Constrained low rank matrix recovery: from efficient algorithms to brain network imaging

建议的研究涉及开发高效的计算算法来将矩阵数据分解成低阶表示，其中因子满足几个约束。有两个场景是有意义的：1)整个数据矩阵是已知的，目标是将数据分解成揭示底层数据结构的解释性组件。2)数据只是部分观察到的，分解也用于恢复未观察到的数据。1)通常用于去除数据中的噪声(例如，清理图像)或将数据分解为几个不同的组件(例如，用于分离录音中的不同扬声器)，而2)例如，在线零售商使用推荐系统根据以前的购买或医学成像来推荐产品，在这些情况下，我们想要减少患者的辐射暴露。一般而言，更好的矩阵分解技术将使我们能够a)更快、更安全、更便宜地获取数据；b)以更高的分辨率获取数据；以及c)从有意义的潜在因素方面找到更好的数据解释。这些改进将通过开发和利用更好的数据模型来实现。特别是，我们将开发能够利用一系列非凸约束的模型和算法，例如稀疏性、光滑性、邻接性、块结构和低阶。这些约束中的每一个以前都被单独利用过，而且每个约束都被发现能够捕获不同的数据特征。例如，稀疏数据模型对数据恢复的有用性已经引起了人们的极大关注(例如，在医学成像中)，而对于矩阵数据，低阶模型现在正变得广泛使用(例如，在推荐系统中)。我们在这里建立在我们之前关于数据的有效恢复和因式分解的工作的基础上，并开发可以利用不止一个这些约束的算法。我们将开发使我们能够有效地联合利用几个约束的方法，而不是强加稀疏性或低排名。这将对许多应用程序产生变革性的影响，在这些应用程序中，可以使用几个约束来捕获数据结构，但每个单一约束都不足以提供实质性的好处。例如，在射电天文学中，由于无法监测天空的某些区域，或者由于无法实际存储现代射电观测站产生的大量数据，观测可能会丢失。这些数据中的结构只能被任何一个约束条件部分捕获，因此不能用目前的方法完全恢复。在这里，我们特别受到我们目前在功能脑成像方面的工作的启发。磁共振成像(MRI)技术可以用来测量人在休息时的大脑活动。这种类型的数据提供了对活着的人脑中信息处理机制的关键见解，也可以用来揭示许多大脑疾病背后的神经机制。矩阵分解方法已经被用作分析这些数据集的主要工具之一。目前的方法构建了时空数据矩阵的低阶近似，描述了展示联合神经活动的空间区域，从而揭示了几个不同的连接大脑区域的网络。我们的新方法将显著改进现有方法。先进的数据模型将使我们能够更好地估计功能神经解剖学，并将使用更少的测量更好地恢复采样不足的fMRI数据。这将加快数据采集速度，降低成本，并提供更高质量的数据。这反过来将使我们能够开发更好的技术来研究健康的人脑，以及检测和研究不同大脑疾病背后的神经过程。

The proposed research concerns the development of efficient computational algorithms to factorise matrix data into a low-rank representation, where the factors satisfy several constraints. Two scenarios are of interest:1) The entire data-matrix is known and the goal is the decomposition of the data into explanatory components that reveal underlying data structure. 2) The data is only partially observed and the decomposition is also used to recover the un-observed data.1) is often used to remove noise from data (e.g. to clean up images ) or to decompose data into several distinct components (e.g. to separation different speakers in a recording), while 2) is used, for example, by online retailers, who use recommender system to recommend products based on previous purchases or in medical imaging, where we want to reduce a patient's exposure to radiation. In general, better matrix factorisation techniques will enable us to a) acquire data faster, safer and cheaper; b) acquire data at a higher resolution; and c) find better interpretations of data in terms of meaningful underlying factors. These improvements will be made possible through the development and exploitation of better data models. In particular, we will develop models and algorithms that are able to utilize a range of non-convex constraints, such as sparseness, smoothness, contiguity, block structure and low-rank. Each of these constraints has been individually exploited previously and each was found to be able to capture distinct data features. For example, the usefulness of sparse data models for data recovery has attracted significant attention (e.g. in medical imaging), whilst for matrix data, low-rank models are now becoming widely used (e.g. in recommender systems). We here build on our previous work on the efficient recovery and factorization of data and develop algorithms that can exploit more than one of these constraints. Instead of imposing either sparsity or low-rank, we will develop methods that will enable us to efficiently exploit several constraints jointly. This will have a transformative impact on many applications where data structure can be captured using several constraints, but where each single constraint is not strong enough to offer substantial benefits. For example, in radio astronomy, observations might be missing, either due to inability to monitor certain regions of the sky or due to inability to physically store the vast amount of data generated by modern radio observatories. The structure in this data is only partially captured by any one constraint and can thus not be fully recovered with current approaches.Here we are particularly inspired by our current work in functional brain imaging. Magnetic Resonance Imaging (MRI) techniques can be used to measure human brain activity whilst a person is at rest. This type of data provides crucial insights into information processing mechanisms in the living human brain and can also be used to reveal neural mechanisms underlying many brain disorders. Matrix factorization methods are already used as one of the main tools to analyse these data-sets. Current methods construct a low-rank approximation of the spatio-temporal data matrix, describing spatial regions that exhibit joint neural activity, thus revealing several distinct networks of connected brain regions.Our new methods will significantly improve on current approaches. Advanced data models will allow us to better estimate functional neuro-anatomy and will provide better recovery of under-sampled fMRI data using far fewer measurements. This will speed up data acquisition, reduce cost and provide data of higher quality. This in turn will enable us to develop better techniques to study the healthy human brain as well as to detect and study neural processes that underlie different brain diseases.