# Statistical limits of spiked tensor models

arXiv: https://arxiv.org/abs/1612.07728 · HTML: https://mlsanity.com/paper/1612.07728/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2016.

## Built on by (most cited first, 28 of 28)

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- [The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics](https://mlsanity.com/paper/2205.09727/) (2022)
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- [Sampling with flows, diffusion and autoregressive neural networks: A spin-glass perspective](https://mlsanity.com/paper/2308.14085/) (2023)
- [Bayes-optimal learning of an extensive-width neural network from quadratically many samples](https://mlsanity.com/paper/2408.03733/) (2024)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [A Random Matrix Perspective on Random Tensors](https://mlsanity.com/paper/2108.00774/) (2021)
- [Long Random Matrices and Tensor Unfolding](https://mlsanity.com/paper/2110.10210/) (2021)
- [When Random Tensors meet Random Matrices](https://mlsanity.com/paper/2112.12348/) (2021)
- [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)
- [On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions](https://mlsanity.com/paper/2209.02001/) (2022)
- [Higher degree sum-of-squares relaxations robust against oblivious outliers](https://mlsanity.com/paper/2211.07327/) (2022)
- [Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model](https://mlsanity.com/paper/2506.02664/) (2025)
- [Optimizing Orthogonalized Tensor Deflation via Random Tensor Theory](https://mlsanity.com/paper/2302.05798/) (2023)
- [Learning from higher-order statistics, efficiently: hypothesis tests, random features, and neural networks](https://mlsanity.com/paper/2312.14922/) (2023)
- [A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation](https://mlsanity.com/paper/2402.03169/) (2024)
- [Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test](https://mlsanity.com/paper/2406.18397/) (2024)
- [On the Accuracy of Hotelling-Type Tensor Deflation: A Random Tensor Analysis](https://mlsanity.com/paper/2211.09004/) (2022)
- [A Nested Matrix-Tensor Model for Noisy Multi-view Clustering](https://mlsanity.com/paper/2305.19992/) (2023)
- [On maximum-likelihood estimation in the all-or-nothing regime](https://mlsanity.com/paper/2101.09994/) (2021)
- [Selective Multiple Power Iteration: from Tensor PCA to gradient-based exploration of landscapes](https://mlsanity.com/paper/2112.12306/) (2021)
- [On the Accuracy of Hotelling-Type Asymmetric Tensor Deflation: A Random Tensor Analysis](https://mlsanity.com/paper/2310.18717/) (2023)
- [Permutation recovery of spikes in noisy high-dimensional tensor estimation](https://mlsanity.com/paper/2412.14650/) (2024)
- [Statistical Limits in Random Tensors with Multiple Correlated Spikes](https://mlsanity.com/paper/2503.03356/) (2025)
- [Mild Over-Parameterization Benefits Asymmetric Tensor PCA](https://mlsanity.com/paper/2604.10208/) (2026)
- [Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field: Tensor PCA and Spherical k-Spin Complexity](https://mlsanity.com/paper/2606.17665/) (2026)
- [Precise sample covariance spectral norm error -- an RDT view](https://mlsanity.com/paper/2607.14460/) (2026)

## Abstract (arXiv)

We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor. We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the planted vector: (i) a uniformly sampled unit vector, (ii) i.i.d. ± 1 entries, and (iii) a sparse vector where a constant fraction ρ of entries are i.i.d. ± 1 and the rest are zero. For each of these cases, our upper and lower bounds match up to a 1+o(1) factor as the order d of the tensor becomes large. For sparse signals (iii), our bounds are also asymptotically tight in the sparse limit ρ → 0 for any fixed d (including the d=2 case of sparse PCA). Our upper bounds for (i) demonstrate a phenomenon reminiscent of the work of Baik, Ben Arous and P'ech'e: an `eigenvalue' of a perturbed tensor emerges from the bulk at a strictly lower signal-to-noise ratio than when the perturbation itself exceeds the bulk; we quantify the size of this effect. We also provide some general results for larger classes of priors. In particular, the large d asymptotics of the threshold location differs between problems with discrete priors versus continuous priors. Finally, for priors (i) and (ii) we carry out the replica prediction from statistical physics, which is conjectured to give the exact information-theoretic threshold for any fixed d. Of independent interest, we introduce a new improvement to the second moment method for contiguity, on which our lower bounds are based. Our technique conditions away from rare `bad' events that depend on interactions between the signal and noise. This enables us to close √2-factor gaps present in several previous works.