Related papers: Tensors in computations
We provide a systematic approach to quantum mechanics from an information-theoretic perspective using the language of tensor networks. Our formulation needs only a single kind of object, so-called positive *-tensors. Physical models…
Square matrices represent linear self-maps of vector spaces, and their eigenpoints are the fixed points of the induced map on projective space. Likewise, polynomial self-maps of a projective space are represented by tensors. We study the…
We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a…
We introduce tensor field neural networks, which are locally equivariant to 3D rotations, translations, and permutations of points at every layer. 3D rotation equivariance removes the need for data augmentation to identify features in…
We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly…
This paper introduces the continuous tensor abstraction, allowing indices to take real-number values (for example, A[3.14]). It also presents continuous tensor algebra expressions, such as C(x,y) = A(x,y) * B(x,y), where indices are defined…
Large-scale neuroimaging studies have been collecting brain images of study individuals, which take the form of two-dimensional, three-dimensional, or higher dimensional arrays, also known as tensors. Addressing scientific questions arising…
The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper…
Distributed representations of words have boosted the performance of many Natural Language Processing tasks. However, usually only one representation per word is obtained, not acknowledging the fact that some words have multiple meanings.…
We define a model for linear logic based on two well-known ingredients: games and simulations. This model is interesting in the following respect: while it is obvious that the objects interpreting formulas are games and that everything is…
This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study…
In mathematics, many notations have been invented for the concise representation of mathematical formulae. Tensor index notation is one of such notations and has been playing a crucial role in describing formulae in mathematical physics.…
An artificial neural network (ANN) is a numerical method used to solve complex classification problems. Due to its high classification power, the ANN method often outperforms other classification methods in terms of accuracy. However, an…
We prove the following theorem: the Dirac equation for an electron (invented by P.A.M.Dirac in 1928) can be written as a tensor equation. An equation is called a tensor equation if all values in it are tensors and all operations in it take…
We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors and we…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Scientific computations or measurements may result in huge volumes of data. Often these can be thought of representing a real-valued function on a high-dimensional domain, and can be conceptually arranged in the format of a tensor of high…
It is a significant challenge to design probabilistic programming systems that can accommodate a wide variety of inference strategies within a unified framework. Noting that the versatility of modern automatic differentiation frameworks is…
Real or complex tensor model observables, the backbone of the tensor theory space, are classical (unitary, orthogonal, symplectic) Lie group invariants. These observables represent as colored graphs, and that representation gives an handle…
Classical and exceptional Lie algebras and their representations are among the most important tools in the analysis of symmetry in physical systems. In this letter we show how the computation of tensor products and branching rules of…