Tensor Decompositions for Signal Processing Applications

背景和研究问题：最近产生的可访问数据数量不断增加，传感器技术的广泛使用为探索事件和自然事件提供了机会，但需要充分解释。这些数据属于被戏称为大数据的类别，即以4V为特征的数据，即(I)海量(数据量)、(Ii)种类繁多(不同形式的数据)、(Iii)高速(数据流)、(Iv)大准确性(数据的不确定性)。这些数据集暴露了经典线性代数的局限性以及相关的矩阵和矢量模型的平面图运算，一些最相关的挑战包括：-非平稳多维数据集的统计分析-找到以前未知的多峰潜在模式-提取的隐藏成分的可解释性因此大数据远非“简单”，分析它是一项非平凡的任务，需要能够评估其所包含信息的复杂技术。方法和新工程：为了解决上述挑战，我们提出了张量方法。这需要通过一种称为张量化的过程，以张量或多路数组的形式重新排列可用的数据。在张量格式中，可以应用分解形式的特征提取算法，例如典型多元分解(CPD)或塔克分解(TKD)。它们类似于矩阵奇异值分解(SVD)，但由于利用了多峰相关性，可以更好地捕捉数据的潜在特征。这种新颖的框架基于最基本的数字信号处理原理，但同时也超越了标准的线性代数工具。由于计算能力的出现，张量分解在过去的十年中得到了极大的关注，并被应用于化学计量学、心理计量学、盲源分离和电信等领域，仅举几例。在他的整个博士学位期间，学生将致力于张量的研究，特别是张量分解，尽管社区感兴趣，但仍可被认为处于初级阶段。更具体地说，学生将专注于：-分析张量网络(TNS)形式的张量分解，并形式化其分类以及将直接对其执行的一组基本操作(即加、减、乘、除)-阐明经典机器学习和信号处理技术对张量域的概括，例如支持向量机(SVM)到支持张量机(STM)-改善现有张量分解的计算复杂性-将张量(和TNS)与神经网络(NNS)和深度学习相结合，目的是解决与NNS相关的“黑箱”问题，从而使训练的模型更易于解释。本博士学位主要是理论上的，因为它涉及一个相对较新的数学领域，其工具仍在开发中。然而，理论发现将通过以下方面的实际应用得到支持：-金融预测，以及伦敦帝国理工学院金融信号处理实验室的额外支持-图像分类-生物医学信号的特征提取和去噪，如脑电或心电

Context and Research Questions:Recent surges in the production of an ever-increasing amount of accessible data and the widespread use of sensor technology gives the opportunity to explore events and natural happenings yet requiring full explanation. Such data falls in the category of that which has been nicknamed Big Data, i.e. data characterized by the 4V's, i.e. (i) massive volume (scale of data), (ii) wide variety (different forms of data), (iii) high velocity (data streams), (iv) large veracity (uncertainty of data). Such data sets have exposed the limitations of classic linear algebra and the associated flat-view operation of matrix and vector models, Some of the most relevant challenges include:- Statistical analysis of non-stationary multi-dimensional datasets- Finding previously unknown multi-modal latent patterns- Interpretability of the extracted hidden componentsBig Data is hence far from "simple", and analyzing it is a nontrivial task which calls for the need of sophisticated techniques able to assess the information it contains. Approach and Novel Engineering:To tackle the above mentioned challenges, we propose a tensor approach. This entails re-arranging the available data in tensors, or multi-way arrays, by a process known as tensorization. In the tensor format it is possible to apply feature extraction algorithms in the form of decompositions, such as the Canonical Polyiadic Decomposition (CPD), or the Tucker Decomposition (TKD). These are analogous to matrix Singular Value Decomposition (SVD), but, due to exploitation of multi-modal correlations, can better capture latent features of data. This novel framework is based on the most fundamental digital signal processing principles but at the same time extends beyond the standard linear algebra tools. Due to the onset of readily available computational power, over the last decade tensor decompositions have gained significant attention and have been applied to fields such as chemometrics, psychometrics, blind source separation, and telecommunications, to name but a few. Throughout his PhD, the student will endeavor upon the study of tensors and, in particular, tensor decompositions, which, despite the community's interest, can still be considered in their infancy. More specifically, the student will focus on:- Analyzing tensor decompositions in the form of Tensor Networks (TNs), and formalize their taxonomy as well as a set of basic operations to be performed directly on them (i.e. addition, subtraction, multiplication, division)- Clarify the generalization of classical machine learning and signal processing techniques to the tensor realm, such as Support Vector Machine (SVM) to the Support Tensor Machine (STM)- Improving computational complexity of existing tensor decompositions- Integrating tensor (and TNs) with Neural Networks (NNs) and Deep Learning, with the goal of tackling the "black-box" issue associated with NNs and hence make trained models more interpretableThis PhD is mostly theoretical as it involves a relatively new mathematical field, the tools of which are still under development. However, theoretical findings will be supported via practical applications in:- Financial forecasting, with the additional support of the Financial Signal Processing Lab at Imperial College London- Image classification- Feature extraction and denoising of biomedical signals, such as EEG or ECG