# Robust Polynomial Regression up to the Information Theoretic Limit

@article{Kane2017RobustPR, title={Robust Polynomial Regression up to the Information Theoretic Limit}, author={Daniel M. Kane and Sushrut Karmalkar and Eric Price}, journal={2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)}, year={2017}, pages={391-402} }

We consider the problem of robust polynomial regression, where one receives samples that are usually within a small additive error of a target polynomial, but have a chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate the target polynomial only when the outlier probability was subconstant in the degree of the target polynomial. We give an algorithm that works for the entire feasible range of outlier probabilities, while simultaneously improving…

## 9 Citations

### Approximate optimization of convex functions with outlier noise

- Computer Science, MathematicsNeurIPS
- 2021

A lower bound result is proved showing that, somewhat surprisingly, one cannot hope to approximate the minimizer nearly as well as one might expect, even if one is allowed an unbounded number of queries to the oracle.

### Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers

- Computer Science, MathematicsITCS
- 2018

This work introduces a criterion, resilience, which allows properties of a dataset to be robustly computed, even in the presence of a large fraction of arbitrary additional data, and provides new information-theoretic results on robust distribution learning, robust estimation of stochastic block models, and robust mean estimation under bounded kth moments.

### J ul 2 01 9 Reconstruction under outliers for Fourier-sparse functions

- Computer Science
- 2019

It is shown that over the torus, assuming that the Fourier transform satisfies a certain granularity condition, there is a sample efficient algorithm to tolerate ρ = Ω(1) fraction of outliers and further, that this is not possible without such a granular condition.

### Fast Regression for Structured Inputs

- Computer Science, MathematicsICLR
- 2022

This work gives an algorithm for `p regression on Vandermonde matrices that runs in time O(n log n+(dp) ·polylogn), where ω is the exponent of matrix multiplication.

### Reconstruction under outliers for Fourier-sparse functions

- Computer Science, MathematicsSODA
- 2020

Over the torus, assuming that the Fourier transform satisfies a certain \emph{granularity} condition, there is a sample efficient algorithm to tolerate $\rho =\Omega(1)$ fraction of outliers and further, that this is not possible without such a granularity condition.

### Model Order Selection From Noisy Polynomial Data Without Using Any Polynomial Coefficients

- MathematicsIEEE Access
- 2020

It is experimentally observed that the root-mean square prediction errors and the variation of the RMS prediction errors appear to scale linearly with the standard deviations of the noise for each degree of a polynomial.

### Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent

- Computer Science, MathematicsArXiv
- 2022

The hardness of computing additive approximations to output probabilities of random quantum circuits is considered and it is shown that approximating the Ising partition function with imaginary couplings to an additive error of 2 − O ( n ) is hard even in the average-case, which extends prior work on worst-case hardness of multiplicative approximation toIsing partition functions.

### Efficient and robust estimation of many-qubit Hamiltonians

- Economics
- 2022

Characterizing the interactions and dynamics of quantum mechanical systems is an essential task in the development of quantum technologies. We propose an eﬃcient protocol based on the estimation of…

### Study of mathematics modeling on ginger geometric changes during drying using image analysis

- Physics
- 2021

Many studies have examined the drying process, but only a few studies have examined shrinkage as one of the parameters that affect drying. Depreciation of agricultural products could have an impact…

## References

SHOWING 1-10 OF 21 REFERENCES

### Agnostic Estimation of Mean and Covariance

- Computer Science, Mathematics2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

This work presents polynomial-time algorithms to estimate the mean and covariance of a distribution from i.i.d. samples in the presence of a fraction of malicious noise with error guarantees in terms of information-theoretic lower bounds.

### Robust Estimators in High Dimensions without the Computational Intractability

- Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

This work obtains the first computationally efficient algorithms for agnostically learning several fundamental classes of high-dimensional distributions: a single Gaussian, a product distribution on the hypercube, mixtures of two product distributions (under a natural balancedness condition), and k Gaussians with identical spherical covariances.

### Learning from untrusted data

- Computer ScienceSTOC
- 2017

An algorithm for robust learning in a very general stochastic optimization setting is provided that has immediate implications for robustly estimating the mean of distributions with bounded second moments, robustly learning mixtures of such distributions, and robustly finding planted partitions in random graphs.

### Agnostically learning halfspaces

- Computer Science46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005

We give the first algorithm that (under distributional assumptions) efficiently learns halfspaces in the notoriously difficult agnostic framework of Kearns, Schapire, & Sellie, where a learner is…

### Robust Fourier and Polynomial Curve Fitting

- Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

The robust curve fitting problem, for both algebraic and Fourier (trigonometric) polynomials, in the presence of outliers is considered, and it is shown that there are polynomial-time algorithms in both settings that recover q up to l∞ error O(δ).

### Fitting algebraic curves to noisy data

- Mathematics, Computer ScienceSTOC '02
- 2002

A rigorous analysis of a brute force algorithm for the version of this problem where the data is generated from a mixture of polynomials when the mixing weights are "nondegenerate" is given.

### Improved decoding of Reed-Solomon and algebraic-geometric codes

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometric codes is presented, including a solution to a weighted curve fitting problem, which is of use in soft-decision decoding algorithms for Reed- Solomon codes.

### Direct least-squares fitting of algebraic surfaces

- MathematicsSIGGRAPH
- 1987

These methods all require (am+bn)n2 operations for fitting a surface of order n to m points, with a = 2 and b = 1/3 typically, except for spherical fit where b is larger due to the need to extract eigenvectors.

### Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography

- Computer ScienceCACM
- 1981

New results are derived on the minimum number of landmarks needed to obtain a solution, and algorithms are presented for computing these minimum-landmark solutions in closed form that provide the basis for an automatic system that can solve the Location Determination Problem under difficult viewing.

### Theory of Approximation

- MathematicsThe Mathematical Gazette
- 1960

Einfuhrung in Theorie und Anwendungen der Laplace-Transformation. By G. D oetsch . Pp. 301. Fr./D.M. 39.40. 1958. (Birkhauser, Basel und Stuttgart) This is an account of those properties of the…