Tensorized LSSVMs for Multitask Regression

Multitask learning (MTL) can utilize the relatedness between multiple tasks for performance improvement. The advent of multimodal data allows tasks to be referenced by multiple indices. High-order tensors are capable of providing efficient representations for such tasks, while preserving structural task-relations. In this paper, a new MTL method is proposed by leveraging low-rank tensor analysis and constructing tensorized Least Squares Support Vector Machines, namely the tLSSVM-MTL, where multilinear modelling and its nonlinear extensions can be flexibly exerted. We employ a high-order tensor for all the weights with each mode relating to an index and factorize it with CP decomposition, assigning a shared factor for all tasks and retaining task-specific latent factors along each index. Then an alternating algorithm is derived for the nonconvex optimization, where each resulting subproblem is solved by a linear system. Experimental results demonstrate promising performances of our tLSSVM-MTL

A dual framework for low-rank tensor completion

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on Synergies in Geometric Data Analysis 201

Multitask and transfer learning for multi-aspect data

Supervised learning aims to learn functional relationships between inputs and outputs. Multitask learning tackles supervised learning tasks by performing them simultaneously to exploit commonalities between them. In this thesis, we focus on the problem of eliminating negative transfer in order to achieve better performance in multitask learning. We start by considering a general scenario in which the relationship between tasks is unknown. We then narrow our analysis to the case where data are characterised by a combination of underlying aspects, e.g., a dataset of images of faces, where each face is determined by a person's facial structure, the emotion being expressed, and the lighting conditions. In machine learning there have been numerous efforts based on multilinear models to decouple these aspects but these have primarily used techniques from the field of unsupervised learning. In this thesis we take inspiration from these approaches and hypothesize that supervised learning methods can also benefit from exploiting these aspects. The contributions of this thesis are as follows: 1. A multitask learning and transfer learning method that avoids negative transfer when there is no prescribed information about the relationships between tasks. 2. A multitask learning approach that takes advantage of a lack of overlapping features between known groups of tasks associated with different aspects. 3. A framework which extends multitask learning using multilinear algebra, with the aim of learning tasks associated with a combination of elements from different aspects. 4. A novel convex relaxation approach that can be applied both to the suggested framework and more generally to any tensor recovery problem. Through theoretical validation and experiments on both synthetic and real-world datasets, we show that the proposed approaches allow fast and reliable inferences. Furthermore, when performing learning tasks on an aspect of interest, accounting for secondary aspects leads to significantly more accurate results than using traditional approaches

Tensor Regression

Regression analysis is a key area of interest in the field of data analysis and machine learning which is devoted to exploring the dependencies between variables, often using vectors. The emergence of high dimensional data in technologies such as neuroimaging, computer vision, climatology and social networks, has brought challenges to traditional data representation methods. Tensors, as high dimensional extensions of vectors, are considered as natural representations of high dimensional data. In this book, the authors provide a systematic study and analysis of tensor-based regression models and their applications in recent years. It groups and illustrates the existing tensor-based regression methods and covers the basics, core ideas, and theoretical characteristics of most tensor-based regression methods. In addition, readers can learn how to use existing tensor-based regression methods to solve specific regression tasks with multiway data, what datasets can be selected, and what software packages are available to start related work as soon as possible. Tensor Regression is the first thorough overview of the fundamentals, motivations, popular algorithms, strategies for efficient implementation, related applications, available datasets, and software resources for tensor-based regression analysis. It is essential reading for all students, researchers and practitioners of working on high dimensional data.Comment: 187 pages, 32 figures, 10 table