### Abstract

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.

Original language | English (US) |
---|---|

Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21861

**The Landscape of the Spiked Tensor Model.** / Ben Arous, Gerard; Mei, Song; Montanari, Andrea; Nica, Mihai.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21861

}

TY - JOUR

T1 - The Landscape of the Spiked Tensor Model

AU - Ben Arous, Gerard

AU - Mei, Song

AU - Montanari, Andrea

AU - Nica, Mihai

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 λBayes = 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.

AB - 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 λBayes = 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.

UR - http://www.scopus.com/inward/record.url?scp=85070771190&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85070771190&partnerID=8YFLogxK

U2 - 10.1002/cpa.21861

DO - 10.1002/cpa.21861

M3 - Article

AN - SCOPUS:85070771190

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -