Corpus ID: 221090014

# Statistical Query Lower Bounds for Tensor PCA

@article{Dudeja2020StatisticalQL,
title={Statistical Query Lower Bounds for Tensor PCA},
author={Rishabh Dudeja and Daniel J. Hsu},
journal={arXiv: Statistics Theory},
year={2020}
}
• Published 2020
• Mathematics
• arXiv: Statistics Theory
In the Tensor PCA problem introduced by Richard and Montanari (2014), one is given a dataset consisting of $n$ samples $\mathbf{T}_{1:n}$ of i.i.d. Gaussian tensors of order $k$ with the promise that $\mathbb{E}\mathbf{T}_1$ is a rank-1 tensor and $\|\mathbb{E} \mathbf{T}_1\| = 1$. The goal is to estimate $\mathbb{E} \mathbf{T}_1$. This problem exhibits a large conjectured hard phase when $k>2$: When $d \lesssim n \ll d^{\frac{k}{2}}$ it is information theoretically possible to estimate… Expand
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#### References

SHOWING 1-10 OF 36 REFERENCES
The landscape of the spiked tensor model
• Mathematics
• 2017
We consider the problem of estimating a large rank-one tensor ${\boldsymbol u}^{\otimes k}\in({\mathbb R}^{n})^{\otimes k}$, $k\ge 3$ in Gaussian noise. Earlier work characterized a criticalExpand
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere
• Mathematics, Computer Science
• APPROX-RANDOM
• 2017
The above bound is the best possible up to lower order terms, namely the optimum of the level-$q$ SoS relaxation is at least A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . Expand
On Mean Estimation for General Norms with Statistical Queries
• Mathematics, Computer Science
• COLT
• 2019
Sharp upper and lower bounds are obtained for the statistical query complexity of this problem when the the underlying norm is symmetric as well as for Schatten-$p norms, answering two questions raised by Feldman, Guzman, and Vempala (SODA 2017). Expand Tensor principal component analysis via sum-of-square proofs • Mathematics, Computer Science • COLT • 2015 It is shown that degree-$4 sum-of-squares relaxations break down for $\tau \leq O(n^{3/4}/\log(n)^{1/4})$, which demonstrates that improving the current guarantees would require new techniques or might even be intractable. Expand
Interpolating Convex and Non-Convex Tensor Decompositions via the Subspace Norm
• Computer Science, Mathematics
• NIPS
• 2015
A new norm is proposed called the subspace norm, which is based on the Kronecker products of factors obtained by the proposed simple estimator, and empirically demonstrates that the sub space norm achieves the nearly ideal denoising performance even with $H=O(1)$. Expand
Efficient Algorithms and Lower Bounds for Robust Linear Regression
• Computer Science, Mathematics
• SODA
• 2019
Any polynomial time SQ learning algorithm for robust linear regression (in Huber's contamination model) with estimation complexity, must incur an error of $\Omega(\sqrt{\epsilon} \sigma)$. Expand
Dealing with Range Anxiety in Mean Estimation via Statistical Queries
Algorithms for high dimensional mean estimation and stochastic convex optimization in these models that work in more general settings than previously known solutions are obtained. Expand
A General Characterization of the Statistical Query Complexity
This work demonstrates that the complexity of solving general problems over distributions using SQ algorithms can be captured by a relatively simple notion of statistical dimension that is introduced, and is also the first to precisely characterize the necessary tolerance of queries. Expand
On the Limitation of Spectral Methods: From the Gaussian Hidden Clique Problem to Rank One Perturbations of Gaussian Tensors
• Computer Science, Mathematics
• IEEE Transactions on Information Theory
• 2017
A lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected is established on the basis of a more general result on rank- one perturbations of the Gaussian tensors. Expand
A statistical model for tensor PCA
• Computer Science, Mathematics
• NIPS
• 2014
It turns out that the Principal Component Analysis problem for large tensors of arbitrary order $k$ under a single-spike (or rank-one plus noise) model is possible as soon as the signal-to-noise ratio $\beta$ becomes larger than $C\sqrt{k\log k}$ (and in particular \$\beta can remain bounded as the problem dimensions increase). Expand