We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb R^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.

We study the task of list-decodable linear regression using batches, recently introduced by Das et al. 2023.. In this setting, we are given $m$ batches with each batch containing $n$ points in $\mathbb R^d$. A batch is called clean if the points it contains are i.i.d. samples from an unknown linear regression distribution. For a parameter $\alpha \in (0, 1/2)$, an unknown $\alpha$-fraction of the batches are clean and no assumptions are made on the remaining batches. The goal is to output a small list of vectors at least one of which is close to the true regressor vector in $\ell_2$-norm. Das et al. 2023 gave an efficient algorithm for this task, under natural distributional assumptions, with the following guarantee. Under the assumption that the batch size satisfies $n \geq \tilde{\Omega}(\alpha^{-1})$ and the total number of batches is $m = \text{poly}(d, n, 1/\alpha)$, their algorithm runs in polynomial time and outputs a list of $O(1/\alpha^2)$ vectors at least one of which is $\tilde{O}(\alpha^{-1/2}/\sqrt{n})$ close to the target regressor. Here we design a new polynomial-time algorithm for this task with significantly stronger guarantees under the assumption that the low-degree moments of the covariates distribution are Sum-of-Squares (SoS) certifiably bounded. Specifically, for any constant $\delta>0$, as long as the batch size is $n \geq \Omega_{\delta}(\alpha^{-\delta})$ and the degree-$\Theta(1/\delta)$ moments of the covariates are SoS certifiably bounded, our algorithm uses $m = \text{poly}((dn)^{1/\delta}, 1/\alpha)$ batches, runs in polynomial-time, and outputs an $O(1/\alpha)$-sized list of vectors one of which is $O(\alpha^{-\delta/2}/\sqrt{n})$ close to the target. That is, our algorithm substantially improves both the minimum batch size and the final error guarantee, while achieving the optimal list size. Our approach leverages higher-order moment information by carefully combining the SoS paradigm interleaved with an iterative method and a novel list pruning procedure for this setting. In the process, we give an SoS proof of the Marcinkiewicz-Zygmund inequality that may be of broader applicability.

Non-Gaussian Component Analysis (NGCA) is the statistical task of finding a non-Gaussian direction in a high-dimensional dataset. Specifically, given i.i.d. samples from a distribution$P_{v}^{A}$on$\mathbb{R}^{n}$that behaves like a known distribution$A$in a hidden direction$v$and like a standard Gaussian in the orthogonal complement, the goal is to approximate the hidden direction. The standard formulation posits that the first$k$- moments of$A$match those of the standard Gaussian and the$k$-th moment differs. Under mild assumptions, this problem has sample complexity$O(n)$. On the other hand, all known efficient algorithms require$\Omega(n^{k/2})$samples. Prior work developed sharp Statistical Query and low-degree testing lower bounds suggesting an information-computation tradeoff for this problem. Here we study the complexity of NGCA in the Sum-of-Squares (SoS) framework. Our main contribution is the first super-constant degree SoS lower bound for NGCA. Specifically, we show that if the non-Gaussian distribution$A$matches the first$(k-1)$moments of$\mathrm{N}(\mathrm{O},\ 1)$and satisfies other mild conditions, then with fewer than$n^{(1-\varepsilon)k/2}$many samples from the normal distribution, with high probability, degree$(\log n)^{\frac{1}{2}-o_{n}(1)}\mathbf{SoS}$fails to refute the existence of such a direction$v$. Our result significantly strengthens prior work by establishing a super-polynomial information-computation tradeoff against a broader family of algorithms. As corollaries, we obtain SoS lower bounds for several problems in robust statistics and the learning of mixture models. Our SoS lower bound proof introduces a novel technique’ that we believe may be of broader interest, and a number of refinements over existing methods. As in previous work, we use the framework of [Barak et al. FOCS 2016], where we express the moment matrix$M$as a sum of graph matrices, find a factorization$M\approx LQL^{T}$using minimum vertex separators, and show that with high probability$Q$is positive semidefinite (PSD) while the errors are small. Our technical innovations involve the following. First, instead of the minimum weight separator used in prior work, we crucially make use of the minimum square separator. Second, proving that$Q$is PSD poses significant challenges due to an intrinsic reason. In all prior work, the major part of$Q$was always a constant term, meaning a matrix whose entries are constant functions of the input. Here, however, even after removing a small error term,$Q$remains a nontrivial linear combination of non-constant, equally dominating terms. We develop an algebraic method to address this difficulty, which may have wider applications. Specifically, we model the multiplications between the “important” graph matrices by an R.-algebra, construct a representation of this algebra, and use it to analyze$Q$. Via this approach, we show that the PSDness of$Q$boils down to the multiplicative identities of Hermite polynomials.

We study Gaussian sparse estimation tasks in Huber's contamination model with a focus on mean estimation, PCA, and linear regression. For each of these tasks, we give the first sample and computationally efficient robust estimators with optimal error guarantees, within constant factors. All prior efficient algorithms for these tasks incur quantitatively suboptimal error. Concretely, for Gaussian robust $k$-sparse mean estimation on $\mathbb{R}^d$ with corruption rate $\epsilon>0$, our algorithm has sample complexity $(k^2/\epsilon ^2)\mathrm{polylog}(d/\epsilon)$, runs in sample polynomial time, and approximates the target mean within $\ell_2$-error $O(\epsilon)$. Previous efficient algorithms inherently incur error $\Omega(\epsilon \sqrt{\log(1/\epsilon)})$. At the technical level, we develop a novel multidimensional filtering method in the sparse regime that may find other applications.

We investigate a variation of the 3D registration problem, named multi-model 3D registration. In the multi-model registration problem, we are given two point clouds picturing a set of objects at different poses (and possibly including points belonging to the background) and we want to simultaneously reconstruct how all objects moved between the two point clouds. This setup generalizes standard 3D registration where one wants to reconstruct a single pose, e.g., the motion of the sensor picturing a static scene. Moreover, it provides a mathematically grounded formulation for relevant robotics applications, e.g., where a depth sensor onboard a robot perceives a dynamic scene and has the goal of estimating its own motion (from the static portion of the scene) while simultaneously recovering the motion of all dynamic objects. We assume a correspondence-based setup where we have putative matches between the two point clouds and consider the practical case where these correspondences are plagued with outliers. We then propose a simple approach based on Expectation-Maximization (EM) and establish theoretical conditions under which the EM approach converges to the ground truth. We evaluate the approach in simulated and real datasets ranging from table-top scenes to self-driving scenarios and demonstrate its effectiveness when combined with state-of-the-art scene flow methods to establish dense correspondences.

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial distribution shifts, where the labels can be arbitrary, and the goal is to find a "best-fit" function. More precisely, given training samples from a reference distribution $p_0$, the goal is to approximate the vector $\mathbf{w}^*$ which minimizes the squared loss with respect to the worst-case distribution that is close in $\chi^2$-divergence to $p_{0}$. We design a computationally efficient algorithm that recovers a vector $ \hat{\mathbf{w}}$ satisfying $\mathbb{E}\_{p^*} (\sigma(\hat{\mathbf{w}} \cdot \mathbf{x}) - y)^2 \leq C \hspace{0.2em} \mathbb{E}\_{p^*} (\sigma(\mathbf{w}^* \cdot \mathbf{x}) - y)^2 + \epsilon$, where $C>1$ is a dimension-independent constant and $(\mathbf{w}^*, p^*)$ is the witness attaining the min-max risk $\min_{\mathbf{w}:\|\mathbf{w}\| \leq W} \max\_{p} \mathbb{E}\_{(\mathbf{x}, y) \sim p} (\sigma(\mathbf{w} \cdot \mathbf{x}) - y)^2 - \nu \chi^2(p, p_0)$. Our algorithm follows the primal-dual framework and is designed by directly bounding the risk with respect to the original, nonconvex $L_2^2$ loss. From an optimization standpoint, our work opens new avenues for the design of primal-dual algorithms under structured nonconvexity.

We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$. In particular, $y = g(w^* \cdot x) + \xi + \eps$ where $\xi$ is the oblivious noise drawn independently of $x$, satisfying $\Pr[\xi = 0] \geq o(1)$, and $\eps \sim \cN(0, \sigma^2)$. Our goal is to accurately recover a function $g(w \cdot x)$ with arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$. We present an algorithm that tackles the problem in its most general distribution-independent setting, where the solution may not be identifiable. The algorithm is designed to return the solution if it is identifiable, and otherwise return a small list of candidates, one of which is close to the true solution. Furthermore, we characterize a necessary and sufficient condition for identifiability, which holds in broad settings. The problem is identifiable when the quantile at which $\xi + \eps = 0$ is known, or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$ for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$. This is the first result for GLM regression which can handle more than half the samples being arbitrarily corrupted. Prior work focused largely on the setting of linear regression with oblivious noise, and giving algorithms under more restrictive assumptions.

We initiate the study of stochastic optimization with oblivious noise, broadly generalizing the standard heavy-tailed noise setup. In our setting, in addition to random observation noise, the stochastic gradient may be subject to independent \emph{oblivious noise}, which may not have bounded moments and is not necessarily centered. Specifically, we assume access to a noisy oracle for the stochastic gradient of $f$ at $x$, which returns a vector $\nabla f(\gamma, x) + \xi$, where $\gamma$ is the bounded variance observation noise and $\xi$ is the oblivious noise that is independent of $\gamma$ and $x$. The only assumption we make on the oblivious noise $\xi$ is that $\Pr[\xi = 0] \ge \alpha$, for some $\alpha \in (0, 1)$. In this setting, it is not information-theoretically possible to recover a single solution close to the target when the fraction of inliers $\alpha$ is less than $1/2$. Our main result is an efficient {\em list-decodable} learner that recovers a small list of candidates at least one of which is close to the true solution. On the other hand, if $\alpha = 1-\epsilon$, where $0< \epsilon < 1/2$ is sufficiently small constant, the algorithm recovers a single solution. Along the way, we develop a rejection-sampling-based algorithm to perform noisy location estimation, which may be of independent interest.

We study the problem of high-dimensional sparse mean estimation in the presence of an $\epsilon$-fraction of adversarial outliers. Prior work obtained sample and computationally efficient algorithms for this task for identity-covariance subgaussian distributions. In this work, we develop the first efficient algorithms for robust sparse mean estimation without a priori knowledge of the covariance. For distributions on $\mathbb{R}^d$ with ‘certifiably bounded’ $t$-th moments and sufficiently light tails, our algorithm achieves error of $O(\epsilon^{1-1/t})$ with sample complexity $m = (k\log(d))^{O(t)}/\epsilon^{2-2/t}$. For the special case of the Gaussian distribution, our algorithm achieves near-optimal error of $\tilde O(\epsilon)$ with sample complexity $m = O(k^4 \mathrm{polylog}(d))/\epsilon^2$. Our algorithms follow the Sum-of-Squares based proofs to algorithms approach. We complement our upper bounds with Statistical Query and low-degree polynomial testing lower bounds, providing evidence that the sample-time-error tradeoffs achieved by our algorithms are qualitatively best possible.

We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $\alpha \in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor \alpha m \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $\mu$. No assumptions are made on the remaining points, which form the majority of the dataset. The goal is to return a small list of candidates containing a vector $\hat \mu$ such that $\|\hat \mu - \mu\|_2$ is small. Prior work had studied the problem of list-decodable mean estimation in the dense setting. In this work, we develop a novel, conceptually simpler technique for list-decodable mean estimation. As the main application of our approach, we provide the first sample and computationally efficient algorithm for list-decodable sparse mean estimation. In particular, for distributions with ``certifiably bounded'' $t$-th moments in $k$-sparse directions and sufficiently light tails, our algorithm achieves error of $(1/\alpha)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/\alpha$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee $\Theta (\sqrt{\log(1/\alpha)})$ with quasi-polynomial complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds.

We characterize the measurement complexity of compressed sensing of signals drawn from a known prior distribution, even when the support of the prior is the entire space (rather than, say, sparse vectors). We show for Gaussian measurements and \emph{any} prior distribution on the signal, that the posterior sampling estimator achieves near-optimal recovery guarantees. Moreover, this result is robust to model mismatch, as long as the distribution estimate (e.g., from an invertible generative model) is close to the true distribution in Wasserstein distance. We implement the posterior sampling estimator for deep generative priors using Langevin dynamics, and empirically find that it produces accurate estimates with more diversity than MAP.

This work tackles the issue of fairness in the context of generative procedures, such as image super-resolution, which entail different definitions from the standard classification setting. Moreover, while traditional group fairness definitions are typically defined with respect to specified protected groups – camouflaging the fact that these groupings are artificial and carry historical and political motivations – we emphasize that there are no ground truth identities. For instance, should South and East Asians be viewed as a single group or separate groups? Should we consider one race as a whole or further split by gender? Choosing which groups are valid and who belongs in them is an impossible dilemma and being “fair” with respect to Asians may require being “unfair” with respect to South Asians. This motivates the introduction of definitions that allow algorithms to be \emph{oblivious} to the relevant groupings. We define several intuitive notions of group fairness and study their incompatibilities and trade-offs. We show that the natural extension of demographic parity is strongly dependent on the grouping, and \emph{impossible} to achieve obliviously. On the other hand, the conceptually new definition we introduce, Conditional Proportional Representation, can be achieved obliviously through Posterior Sampling. Our experiments validate our theoretical results and achieve fair image reconstruction using state-of-the-art generative models.

We consider the fundamental problem of ReLU regression, where the goal is to output the best fitting ReLU with respect to square loss given access to draws from some unknown distribution. We give the first efficient, constant-factor approximation algorithm for this problem assuming the underlying distribution satisfies some weak concentration and anti-concentration conditions (and includes, for example, all log-concave distributions). This solves the main open problem of Goel et al., who proved hardness results for any exact algorithm for ReLU regression (up to an additive $\epsilon$). Using more sophisticated techniques, we can improve our results and obtain a polynomial-time approximation scheme for any subgaussian distribution. Given the aforementioned hardness results, these guarantees can not be substantially improved. Our main insight is a new characterization of {\em surrogate losses} for nonconvex activations. While prior work had established the existence of convex surrogates for monotone activations, we show that properties of the underlying distribution actually induce strong convexity for the loss, allowing us to relate the global minimum to the activation’s {\em Chow parameters}.

We give the first outlier-robust efficient algorithm for clustering a mixture of k statistically separated d - dimensional Gaussians ( k-GMMs). Concretely, our algorithm takes input an ε-corrupted sample from a k-GMM and outputs an approximate clustering that misclassifies at most kO(k)(ε+η) fraction of the points whenever every pair of mixture components are separated by 1-exp(-poly(k/η)) in total variation distance. This is the statistically weakest possible notion of separation and allows, for e.g., clustering of mixtures with components with the same mean with covariances differing in a single unknown direction or separated in Frobenius distance. The running time of our algorithm is dpoly(k/η). Such results were not known prior to our work, even for k=2. More generally, our algorithms succeed for mixtures of any distribution that satisfies two well-studied analytic assumptions - sum-of-squares certifiable hypercontractivity and anti-concentration. As an immediate corollary, they extend to clustering mixtures of arbitrary affine transforms of the uniform distribution on the d-dimensional unit sphere. Even the information theoretic clusterability of separated distributions satisfying our analytic assumptions was not known and is likely to be of independent interest. Our algorithms build on the recent flurry of work relying on certifiable anti-concentration first introduced in [1], [2]. Our techniques expand the sum-of-squares toolkit to show robust certifiability of TV-separated Gaussian clusters in data. This involves giving a low-degree sum-of-squares proof of statements that relate parameter (i.e. mean and covariances) distance to total variation distance by relying only on hypercontractivity and anti-concentration.

We give the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution for a broad class of algorithms. In the regression setting, we prove that gradient descent run on any classifier with respect to square loss will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes and sufficiently smooth classifiers. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm will fail to achieve small test error in polynomial time. Our lower bounds hold for commonly used activations such as ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions.

The goal of compressed sensing is to learn a structured signal $x$ from a limited number of noisy linear measurements $y \approx Ax$. In traditional compressed sensing, “structure” is represented by sparsity in some known basis. Inspired by the success of deep learning in modeling images, recent work starting with Bora-Jalal-Price-Dimakis’17 has instead considered structure to come from a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. We present two results establishing the difficulty and strength of this latter task, showing that existing bounds are tight: First, we provide a lower bound matching the Bora et.al upper bound for compressed sensing with $L$-Lipschitz generative models $G$ which holds even for the more relaxed goal of \emph{non-uniform} recovery. Second, we show that generative models generalize sparsity as a representation of structure by constructing a ReLU-based neural network with $2$ hidden layers and $O(n)$ activations per layer whose range is precisely the set of all $k$-sparse vectors.

We study high-dimensional sparse estimation tasks in a robust setting where a constant fraction of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust sparse mean estimation and robust sparse PCA. We give the first practically viable robust estimators for these problems. In more detail, our algorithms are sample and computationally efficient and achieve near-optimal robustness guarantees. In contrast to prior provable algorithms which relied on the ellipsoid method, our algorithms use spectral techniques to iteratively remove outliers from the dataset. Our experimental evaluation on synthetic data shows that our algorithms are scalable and significantly outperform a range of previous approaches, nearly matching the best error rate without corruptions.

We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\opt < 1$ be the population loss of the best-fitting ReLU. We prove: \begin{itemize} \item Finding a ReLU with square-loss $\opt + \epsilon$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply --{\em unconditionally}-- that gradient descent cannot converge to the global minimum in polynomial time. \item There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error $O(\opt^{2/3})$. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0/1$ loss. \end{itemize} Prior work due to Soltanolkotabi \cite{soltanolkotabi2017learning} showed that gradient descent {\em can} find the best-fitting ReLU with respect to Gaussian marginals, if the training set is {\em exactly} labeled by a ReLU.

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 \alpha < 1, our algorithm takes as input a sample {(xi,yi)}{i \leq n} of n linear equations where \alpha n of the equations satisfy yi = \langle x_i,\ell^\rangle +\zeta for some small noise \zeta and (1-\alpha) n of the equations are {\em arbitrarily} chosen. It outputs a list L of size O(1/\alpha) - a fixed constant - that contains an \ell that is close to \ell^. Our algorithm succeeds whenever the inliers are chosen from a certifiably anti-concentrated distribution D. In particular, this gives a (d/\alpha)^{O(1/\alpha^8)} time algorithm to find a O(1/\alpha) size list when the inlier distribution is a standard Gaussian. For discrete product distributions that are anti-concentrated only in regular directions, we give an algorithm that achieves similar guarantee under the promise that \ell^* has all coordinates of the same magnitude. To complement our result, we prove that the anti-concentration assumption on the inliers is information-theoretically necessary. To solve the problem we introduce a new framework for list-decodable learning that strengthens the ``identifiability to algorithms'' paradigm based on the sum-of-squares method.

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 other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a 1.09 approximation is impossible even with infinitely many samples.