689 research outputs found
A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem
The canonical tensor rank approximation problem (TAP) consists of
approximating a real-valued tensor by one of low canonical rank, which is a
challenging non-linear, non-convex, constrained optimization problem, where the
constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian
Gauss-Newton method with trust region for solving small-scale, dense TAPs. The
novelty of our approach is threefold. First, we parametrize the constraint set
as the Cartesian product of Segre manifolds, hereby formulating the TAP as a
Riemannian optimization problem, and we argue why this parametrization is among
the theoretically best possible. Second, an original ST-HOSVD-based retraction
operator is proposed. Third, we introduce 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. Numerical experiments show improvements
of up to three orders of magnitude in terms of the expected time to compute a
successful solution over existing state-of-the-art methods
Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number
We obtain estimates of the multiplicative constants appearing in local
convergence results of the Riemannian Gauss-Newton method for least squares
problems on manifolds and relate them to the geometric condition number of [P.
B\"urgisser and F. Cucker, Condition: The Geometry of Numerical Algorithms,
2013]
The condition number of join decompositions
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 to a set of ill-posed points
in a supplementary product of Grassmannians. We prove that this condition
number can be computed efficiently as the smallest singular value of an
auxiliary matrix. For some special join sets, we characterized the behavior of
sequences in the join set converging to the latter's boundary points. Finally,
we specialize our discussion to the tensor rank and Waring decompositions and
provide several numerical experiments confirming the key results
On the average condition number of tensor rank decompositions
We compute the expected value of powers of the geometric condition number of
random tensor rank decompositions. It is shown in particular that the expected
value of the condition number of tensors with a random
rank- decomposition, given by factor matrices with independent and
identically distributed standard normal entries, is infinite. This entails that
it is expected and probable that such a rank- decomposition is sensitive to
perturbations of the tensor. Moreover, it provides concrete further evidence
that tensor decomposition can be a challenging problem, also from the numerical
point of view. On the other hand, we provide strong theoretical and empirical
evidence that tensors of size with all have a finite average condition number. This suggests there exists a gap
in the expected sensitivity of tensors between those of format and other order-3 tensors. For establishing these results, we show
that a natural weighted distance from a tensor rank decomposition to the locus
of ill-posed decompositions with an infinite geometric condition number is
bounded from below by the inverse of this condition number. That is, we prove
one inequality towards a so-called condition number theorem for the tensor rank
decomposition
The average condition number of most tensor rank decomposition problems is infinite
The tensor rank decomposition, or canonical polyadic decomposition, is the
decomposition of a tensor into a sum of rank-1 tensors. The condition number of
the tensor rank decomposition measures the sensitivity of the rank-1 summands
with respect to structured perturbations. Those are perturbations preserving
the rank of the tensor that is decomposed. On the other hand, the angular
condition number measures the perturbations of the rank-1 summands up to
scaling.
We show for random rank-2 tensors with Gaussian density that the expected
value of the condition number is infinite. Under some mild additional
assumption, we show that the same is true for most higher ranks as
well. In fact, as the dimensions of the tensor tend to infinity, asymptotically
all ranks are covered by our analysis. On the contrary, we show that rank-2
Gaussian tensors have finite expected angular condition number.
Our results underline the high computational complexity of computing tensor
rank decompositions. We discuss consequences of our results for algorithm
design and for testing algorithms that compute the CPD. Finally, we supply
numerical experiments
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