Giuseppe G. Calvi

Yao Lei Xu, Giuseppe G. Calvi, Danilo P. Mandic

Recurrent Neural Networks (RNNs) represent the de facto standard machine learning tool for sequence modelling, owing to their expressive power and memory. However, when dealing with large dimensional data, the corresponding exponential increase in the number of parameters imposes a computational bottleneck. The necessity to equip RNNs with the ability to deal with the curse of dimensionality, such as through the parameter compression ability inherent to tensors, has led to the development of the Tensor-Train RNN (TT-RNN). Despite achieving promising results in many applications, the full potential of the TT-RNN is yet to be explored in the context of interpretable financial modelling, a notoriously challenging task characterized by multi-modal data with low signal-to-noise ratio. To address this issue, we investigate the potential of TT-RNN in the task of financial forecasting of currencies. We show, through the analysis of TT-factors, that the physical meaning underlying tensor decomposition, enables the TT-RNN model to aid the interpretability of results, thus mitigating the notorious "black-box" issue associated with neural networks. Furthermore, simulation results highlight the regularization power of TT decomposition, demonstrating the superior performance of TT-RNN over its uncompressed RNN counterpart and other tensor forecasting methods.

Giuseppe G. Calvi, Ahmad Moniri, Mahmoud Mahfouz et al.

This work aims to help resolve the two main stumbling blocks in the application of Deep Neural Networks (DNNs), that is, the exceedingly large number of trainable parameters and their physical interpretability. This is achieved through a tensor valued approach, based on the proposed Tucker Tensor Layer (TTL), as an alternative to the dense weight-matrices of DNNs. This allows us to treat the weight-matrices of general DNNs as a matrix unfolding of a higher order weight-tensor. By virtue of the compression properties of tensor decompositions, this enables us to introduce a novel and efficient framework for exploiting the multi-way nature of the weight-tensor in order to dramatically reduce the number of DNN parameters. We also derive the tensor valued back-propagation algorithm within the TTL framework, by extending the notion of matrix derivatives to tensors. In this way, the physical interpretability of the Tucker decomposition is exploited to gain physical insights into the NN training, through the process of computing gradients with respect to each factor matrix. The proposed framework is validated on both synthetic data, and the benchmark datasets MNIST, Fashion-MNIST, and CIFAR-10. Overall, through the ability to provide the relative importance of each data feature in training, the TTL back-propagation is shown to help mitigate the "black-box" nature inherent to NNs. Experiments also illustrate that the TTL achieves a 66.63-fold compression on MNIST and Fashion-MNIST, while, by simplifying the VGG-16 network, it achieves a 10\% speed up in training time, at a comparable performance.

Ilia Kisil, Giuseppe G. Calvi, Danilo P. Mandic

A novel method for common and individual feature analysis from exceedingly large-scale data is proposed, in order to ensure the tractability of both the computation and storage and thus mitigate the curse of dimensionality, a major bottleneck in modern data science. This is achieved by making use of the inherent redundancy in so-called multi-block data structures, which represent multiple observations of the same phenomenon taken at different times, angles or recording conditions. Upon providing an intrinsic link between the properties of the outer vector product and extracted features in tensor decompositions (TDs), the proposed common and individual information extraction from multi-block data is performed through imposing physical meaning to otherwise unconstrained factorisation approaches. This is shown to dramatically reduce the dimensionality of search spaces for subsequent classification procedures and to yield greatly enhanced accuracy. Simulations on a multi-class classification task of large-scale extraction of individual features from a collection of partially related real-world images demonstrate the advantages of the "blessing of dimensionality" associated with TDs.