Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Geometric Expression Invariant 3D Face Recognition using Statistical Discriminant Models

Currently there is no complete face recognition system that is invariant to all facial expressions. Although humans find it easy to identify and recognise faces regardless of changes in illumination, pose and expression, producing a computer system with a similar capability has proved to be particularly di cult. Three dimensional face models are geometric in nature and therefore have the advantage of being invariant to head pose and lighting. However they are still susceptible to facial expressions. This can be seen in the decrease in the recognition results using principal component analysis when expressions are added to a data set. In order to achieve expression-invariant face recognition systems, we have employed a tensor algebra framework to represent 3D face data with facial expressions in a parsimonious space. Face variation factors are organised in particular subject and facial expression modes. We manipulate this using single value decomposition on sub-tensors representing one variation mode. This framework possesses the ability to deal with the shortcomings of PCA in less constrained environments and still preserves the integrity of the 3D data. The results show improved recognition rates for faces and facial expressions, even recognising high intensity expressions that are not in the training datasets. We have determined, experimentally, a set of anatomical landmarks that best describe facial expression e ectively. We found that the best placement of landmarks to distinguish di erent facial expressions are in areas around the prominent features, such as the cheeks and eyebrows. Recognition results using landmark-based face recognition could be improved with better placement. We looked into the possibility of achieving expression-invariant face recognition by reconstructing and manipulating realistic facial expressions. We proposed a tensor-based statistical discriminant analysis method to reconstruct facial expressions and in particular to neutralise facial expressions. The results of the synthesised facial expressions are visually more realistic than facial expressions generated using conventional active shape modelling (ASM). We then used reconstructed neutral faces in the sub-tensor framework for recognition purposes. The recognition results showed slight improvement. Besides biometric recognition, this novel tensor-based synthesis approach could be used in computer games and real-time animation applications