Motivated by the problem of robust factorization of a low-rank tensor, we study the question of sparse and low-rank tensor decomposition. We present an efficient computational algorithm that modifies Leurgans’ algoirthm for tensor factorization. Our method relies on a reduction of the problem to sparse and low-rank matrix decomposition via the notion of tensor contraction. We use well-understoo...

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product ⊗n s,μX, for any symmetric tensor norm μ. In addi...

In symmetric tensor decomposition one expresses a given symmetric tensor T a sum of tensor powers of a number of vectors: T = v⊗d 1 + · · · + v ⊗d k . Orthogonal decomposition is a special type of symmetric tensor decomposition in which in addition the vectors v1, ..., vk are required to be pairwise orthogonal. We study the properties of orthogonally decomposable tensors. In particular, we give...

Mechanical vibration signal mapped into a high-dimensional space tends to exhibit a special distribution and movement characteristics, which can further reveal the dynamic behavior of the original time series. As the most natural representation of high-dimensional data, tensor can preserve the intrinsic structure of the data to the maximum extent. Thus, the tensor decomposition algorithm has br...

Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximations (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, an...

Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximations (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, an...

The widespread use of multisensor technology and the emergence of big datasets have created the need to develop tools to reduce, approximate, and classify large and multimodal data such as higher-order tensors. While early approaches focused on matrix and vector based methods to represent these higher-order data, more recently it has been shown that tensor decomposition methods are better equip...

Tensor decomposition has recently become an invaluable algorithmic primitive. It has seen much use in new algorithms with provable guarantees for fundamental statistics and machine learning problems. In these settings, some low-rank k-tensor A ∑r i 1 a ⊗k i which wewould like to decompose into components a1, . . . , ar ∈ ’n is often not directly accessible. This could happen for many reasons; a...

Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of mod kG. It is shown that every thick tensor-ideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C = ∐ i∈I Ci into indecomposable thick tensor-ideals. Th...