# List Decodable Subspace Recovery

@article{Raghavendra2020ListDS, title={List Decodable Subspace Recovery}, author={Prasad Raghavendra and Morris Yau}, journal={ArXiv}, year={2020}, volume={abs/2002.03004} }

Learning from data in the presence of outliers is a fundamental problem in statistics. In this work, we study robust statistics in the presence of overwhelming outliers for the fundamental problem of subspace recovery. Given a dataset where an $\alpha$ fraction (less than half) of the data is distributed uniformly in an unknown $k$ dimensional subspace in $d$ dimensions, and with no additional assumptions on the remaining data, the goal is to recover a succinct list of $O(\frac{1}{\alpha…

## 15 Citations

### List Decodable Mean Estimation in Nearly Linear Time

- Computer Science2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

This paper considers robust statistics in the presence of overwhelming outliers where the majority of the dataset is introduced adversarially and develops an algorithm for list decodable mean estimation in the same setting achieving up to constants the information theoretically optimal recovery, optimal sample complexity, and in nearly linear time up to polylogarithmic factors in dimension.

### List-Decodable Sparse Mean Estimation

- Computer ScienceArXiv
- 2022

The main contribution is the first polynomial-time algorithm that enjoys sample complexity O i.e. poly-logarithmic in the dimension, using low-degree sparse polynomials to connect outliers, which may be of independent interest.

### List-Decodable Subspace Recovery: Dimension Independent Error in Polynomial Time

- Computer ScienceSODA
- 2021

Apoly(1/\alpha) d^{O(1)} time algorithm is given that outputs a list containing a list of candidate covariances that contains a $\hat{\Pi}$ satisfying $\|\hat{Pi} -\Pi-\Pi_*\|_F \leq \eta$ for any arbitrary $\eta > 0$ in $d^{O(\alpha) + \log ( 1/\eta))}$ time.

### List-Decodable Mean Estimation via Iterative Multi-Filtering

- Computer Science, MathematicsNeurIPS
- 2020

The main technical innovation is the design of a soft outlier removal procedure for high-dimensional heavy-tailed datasets with a majority of outliers with information-theoretically near-optimal error.

### List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering

- Computer ScienceArXiv
- 2022

This work develops a novel, conceptually simpler technique for list-decodable mean estimation that achieves the optimal error guarantee of Θ( √ log(1/α) with quasi-polynomial sample and computational complexity and complements the authors' upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds.

### Learning a mixture of two subspaces over finite fields

- Mathematics, Computer ScienceALT
- 2021

These algorithms imply computational tractability of the problem of learning mixtures of two subspaces, except in the degenerate setting captured by learning parities with noise.

### Statistical Query Lower Bounds for List-Decodable Linear Regression

- Computer Science, MathematicsNeurIPS
- 2021

The main result is a Statistical Query (SQ) lower bound of d, which qualitatively matches the performance of previously developed algorithms, providing evidence that current upper bounds for this task are nearly best possible.

### List-decodable covariance estimation

- Computer Science, MathematicsSTOC
- 2022

The first polynomial time algorithm for list-decodable covariance estimation is given, and this algorithm works more generally for any distribution D that possesses low-degree sum-of-squares certificates of two natural analytic properties: 1) anti-concentration of one-dimensional marginals and 2) hypercontractivity of degree 2 polynomials.

### Privately Learning Mixtures of Axis-Aligned Gaussians

- Computer ScienceNeurIPS
- 2021

It is proved that Õ(kd log(1/δ)/αε) samples are sufficient to learn a mixture of k axis-aligned Gaussians in R to within total variation distance α while satisfying (ε, δ)-differential privacy, the first result for privately learning mixtures of unbounded axis- aligned (or even unbounded univariate)Gaussians.

### Polynomial-Time Sum-of-Squares Can Robustly Estimate Mean and Covariance of Gaussians Optimally

- Computer Science, MathematicsALT
- 2022

This work revisits the problem of estimating the mean and covariance of an unknown d dimensional Gaussian distribution in the presence of an ε -fraction of adversarial outliers and gives a new, simple analysis of the same canonical sum-of-squares relaxation used in Kothari and Steurer (2017) and Bakshi and Kotharis (2020) and shows that their algorithm achieves the same error, sample complexity and running time guarantees.

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- 2020

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