Poisson distribution

Matrix polynomial

Statistics

Combinatorics

Physics

Mathematical analysis

Gaussian binomial coefficient

Function (biology)

Exponential function

Tensor (intrinsic definition)

Gaussian

Rank (graph theory)

Homogeneous polynomial

Estimator

Maxima and minima

Negative binomial distribution

Programming language

Pure mathematics

Constant (computer programming)

Applied mathematics

Mathematics

Computer science

Art history

Performance art

Quantum mechanics

Exponential growth

Evolutionary biology

Unit vector

Biology

Polynomial

Maxima

Communications on Pure and Applied Mathematics

We consider the problem of estimating a large rank‐one tensor u ⊗ k ∈ ( ℝ n ) ⊗ 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.

The Landscape of the Spiked Tensor Model