Decompose et Impera: tensor methods in high-dimensional data

Abstract

This thesis is written with the scope of exploring multiway data. Multiway data, also referred to as tensor data, is a collection of data points in multidimensional matrices. At a first glance one may think that these objects are only a convenient representation of a datasets. They are not just a col- lection of data, they have their own structure. For this reason, multiway data need specific models to be correctly analysed. In this spirit, I developed my personal idea on data analysis which can be represented by following statement: \It is not the data that should fit models, but models that should fit the data" However, this should not be taken literary I do think that models are im- portant: giving a structure to our techniques is necessary. Nevertheless, I do think that data should be the main driver.This means that instead of trimming data at our necessity to fit existing models, researchers should develop new models to re ect the complexity of the data. The purpose of this work is to provide an overview of tensor methods applied to Economics and Finance. Yet, the most important aspect of this thesis are ideas and applications rather than the mathematical content. New models are proposed and fitted to data in order to test their performance and get insights from the datasets analysed. The description of the tensor methods provided in this thesis is not intended to be complete but rather restricted to the model applicable to the analysed data.This thesis is written with the scope of exploring multiway data. Multiway data, also referred to as tensor data, is a collection of data points in multidimensional matrices. At a first glance one may think that these objects are only a convenient representation of a datasets. They are not just a col- lection of data, they have their own structure. For this reason, multiway data need specific models to be correctly analysed. In this spirit, I developed my personal idea on data analysis which can be represented by following statement: \It is not the data that should fit models, but models that should fit the data" However, this should not be taken literary I do think that models are im- portant: giving a structure to our techniques is necessary. Nevertheless, I do think that data should be the main driver.This means that instead of trimming data at our necessity to fit existing models, researchers should develop new models to re ect the complexity of the data. The purpose of this work is to provide an overview of tensor methods applied to Economics and Finance. Yet, the most important aspect of this thesis are ideas and applications rather than the mathematical content. New models are proposed and fitted to data in order to test their performance and get insights from the datasets analysed. The description of the tensor methods provided in this thesis is not intended to be complete but rather restricted to the model applicable to the analysed data.LUISS PhD Thesi