Related papers: Symmetric tensors: rank, Strassen's conjecture and…
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 $n\times n\times n$…
In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been…
High-dimensional data in the form of tensors are challenging for kernel classification methods. To both reduce the computational complexity and extract informative features, kernels based on low-rank tensor decompositions have been…
Tensor contraction (TC) is an important computational kernel widely used in numerous applications. It is a multi-dimensional generalization of matrix multiplication (GEMM). While Strassen's algorithm for GEMM is well studied in theory and…
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,…
Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-$N$ tensor from exponential…
The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional geometrical domain in the tensor train format; secondly, to develop a new algorithm…
We define new norms for symmetric tensors over ordered normed spaces; these norms are defined by considering linear combinations of tensor products or powers of positive elements only. Relations between the different norms are studied. The…
We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call…
Tensor network contraction is a powerful computational tool in quantum many-body physics, quantum information and quantum chemistry. The complexity of contracting a tensor network is thought to mainly depend on its entanglement properties,…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
We present an algorithm for low rank decomposition of tensors of any symmetry type, from fully asymmetric to fully symmetric. It recovers the decomposition one summand at a time via the higher-order power method. This approach is known to…
We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector…
We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations…
We relate the condition numbers of computing three decompositions of symmetric tensors: the canonical polyadic decomposition, the Waring decomposition, and a Tucker-compressed Waring decomposition. Based on this relation we can speed up the…
Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…
Accurate contraction of tensor networks beyond one dimension is essential in various fields including quantum many-body physics. Existing approaches typically rely on approximate contraction schemes and do not provide certified error bars.…
We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and…
We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…
We propose a new sufficient condition for verifying whether generic rank-r complex tensors of arbitrary order admit a unique decomposition as a linear combination of rank-1 tensors. A practical algorithm is proposed for verifying this…