The Condition Number of Join Decompositions

@article{Breiding2016TheCN,
  title={The Condition Number of Join Decompositions},
  author={Paul Breiding and Nick Vannieuwenhoven},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2016},
  volume={39},
  pages={287-309}
}
The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, partially symmetric rank and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance… 
[PDF] Semantic Reader
28 Citations
Highly Influential Citations
3
Background Citations
15
Methods Citations
5

Figures and Tables from this paper

28 Citations

Towards a condition number theorem for the tensor rank decomposition

We show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions (i.e., decompositions with unbounded geometric condition number, derived in [P.…

The condition number of many tensor decompositions is invariant under Tucker compression

This work characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor, and shows that the condition number of all these decomposition is invariant under Tucker compression.

The average condition number of most tensor rank decomposition problems is infinite

The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations, and it is shown that rank-2 Gaussian tensors have finite expected angular condition number.

Numerical and statistical aspects of tensor decompositions

This work studies numerical and statistical properties of tensor decompositions, namely the canonical-polyadic decomposition— commonly known as tensor-rank decomposition)—and the computation of Tensor eigenpairs, and constructs an efficient (average polynomial-time) homotopy-method to solve for tensor eignepairs.

A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

A Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs and a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions.

On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition

Tensor completion using geodesics on Segre manifolds

This work proposes a Riemannian conjugate gradient algorithm for approximating incomplete tensors by canonical polyadic decompositions of low rank and applies this method to movie rating predictions in a recommender system for the MovieLens dataset, and for identifying pure fluorophores via fluorescent spectroscopy with missing data.

Perturbations of the Tcur Decomposition for Tensor Valued Data in the Tucker Format

A thorough investigation of what happens to the approximations in the presence of noise is performed and two forms of the tensor CUR decomposition are presented and deduce the errors of the approximation.

The Condition Number of Riemannian Approximation Problems

The main results are validated through experiments with the $n$-camera triangulation problem in computer vision and the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem is characterized.

Scaling Probabilistic Tensor Canonical Polyadic Decomposition to Massive Data

The idea of stochastic optimization is introduced into the probabilistic tensor CPD, rendering a scalable algorithm that only processes mini-batch data at a time.

References

SHOWING 1-10 OF 53 REFERENCES

Generic Uniqueness Conditions for the Canonical Polyadic Decomposition and INDSCAL

We find conditions that guarantee that a decomposition of a generic third-order tensor in a minimal number of rank-$1$ tensors (canonical polyadic decomposition (CPD)) is unique up to a permutation…

Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness

  • L. Lathauwer
  • Mathematics, Computer Science
    SIAM J. Matrix Anal. Appl.
  • 2008
A new class of tensor decompositions is introduced, where the size is characterized by a set of mode-$n$ ranks, and conditions under which essential uniqueness is guaranteed are derived.

On generic identifiability of symmetric tensors of subgeneric rank

We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r…

Orthogonal and unitary tensor decomposition from an algebraic perspective

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition.…

On partial and generic uniqueness of block term tensor decompositions

We present several conditions for generic uniqueness of tensor decompositions of multilinear rank $$(1,\ L_{1},\ L_{1}),\cdots ,(1,\ L_{R},\ L_{R})$$(1,L1,L1),⋯,(1,LR,LR) terms. In geometric…

A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

A Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs and a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions.

Semialgebraic Geometry of Nonnegative Tensor Rank

It is shown that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank.

Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem

It is argued that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations, and a natural way of overcoming the ill-posedness of the low-rank approximation problem is proposed by using weak solutions when true solutions do not exist.

On Generic Identifiability of 3-Tensors of Small Rank

An inductive method is introduced for the study of the uniqueness of decompositions of tensors, by means of Tensors of rank $1, based on the geometric notion of weak defectivity, which proves that the decomposition is unique for three-dimensional tensors of type a,b,c.

Kruskal's permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models with constant modulus constraints

Kruskal's permutation lemma is revisited, and a similar necessary and sufficient condition for unique bilinear factorization under constant modulus (CM) constraints is derived, thus providing an interesting link to (and unification with) CP.
...