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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
Convexity properties of the condition number
We define in the space of n by m matrices of rank n, n less or equal than m,
the condition Riemannian structure as follows: For a given matrix A the tangent
space of A is equipped with the Hermitian inner product obtained by multiplying
the usual Frobenius inner product by the inverse of the square of the smallest
singular value of A denoted sigma_n(A). When this smallest singular value has
multiplicity 1, the function A -> log (sigma_n(A)^(-2)) is a convex function
with respect to the condition Riemannian structure that is t -> log
(sigma_n(A(t))^(-2)) is convex, in the usual sense for any geodesic A(t). In a
more abstract setting, a function alpha defined on a Riemannian manifold
(M,) is said to be self-convex when log alpha (gamma(t)) is convex for any
geodesic in (M,). Necessary and sufficient conditions for self-convexity are
given when alpha is C^2. When alpha(x) = d(x,N)^(-2) where d(x,N) is the
distance from x to a C^2 submanifold N of R^j we prove that alpha is
self-convex when restricted to the largest open set of points x where there is
a unique closest point in N to x. We also show, using this more general notion,
that the square of the condition number ||A|||_F / sigma_n(A) is self-convex in
projective space and the solution variety.Comment: This article was improved for readbility, following referee
suggestion
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