Related papers: The Tensor Track VIII: Stochastic Analysis
Matrix models are a highly successful framework for the analytic study of random two dimensional surfaces with applications to quantum gravity in two dimensions, string theory, conformal field theory, statistical physics in random geometry,…
Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a…
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…
These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in…
We give a motivation and gentle introduction into the regularity structure and model introduced by Otto, Sauer, Smith and Weber, which fall into the framework of Hairer, but have a greedier index set than the one given by trees. We do this…
Random matrix models encode a theory of random two dimensional surfaces with applications to string theory, conformal field theory, statistical physics in random geometry and quantum gravity in two dimensions. The key to their success lies…
We study the XXZ Heisenberg model in a longitudinal magnetic field using a tensor renormalization method. Built into the tensor representation of the XXZ model is the U(1) symmetry, which is systematically maintained at each renormalization…
Unsupervised learning aims at the discovery of hidden structure that drives the observations in the real world. It is essential for success in modern machine learning. Latent variable models are versatile in unsupervised learning and have…
We provide a brief overview of tensor models and group field theories, focusing on their main common features. Both frameworks arose in the context of quantum gravity research, and can be understood as higher-dimensional generalizations of…
The interest in machine learning with tensor networks has been growing rapidly in recent years. We show that tensor-based methods developed for learning the governing equations of dynamical systems from data can, in the same way, be used…
Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of…
The tensor track is a background-independent discretization of quantum gravity which includes a sum over all topologies. We discuss how to define a functional renormalization group flow and the Wetterich equation in the corresponding theory…
Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a…
Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity…
We theoretically and experimentally investigate tensor-based regression and classification. Our focus is regularization with various tensor norms, including the overlapped trace norm, the latent trace norm, and the scaled latent trace norm.…
We investigate exceptional generalised diffeomorphisms based on $E_{8(8)}$ in a geometric setting. The transformations include gauge transformations for the dual gravity field. The surprising key result, which allows for a development of a…
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…
This paper studies the local structure of continuous random fields on $\mathbb R^d$ taking values in a complete separable linear metric space ${\mathbb V}$. Extending seminal work of Falconer, we show that the generalized $(1+k)$-th order…
Leveraging advances in variational inference, we propose to enhance recurrent neural networks with latent variables, resulting in Stochastic Recurrent Networks (STORNs). The model i) can be trained with stochastic gradient methods, ii)…
In this paper, we construct a parameter estimation framework for robust low-rank tensor regression based on a truncation method and Huber loss, specifically focusing on models with random noise having only finite second-order moments.…