Big-Data Compressive Sensing: Fast, Parallelised and Distributed Algorithms

新兴的大数据时代将带来挑战和机遇。如果处理大数据并从中提取有意义的信息的挑战能够得到解决，那么从这些数据集中提取出的丰富信息将具有高度的信息性，从而彻底改变许多学术领域和行业。从数据中提取有意义的信息的一种有效方法是提出并解决逆问题，这是一个在学术和工业领域广泛存在的大型且重要的数学问题。压缩感知是信息理论的一项新突破，它有可能彻底改变许多领域的数据采集和分析，通过稀疏正则化解决与高数据欠采样相关的逆问题，为解决大数据挑战提供了一条有前途的途径。虽然这样的方法提供了一个严格的理论框架来解决反问题，这必须由快速算法与有效的实现。许多用Matlab和Python编写的研究代码可以解决这些问题，但是缺乏并行化的专业软件包。我们将通过开发SOPT++来填补这一空白，SOPT++是一个公共开源软件包，用于使用稀疏正则化技术解决逆问题，利用压缩感知的理论发展。SOPT++将为大数据实现新的高度并行和分布式凸优化算法。我们的凸优化算法的结构不仅允许计算分布在多节点架构，但内存和存储要求也。此外，在反问题描述中出现的常见测量和稀疏化运算符（在找到解决方案时会重复应用）将通过矢量化或轻量级线程在多核架构（如GPGPU和Xeon Phi协处理器）上高度并行化。这种分层并行化将允许SOPT++在整个现代高性能计算系统中部署。SOPT++将从算法和实现的角度进行精心设计。前者将确保可以考虑各种稀疏正则化问题公式，而后者将确保SOPT++可以无缝地应用于不同的应用领域。预计SOPT++将被应用于解决广泛领域的逆问题，包括磁共振成像，计算机断层扫描，地震成像，计算机视觉，机器学习，无线电干涉测量和宇宙学，仅举几例，允许研究人员将他们的分析扩展到大数据集。SOPT++的广泛吸收将通过提供一个精心设计的专业软件平台来促进，该平台有很好的文档记录，并包含许多教程和示例。此外，我们将应用SOPT++扩散磁共振成像，神经科学的中心模式。高角分辨率弥散MRI（HARDI）的临床应用需要快速采集序列。我们将利用SOPT++算法的正则化功能，从高度欠采样的数据中实现HARDI，其中欠采样是序列加速的关键。

The emerging era of big-data will provide both challenges and opportunities. If the challenge of handling big-data and extracting meaningful information from it can be met, then the wealth of information teased out of such data-sets will be highly informative, revolutionising numerous academic fields and industries. An effective means to tease meaningful information from data is by posing and solving inverse problems, which are a large and important class of mathematical problem experienced in a broad range of academic and industrial domains. Compressive sensing is a recent breakthrough in information theory that has the potential to revolutionise the acquisition and analysis of data in many fields, providing a promising route to addressing the big-data challenge by solving inverse problems associated with high data under-sampling via sparse regularisation. Although such an approach provides a rigorous theoretical framework to solve inverse problems, this must be complemented by fast algorithms with efficient implementations. Many research codes written in Matlab and Python exist to solve these problems, however, a professional software package that is parallelised is lacking. We will fill this void by developing SOPT++, a public open-source software package for solving inverse problems using sparse regularisation techniques, exploiting theoretical developments from compressive sensing. SOPT++ will implement novel highly parallelised and distributed convex optimisation algorithms for big-data. The structure of our convex optimisation algorithms will not only allow computations to be distributed across multi-node architectures, but memory and storage requirements also. Moreover, common measurement and sparsifying operators that appear in descriptions of inverse problems, which are applied repeatedly when finding a solution, will be highly parallelised on many-core architectures, such as GPGPU and Xeon Phi co-processors, through vectorisation or light-weight threads. This tiered parallelisation will allow SOPT++ to be deployed across the full range of modern high performance computing systems. SOPT++ will be designed carefully from both algorithmic and implementation perspectives. The former will ensure a variety of sparse regularisation problem formulations can be considered, while the latter will ensure that SOPT++ can be applied seamlessly to different domains of application. It is anticipated that SOPT++ will be applied to solve inverse problems in a wide range of fields, including magnetic resonance imaging, computed tomography, seismic imaging, computer vision, machine learning, radio interferometry, and cosmology, to name just a few, allowing researchers to scale their analyses up to big data-sets. Wide uptake of SOPT++ will be facilitated by providing a well designed professional software platform that is well documented and contains numerous tutorials and examples. In addition, we will apply SOPT++ to diffusion magnetic resonance imaging, a central modality for neuroscience. Clinical application of high angular resolution diffusion MRI (HARDI) requires fast acquisition sequences. We will leverage the regularisation power of SOPT++ algorithms to enable HARDI from highly under-sampled data, where under-sampling is key to sequence acceleration.