Tensor Decompositions for Signal Processing Applications

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 components

Big 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 interpretable

This 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