Related papers: Tensor-Train Parameterization for Ultra Dimensiona…

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer…

Tensor train is a hierarchical tensor network structure that helps alleviate the curse of dimensionality by parameterizing large-scale multidimensional data via a set of network of low-rank tensors. Associated with such a construction is a…

Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has…

Tensor train (TT) factorization and corresponding TT rank, which can well express the low-rankness and mode correlations of higher-order tensors, have attracted much attention in recent years. However, TT factorization based methods are…

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…

Tensor decompositions are powerful tools for dimensionality reduction and feature interpretation of multidimensional data such as signals. Existing tensor decomposition objectives (e.g., Frobenius norm) are designed for fitting raw data…

In recent years, the application of tensors has become more widespread in fields that involve data analytics and numerical computation. Due to the explosive growth of data, low-rank tensor decompositions have become a powerful tool to…

Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition…

Many problems in computational neuroscience, neuroinformatics, pattern/image recognition, signal processing and machine learning generate massive amounts of multidimensional data with multiple aspects and high dimensionality. Tensors (i.e.,…

Dimensionality reduction (DR) is a popular method for preparing and analyzing high-dimensional data. Reduced data representations are less computationally intensive and easier to manage and visualize, while retaining a significant…

Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of…

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data. For…

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…

Tensor train (TT) representation has achieved tremendous success in visual data completion tasks, especially when it is combined with tensor folding. However, folding an image or video tensor breaks the original data structure, leading to…

The global optimization of atomic clusters represents a fundamental challenge in computational chemistry and materials science due to the exponential growth of local minima with system size (i.e., the curse of dimensionality). We introduce…

With the increasing availability of high-dimensional data, analysts often rely on exploratory data analysis to understand complex data sets. A key approach to exploring such data is dimensionality reduction, which embeds high-dimensional…

We propose tensorial neural networks (TNNs), a generalization of existing neural networks by extending tensor operations on low order operands to those on high order ones. The problem of parameter learning is challenging, as it corresponds…

In this work, we introduce an interior-point method that employs tensor decompositions to efficiently represent and manipulate the variables and constraints of semidefinite programs, targeting problems where the solutions may not be…

Tensor networks serve as a powerful tool for efficiently representing and manipulating high-dimensional data in applications such as quantum physics, machine learning, and data compression. Tensor Decision Diagrams (TDDs) offer an efficient…

Geometric data pruning methods, while practical for leveraging pretrained models, are fundamentally unstable. Their reliance on extrinsic geometry renders them highly sensitive to latent space perturbations, causing performance to degrade…