Related papers: Learning with tree-based tensor formats
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…
Tensor networks are a compressed format for multi-dimensional data. One-dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum…
We present a tree structure algorithm for optimal control problems with state constraints. We prove a convergence result for a discrete time approximation of the value function based on a novel formulation of the constrained problem. Then…
Supervised machine learning often requires large training sets to train accurate models, yet obtaining large amounts of labeled data is not always feasible. Hence, it becomes crucial to explore active learning methods for reducing the size…
Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…
We use deep sparsely connected neural networks to measure the complexity of a function class in $L^2(\mathbb R^d)$ by restricting connectivity and memory requirement for storing the neural networks. We also introduce representation system -…
The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for…
The widespread deployment of deep nets in practical applications has lead to a growing desire to understand how and why such black-box methods perform prediction. Much work has focused on understanding what part of the input pattern (an…
Mappings to structured output spaces (strings, trees, partitions, etc.) are typically learned using extensions of classification algorithms to simple graphical structures (eg., linear chains) in which search and parameter estimation can be…
We study the recovery of multiple high-dimensional signals from two noisy, correlated modalities: a spiked matrix and a spiked tensor sharing a common low-rank structure. This setting generalizes classical spiked matrix and tensor models,…
Tensor classification has become increasingly crucial in statistics and machine learning, with applications spanning neuroimaging, computer vision, and recommendation systems. However, the high dimensionality of tensors presents significant…
Hierarchical tree structures are common in many real-world systems, from tree roots and branches to neuronal dendrites and biologically inspired artificial neural networks, as well as in technological networks for organizing and searching…
In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal…
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for…
We introduce compositional tensor trains (CTTs) for the approximation of multivariate functions, a class of models obtained by composing low-rank functions in the tensor-train format. This format can encode standard approximation tools,…
Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of…
We consider tensor factorizations based on sparse measurements of the components of relatively high rank tensors. The measurements are designed in a way that the underlying graph of interactions is a random graph. The setup will be useful…
Nonparametric estimation of the conditional distribution of a response given high-dimensional features is a challenging problem. It is important to allow not only the mean but also the variance and shape of the response density to change…
Tensor train (TT) format is a common approach for computationally efficient work with multidimensional arrays, vectors, matrices, and discretized functions in a wide range of applications, including computational mathematics and machine…
Inspired by coarse-graining approaches used in physics, we show how similar algorithms can be adapted for data. The resulting algorithms are based on layered tree tensor networks and scale linearly with both the dimension of the input and…