, 1 October 2024https://doi.org/10.1016/j.jcp.2024.113217Author:Sidi WuSchool of Mathematical Sciences, Peking University, Beijing 100871, ChinaLSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaAiqing ZhuLSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaYifa TangLSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaBenzhuo LuLSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaEmail: bzlu@lsec.cc.ac.cn
The GVW algorithm, one of the most important so-called signaturebased algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm. (c) 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.Publication:Journal of Symbolic Computation Volume 127, March–April 2025https://doi.org/10.1016/j.jsc.2024.102370Author:Dong LuSchool of Mathematics, Southwest Jiaotong University, Chengdu 610031, ChinaEmail: donglu@swjtu.edu.cnDingkang WangKLMM, Academy of Mathematics and Systems Science, CAS, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaEmail: dwang@mmrc.iss.ac.cnFanghui XiaoMOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, ChinaEmail: xiaofanghui@hunnu.edu.cnXiaopeng ZhengCollege of Mathematics and Computer Science, Shantou University, Shantou 515821, ChinaEmail: zhengxiaopeng@amss.ac.cn
This paper focuses on multi-block optimization problems over transport polytopes, which underlie various applications including strongly correlated quantum physics and machine learning. Conventional block coordinate descent-type methods for the general multi-block problems store and operate on the matrix variables directly, resulting in formidable expenditure for large-scale settings. On the other hand, optimal transport problems, as a special case, have attracted extensive attention and numerical techniques that waive the use of the full matrices have recently emerged. However, it remains nontrivial to apply these techniques to the multi-block, possibly nonconvex problems with theoretical guarantees. In this work, we leverage the benefits of both sides and develop novel sampling-based block coordinate descent-type methods, which are equipped with either entropy regularization or Kullback Leibler divergence. Each iteration of these methods solves subproblems restricted on the sampled degrees of freedom. Consequently, they involve only sparse matrices, which amounts to considerable complexity reductions. We explicitly characterize the sampling-induced errors and establish convergence and asymptotic properties for the methods equipped with the entropy regularization. Numerical experiments on typical strongly correlated electron systems corroborate their superior scalability over the methods utilizing full matrices. The advantage also enables the first visualization of approximate optimal transport maps between electron positions in three-dimensional contexts.Publication:MATHEMATICS OF COMPUTATION, August 15,2024http://dx.doi.org/10.1090/mcom/3989Author:YUKUAN HUState Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republicof ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, People’s Republic of ChinaEmail: ykhu@lsec.cc.ac.cnMENGYU LIInstitute of Statistics and Big Data, Renmin University of China, Beijing 100872, People’s Republic of ChinaEmail: limengyu516@ruc.edu.cnXIN LIUState Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of ChinaEmail: liuxin@lsec.cc.ac.cnCHENG MENGCenter for Applied Statistics, Institute of Statistics and Big Data, Renmin University of China, Beijing 100872, People’s Republic of ChinaEmail: chengmeng@ruc.edu.cn
Motivation Gene regulatory networks (GRNs) are vital tools for delineating regulatory relationships between transcription factors and their target genes. The boom in computational biology and various biotechnologies has made inferring GRNs from multi-omics data a hot topic. However, when networks are constructed from gene expression data, they often suffer from false-positive problem due to the transitive effects of correlation. The presence of spurious noise edges obscures the real gene interactions, which makes downstream analyses, such as detecting gene function modules and predicting disease-related genes, difficult and inefficient. Therefore, there is an urgent and compelling need to develop network denoising methods to improve the accuracy of GRN inference. Results In this study, we proposed a novel network denoising method named reverse Network Diffusion On Random walks (RENDOR). RENDOR is designed to enhance the accuracy of GRNs afflicted by indirect effects. RENDOR takes noisy networks as input, models higher-order indirect interactions between genes by transitive closure, eliminates false-positive effects using the inverse network diffusion method, and produces refined networks as output. We conducted a comparative assessment of GRN inference accuracy before and after denoising on simulated networks and real GRNs. Our results emphasized that the network derived from RENDOR more accurately and effectively captures gene interactions. This study demonstrates the significance of removing network indirect noise and highlights the effectiveness of the proposed method in enhancing the signal-to-noise ratio of noisy networks. Availability and implementation The R package RENDOR is provided at https://github.com/Wu-Lab/RENDOR and other source code and data are available at https://github.com/Wu-Lab/RENDOR-reproducePublication:Bioinformatics, Volume 40, Issue 7, July 2024https://doi.org/10.1093/bioinformatics/btae435Author:Jiating YuSchool of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, ChinaIAM, MADIS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaJiacheng LengIAM, MADIS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaZhejiang Lab, Hangzhou 311121, ChinaJiacheng LengIAM, MADIS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaDuanchen SunSchool of Mathematics, Shandong University, Jinan 250100, China.E-mail: dcsun@sdu.edu.cnLing-Yun WuIAM, MADIS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaE-mail: lywu@amss.ac.cn
Radar systems typically employ well-designed deterministic signals for target sensing, while integrated sensing and communications (ISAC) systems have to adopt random signals to convey useful information. This paper analyzes the sensing and ISAC performance relying on random signaling in a multi-antenna system. Towards this end, we define a new sensing performance metric, namely, ergodic linear minimum mean square error (ELMMSE), which characterizes the estimation error averaged over random ISAC signals. Then, we investigate a data-dependent precoding (DDP) scheme to minimize the ELMMSE in sensing-only scenarios, which attains the optimized performance at the cost of high implementation overhead. To reduce the cost, we present an alternative data-independent precoding (DIP) scheme by stochastic gradient projection (SGP). Moreover, we shed light on the optimal structures of both sensing-only DDP and DIP precoders. As a further step, we extend the proposed DDP and DIP approaches to ISAC scenarios, which are solved via a tailored penalty-based alternating optimization algorithm. Our numerical results demonstrate that the proposed DDP and DIP methods achieve substantial performance gains over conventional ISAC signaling schemes that treat the signal sample covariance matrix as deterministic, which proves that random ISAC signals deserve dedicated precoding designs.Publication:IEEE Transactions on Signal Processing ( Volume: 72) 12 July 2024http://dx.doi.org/10.1109/TSP.2024.3427373Author:Shihang LuSchool of System Design and Intelligent Manufacturing (SDIM), Southern University of Science and Technology, Shenzhen 518055, ChinaEmail: lush2021@mail. sustech.edu.cnFuwang DongSchool of System Design and Intelligent Manufacturing (SDIM), Southern University of Science and Technology, Shenzhen 518055, ChinaEmail: dongfw@sustech.edu.cnFan Liu National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, ChinaEmail: f.liu@ieee.orgShi JinNational Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, ChinaEmail: jinshi@seu.edu.cnYifeng XiongSchool of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, ChinaEmail: yifengxiong@bupt.edu.cnJie XuSchool of Science and Engineering (SSE) and the Future Network of Intelligence Institute (FNii), The Chinese University of Hong Kong (Shenzhen), Shenzhen 518172, ChinaEmail:xujie@cuhk.edu.cnYa-Feng LiuState Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/ Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaEmail:yafliu@ lsec.cc.ac.cn
In this article, the output regulation for a 1-D wave equation where the disturbances generated from an unknown finite-dimensional exosystem appear in all possible channels is studied. The system is first transformed into a new system for which the disturbance appears in tracking error only. An adaptive observer approach is adopted in investigation to estimate all possible unknown frequencies that have entered into a transformed new system. By the estimates of the unknown frequencies, we are able to design a tracking-error-based feedback control to achieve output regulation and disturbance rejection for this partial differential equations (PDEs) in two different cases. In the first case, the derivative of the tracking error is allowed to be used in the control design, which leads to the exponential convergence of the tracking error. In the second case, the tracking error is solely used and the asymptotic convergence is achieved. A remarkable characteristic of the problem lies in the fact that the control operator is unbounded and is noncollocated with the regulated output, which represents a difficult situation for output regulation on PDEs. The proposed approach is potentially applicable to other PDEs.Publication:IEEE Transactions on Automatic Control ( Volume: 69, Issue: 5, May 2024) http://dx.doi.org/10.1109/TAC.2023.3303340Author:Ren-Xi ZhaoKey Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaEmail: zhaorenxi@amss.ac.cnBao-Zhu GuoSchool of Mathematics and Physics, North China Electric Power University, Beijing 102206, ChinaKey Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, ChinaEmail: bzguo@iss.ac.cn
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