# The landscape of the spiked tensor model

arXiv: https://arxiv.org/abs/1711.05424 · HTML: https://mlsanity.com/paper/1711.05424/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2017.

## Built on by (most cited first, 30 of 34)

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- [On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors](https://mlsanity.com/paper/2107.09660/) (2021)
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- [The Complexity of Sparse Tensor PCA](https://mlsanity.com/paper/2106.06308/) (2021)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
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- [A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation](https://mlsanity.com/paper/2402.03169/) (2024)
- [Low-Rank Adaptation Redux for Large Models](https://mlsanity.com/paper/2604.21905/) (2026)
- [Sharp Analysis of Power Iteration for Tensor PCA](https://mlsanity.com/paper/2401.01047/) (2024)
- [Fundamental limits of Non-Linear Low-Rank Matrix Estimation](https://mlsanity.com/paper/2403.04234/) (2024)
- [Low-Rank Tensor Decompositions for the Theory of Neural Networks](https://mlsanity.com/paper/2508.18408/) (2025)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Learning with Restricted Boltzmann Machines: Asymptotics of AMP and GD in High Dimensions](https://mlsanity.com/paper/2505.18046/) (2025)
- [Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval](https://mlsanity.com/paper/2602.17779/) (2026)
- [Selective Multiple Power Iteration: from Tensor PCA to gradient-based exploration of landscapes](https://mlsanity.com/paper/2112.12306/) (2021)
- [Large-N dynamics of the spiked tensor model with random initial conditions](https://mlsanity.com/paper/2208.12586/) (2022)
- [Topological complexity of spiked random polynomials and finite-rank spherical integrals](https://mlsanity.com/paper/2312.12323/) (2023)
- [Dynamical loss functions shape landscape topography and improve learning in artificial neural networks](https://mlsanity.com/paper/2410.10690/) (2024)

## Abstract (arXiv)

We consider the problem of estimating a large rank-one tensor boldsymbol uotimes k∈(mathbb Rn)otimes 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≥ C n(k-2)/4 and even powerful semidefinite programming relaxations appear to fail for 1ll λll 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 boldsymbol u, or are confined in a band of width Theta(λ-1/(k-1)) around the maximum circle that is orthogonal to boldsymbol u. For local maxima, this band shrinks to be of size Theta(λ-1/(k-2)). These `uninformative' local maxima are likely to cause the failure of optimization algorithms.