Related papers: Simplifying Tensor Polynomials with Indices

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…

We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order $d \geq 3$ and dimension $2$ and symmetric tensors of order $3$ and dimension $3$, we prove that only finitely many…

Here we study an efficient algorithm for decoding the topological codes. It is based on a simple principle, which should allow straightforward generalization to complex decoding problems. It is benchmarked with the planar code for both…

We develop a framework to analyse invariant decompositions of elements of tensor product spaces. Namely, we define an invariant decomposition with indices arranged on a simplicial complex, and which is explicitly invariant under a group…

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is…

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

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 generalize the method of combinatorial telescoping to the case of multiple summations. We shall demonstrate this idea by giving combinatorial proofs for two identities of Andrews on parity indices of partitions.

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial.…

Many problems in high-dimensional statistics appear to have a statistical-computational gap: a range of values of the signal-to-noise ratio where inference is information-theoretically possible, but (conjecturally) computationally…

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the…

We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and…

Tensors are fundamental in mathematics, computer science, and physics. Their study through algebraic geometry and representation theory has proved very fruitful in the context of algebraic complexity theory and quantum information. In…

In this paper, I present a technique to simplify the tensorial reduction of one-loop integrals with arbitrary internal masses, but at least two massless external legs. By applying the method to rank l tensor integrals, one ends up with at…

The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…

Computational Group Theory is applied to indexed objects (tensors, spinors, and so on) with dummy indices. There are two groups to consider: one describes the intrinsic symmetries of the object and the other describes the interchange of…

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…

We give several new algorithms for dense polynomial multiplication based on the Kronecker substitution method. For moderately sized input polynomials, the new algorithms improve on the performance of the standard Kronecker substitution by a…

We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.