In list-decodable subspace recovery, the input is a collection of $n$ points $αn$ (for some $α\ll 1/2$) of which are drawn i.i.d. from a distribution $\mathcal{D}$ with a isotropic rank $r$ covariance $Π_*$ (the \emph{inliers}) and the rest are arbitrary, potential adversarial outliers. The goal is to recover a $O(1/α)$ size list of candidate covariances that contains a $\hatΠ$ close to $Π_*$. Two recent independent works (Raghavendra-Yau, Bakshi-Kothari 2020) gave the first efficient algorithm for this problem. These results, however, obtain an error that grows with the dimension (linearly in [RY] and logarithmically in BK) at the cost of quasi-polynomial running time) and rely on \emph{certifiable anti-concentration} - a relatively strict condition satisfied essentially only by the Gaussian distribution.
In this work, we improve on these results on all three fronts: \emph{dimension-independent} error via a faster fixed-polynomial running time under less restrictive distributional assumptions. Specifically, we give a $poly(1/α) d^{O(1)}$ time algorithm that outputs a list containing a $\hatΠ$ satisfying $\|\hatΠ -Π_*\|_F \leq O(1/α)$. Our result only needs $\mathcal{D}$ to have \emph{certifiably hypercontractive} degree 2 polynomials. As a result, in addition to Gaussians, our algorithm applies to the uniform distribution on the hypercube and $q$-ary cubes and arbitrary product distributions with subgaussian marginals. Prior work (Raghavendra and Yau, 2020) had identified such distributions as potential hard examples as such distributions do not exhibit strong enough anti-concentration. When $\mathcal{D}$ satisfies certifiable anti-concentration, we obtain a stronger error guarantee of $\|\hatΠ-Π_*\|_F \leq η$ for any arbitrary $η> 0$ in $d^{O(poly(1/α) + \log (1/η))}$ time.
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than $1/2$ fraction of examples. For any $α< 1$, our algorithm takes as input a sample $\{(x_i,y_i)\}_{i \leq n}$ of $n$ linear equations where $αn$ of the equations satisfy $y_i = \langle x_i,\ell^*\rangle +ζ$ for some small noise $ζ$ and $(1-α)n$ of the equations are {\em arbitrarily} chosen. It outputs a list $L$ of size $O(1/α)$ - a fixed constant - that contains an $\ell$ that is close to $\ell^*$. Our algorithm succeeds whenever the inliers are chosen f…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
Traditionally, robust statistics has focused on designing estimators tolerant to a minority of contaminated data. Robust list-decodable learning focuses on the more challenging regime where only a minority $\frac 1 k$ fraction of the dataset is drawn from the distribution of interest, and no assumptions are made on the remaining data. We study the fundamental task of list-decodable mean estimation in high dimensions. Our main result is a new list-decodable mean estimation algorithm for bounded covariance distributions with optimal sample complexity and error rate, running in nearly-PCA time. A…
We give the first polynomial time algorithm for \emph{list-decodable covariance estimation}. For any $α> 0$, our algorithm takes input a sample $Y \subseteq \mathbb{R}^d$ of size $n\geq d^{\mathsf{poly}(1/α)}$ obtained by adversarially corrupting an $(1-α)n$ points in an i.i.d. sample $X$ of size $n$ from the Gaussian distribution with unknown mean $μ_*$ and covariance $Σ_*$. In $n^{\mathsf{poly}(1/α)}$ time, it outputs a constant-size list of $k = k(α)= (1/α)^{\mathsf{poly}(1/α)}$ candidate parameters that, with high probability, contains a $(\hatμ,\hatΣ)$ such that the total variation distan…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set $T$ of $n$ points in $\mathbb{R}^d$ and a parameter $0< α<\frac 1 2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a well-behaved distribution $\mathcal{D}$ and the remaining $(1-α)$-fraction are arbitrary. The goal is to output a small list of vectors, at least one of which is close to the mean of $\mathcal{D}$. We develop new algorithms for list-decodable mean estimation, achieving nearly-optimal statistical guarantees, wi…
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 $α$ 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 $O(\frac{1}α)$ subspaces one of which is nontrivially correlated with the planted subspace. We provide the first polynomial time al…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加州大学圣地亚哥分校）；Web 3 Foundation（Web3基金会）
We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples. Specifically, we are given a set $T$ of labeled examples $(x, y) \in \mathbb{R}^d \times \mathbb{R}$ and a parameter $0< α<1/2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a linear regression model with Gaussian covariates, and the remaining $(1-α)$-fraction of the points are drawn from an arbitrary noise distribution. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector. Our m…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the problem of {\em list-decodable mean estimation} for bounded covariance distributions. Specifically, we are given a set $T$ of points in $\mathbb{R}^d$ with the promise that an unknown $α$-fraction of points in $T$, where $0< α< 1/2$, are drawn from an unknown mean and bounded covariance distribution $D$, and no assumptions are made on the remaining points. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the mean of $D$. We give the first practically viable estimator for this problem. In more detail, our algorithm is sample and co…
In the list-decodable learning setup, an overwhelming majority (say a $1-β$-fraction) of the input data consists of outliers and the goal of an algorithm is to output a small list $\mathcal{L}$ of hypotheses such that one of them agrees with inliers. We develop a framework for list-decodable learning via the Sum-of-Squares SDP hierarchy and demonstrate it on two basic statistical estimation problems {\it Linear regression:} Suppose we are given labelled examples $\{(X_i,y_i)\}_{i \in [N]}$ containing a subset $S$ of $βN$ {\it inliers} $\{X_i \}_{i \in S}$ that are drawn i.i.d. from standard Ga…
Robust Mixture Learning when Outliers Overwhelm Small Groups
ETH Zurich（苏黎世联邦理工学院）；TU Munich（慕尼黑工业大学）；Lucerne School of Computer Science and Information Technology（卢塞恩计算机科学与信息技术学院）；University of Copenhagen（哥本哈根大学）
Learning from data in the presence of outliers is a fundamental problem in statistics. Until recently, no computationally efficient algorithms were known to compute the mean of a high dimensional distribution under natural assumptions in the presence of even a small fraction of outliers. In this paper, we consider robust statistics in the presence of overwhelming outliers where the majority of the dataset is introduced adversarially. With only an $α< 1/2$ fraction of "inliers" (clean data) the mean of a distribution is unidentifiable. However, in their influential work, [CSV17] introduces a po…
University of Southern California（南加州大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the problem of list-decodable Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. {\bf List-Decodable Mean Estimation.} Fix any $d \in \mathbb{Z}_+$ and $0< α<1/2$. We design an algorithm with runtime $O (\mathrm{poly}(n/α)^{d})$ that outputs a list of $O(1/α)$ many candidate vectors such that with high probability one of the candidates is within $\ell_2$-distance $O(α^{-1/(2d)})$ from the true mean. The only …
We give a complete answer to the following basic question: "What is the maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable codes with non-vanishing information rate?" This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best-known result was that a $γ\leq 0.707$ fraction of insertions is tolerable by some binary code family. For any desired $ε> 0$, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of $γ$ fraction of insertions and …
We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in Σ'^n$ with $|Σ'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in Σ^n$, symbol-by-symbol, with $I$ results in a string $S' \in Σ''^n$ where $Σ'' = Σ\times Σ'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the e…
In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$, using $m \ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs wi…
