7 research outputs found
Tensor decomposition and homotopy continuation
A computationally challenging classical elimination theory problem is to
compute polynomials which vanish on the set of tensors of a given rank. By
moving away from computing polynomials via elimination theory to computing
pseudowitness sets via numerical elimination theory, we develop computational
methods for computing ranks and border ranks of tensors along with
decompositions. More generally, we present our approach using joins of any
collection of irreducible and nondegenerate projective varieties
defined over . After computing
ranks over , we also explore computing real ranks. Various examples
are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix
multiplication with zeros. (26 pages, 1 figure
Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems
How many operations do we need on the average to compute an approximate root
of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked
whether a polynomial bound is possible, we prove a quasi-optimal bound
. This improves upon the previously known
bound.
The new algorithm relies on numerical continuation along \emph{rigid
continuation paths}. The central idea is to consider rigid motions of the
equations rather than line segments in the linear space of all polynomial
systems. This leads to a better average condition number and allows for bigger
steps. We show that on the average, we can compute one approximate root of a
random Gaussian polynomial system of~ equations of degree at most in
homogeneous variables with continuation steps. This is a
decisive improvement over previous bounds that prove no better than
continuation steps on the average
Rigid continuation paths II. Structured polynomial systems
We design a probabilistic algorithm that, on input ε>0 and a polynomial system F given by black-box evaluation functions, outputs an approximate zero of F, in the sense of Smale, with probability at least 1-ε. When applying this algorithm to u·F, where u is uniformly random in the product of unitary groups, the algorithm performs poly(n,δ) L(F) ( Γ(F) log Γ(F) + log log 1/ε ) operations on average. Here n is the number of variables, δ the maximum degree, L(F) denotes the evaluation cost of F, and Γ(F) reflects an aspect of the numerical condition of F. Moreover, we prove that for inputs given by random Gaussian algebraic branching programs of size poly(n,δ), the algorithm runs on average in time polynomial in n an δ. Our result may be interpreted as an affirmative answer to a refined version of Smale's 17th question, concerned with systems of structured polynomial equations
Rigid continuation paths II. structured polynomial systems
This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most
in n variables with only
operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem