96.
We consider the problem of estimating a large rank-one tensor u
⊗k ∈ (
ℝn)
⊗k ,
k ≥ 3 , in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio
λ Bayes =
O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial-time algorithm is known that achieved this goal unless
λ ≥
Cn(k − 2)/4 , and even powerful semidefinite programming relaxations appear to fail for 1 ≪
λ ≪
n(k − 2)/4 . In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-
k homogeneous polynomial over the unit sphere in
n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions
n , and give exact formulas for the exponential growth rate. We show that (for
λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ(
λ−1/(k − 1)) around the maximum circle that is orthogonal to u . For local maxima, this band shrinks to be of size Θ(
λ−1/(k − 2)) . These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc.
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