no code implementations • 31 Mar 2024 • Samuel B. Hopkins, Anqi Li
We first confirm a folklore belief that a malicious adversary who can corrupt an inverse-polynomial fraction of the leaves of their choosing makes this inference impossible.
no code implementations • 29 Nov 2023 • Ainesh Bakshi, Vincent Cohen-Addad, Samuel B. Hopkins, Rajesh Jayaram, Silvio Lattanzi
Multi-dimensional Scaling (MDS) is a family of methods for embedding an $n$-point metric into low-dimensional Euclidean space.
no code implementations • 30 Jul 2023 • Daniel Freund, Samuel B. Hopkins
We investigate practical algorithms to find or disprove the existence of small subsets of a dataset which, when removed, reverse the sign of a coefficient in an ordinary least squares regression involving that dataset.
no code implementations • 28 Jan 2023 • Gavin Brown, Samuel B. Hopkins, Adam Smith
Our algorithm runs in time $\tilde{O}(nd^{\omega - 1} + nd/\varepsilon)$, where $\omega < 2. 38$ is the matrix multiplication exponent.
no code implementations • 9 Dec 2022 • Samuel B. Hopkins, Gautam Kamath, Mahbod Majid, Shyam Narayanan
We study the relationship between adversarial robustness and differential privacy in high-dimensional algorithmic statistics.
1 code implementation • 1 Nov 2022 • Kristian Georgiev, Samuel B. Hopkins
We establish a simple connection between robust and differentially-private algorithms: private mechanisms which perform well with very high probability are automatically robust in the sense that they retain accuracy even if a constant fraction of the samples they receive are adversarially corrupted.
no code implementations • 19 May 2022 • Afonso S. Bandeira, Ahmed El Alaoui, Samuel B. Hopkins, Tselil Schramm, Alexander S. Wein, Ilias Zadik
We define a free-energy based criterion for hardness and formally connect it to the well-established notion of low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal.
no code implementations • 5 Mar 2022 • Samuel B. Hopkins, Tselil Schramm, Jonathan Shi
We give a spectral algorithm for decomposing overcomplete order-4 tensors, so long as their components satisfy an algebraic non-degeneracy condition that holds for nearly all (all but an algebraic set of measure $0$) tensors over $(\mathbb{R}^d)^{\otimes 4}$ with rank $n \le d^2$.
no code implementations • 25 Nov 2021 • Samuel B. Hopkins, Gautam Kamath, Mahbod Majid
SoS proofs to algorithms is a key theme in numerous recent works in high-dimensional algorithmic statistics -- estimators which apparently require exponential running time but whose analysis can be captured by low-degree Sum of Squares proofs can be automatically turned into polynomial-time algorithms with the same provable guarantees.
no code implementations • 13 Sep 2020 • Matthew Brennan, Guy Bresler, Samuel B. Hopkins, Jerry Li, Tselil Schramm
Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems.
no code implementations • NeurIPS 2020 • Jingqiu Ding, Samuel B. Hopkins, David Steurer
For the case of Gaussian noise, the top eigenvector of the given matrix is a widely-studied estimator known to achieve optimal statistical guarantees, e. g., in the sense of the celebrated BBP phase transition.
no code implementations • NeurIPS 2020 • Samuel B. Hopkins, Jerry Li, Fred Zhang
In this paper, we provide a meta-problem and a duality theorem that lead to a new unified view on robust and heavy-tailed mean estimation in high dimensions.
no code implementations • 13 May 2020 • Ilias Diakonikolas, Samuel B. Hopkins, Daniel Kane, Sushrut Karmalkar
The key ingredients of this proof are a novel use of SoS-certifiable anti-concentration and a new characterization of pairs of Gaussians with small (dimension-independent) overlap in terms of their parameter distance.
1 code implementation • NeurIPS 2019 • Yihe Dong, Samuel B. Hopkins, Jerry Li
In robust mean estimation the goal is to estimate the mean $\mu$ of a distribution on $\mathbb{R}^d$ given $n$ independent samples, an $\varepsilon$-fraction of which have been corrupted by a malicious adversary.
no code implementations • 19 Sep 2018 • Samuel B. Hopkins
We study polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector.
Statistics Theory Data Structures and Algorithms Statistics Theory
no code implementations • 30 Sep 2017 • Samuel B. Hopkins, David Steurer
in constant average degree graphs---up to what we conjecture to be the computational threshold for this model.
no code implementations • 8 Dec 2015 • Samuel B. Hopkins, Tselil Schramm, Jonathan Shi, David Steurer
For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent $\approx 1. 086$) that approximately recovers a component of a random 3-tensor over $\mathbb R^n$ of rank up to $\tilde \Omega(n^{4/3})$.
no code implementations • 12 Jul 2015 • Samuel B. Hopkins, Jonathan Shi, David Steurer
We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = \tau \cdot v_0^{\otimes 3} + A$, where $\tau \geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover the planted vector $v_0$.