The Landscape of the Spiked Tensor Model

We consider the problem of estimating a large rank-one tensor u(circle times k) is an element of (Double-struck capital R-n)(circle times k), k >= 3, in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio lambda( 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 lambda >= Cn((k - 2)/4), and even powerful semidefinite programming relaxations appear to fail for 1 MUCH LESS-THAN lambda MUCH LESS-THAN 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 lambda larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Theta(lambda(-1/(k - 1))) around the maximum circle that is orthogonal to u. For local maxima, this band shrinks to be of size Theta(lambda(-1/(k - 2))). These "uninformative" local maxima are likely to cause the failure of optimization algorithms. (c) 2019 Wiley Periodicals, Inc.