Related papers: Quotient geometry of tensor ring decomposition

Scientific problems require resolving multi-scale phenomena across different resolutions and learning solution operators in infinite-dimensional function spaces. Neural operators provide a powerful framework for this, using…

We investigate the geometry of a quantum universe with the topology of the four-torus. The study of non-contractible geodesic loops reveals that a typical quantum geometry consists of a small semi-classical toroidal bulk part, dressed with…

Decoherence-free subsystems have been successfully developed as a tool to preserve fragile quantum information against noises. In this letter, we develop a structure theory for decoherence-free subsystems. Based on it, we present an…

The quantum geometric tensor (QGT) of a quantum system in a given parameter space captures both the geometry of the state manifold and the topology of the system. While the local QGT elements have been successfully measured in various…

Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been…

We present a new approach to obtaining the scaling behavior of the entanglement entropy in fractional quantum Hall states from finite-size wavefunctions. By employing the torus geometry and the fact that the torus aspect ratio can be…

This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical…

Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It is generalization of approximate…

In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then…

Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a…

In this paper, we investigate and discuss in detail the structures of quaternion tensor SVD, quaternion tensor rank decomposition, and $\eta$-Hermitian quaternion tensor decomposition with the isomorphic group structures and Einstein…

We review the notion of shape tensor of an embedded manifold, which efficiently combines intrinsic and extrinsic geometry, and allows for intuitive understanding of some basic concepts of classical differential geometry, such as parallel…

Inspired by holographic codes and tensor-network decoders, we introduce tensor-network stabilizer codes which come with a natural tensor-network decoder. These codes can correspond to any geometry, but, as a special case, we generalize…

In this paper we study the effect of non-trivial spatial topology on quantum entanglement by examining the degenerate ground states of a topologically ordered system on torus. Using the string-net (fixed-point) wave-function, we propose a…

Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging…

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…

In this paper, the canonical polyadic (CP) decomposition of tensors that corresponds to matrix multiplications is studied. Finding the rank of these tensors and computing the decompositions is a fundamental problem of algebraic complexity…

Traditional tensor decomposition methods, e.g., two dimensional principal component analysis and two dimensional singular value decomposition, that minimize mean square errors, are sensitive to outliers. To overcome this problem, in this…

Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees,…

The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor…