We study the problem of approximately recovering a probability distribution given noisy measurements of its Chebyshev polynomial moments. This problem arises broadly across algorithms, statistics, and machine learning. By leveraging a \emph{global decay bound} on the coefficients in the Chebyshev expansion of any Lipschitz function, we sharpen prior work, proving that accurate recovery in the Wasserstein distance is possible with more noise than previously known. Our result immediately yields a number of applications: (1) We give a simple “linear query” algorithm for constructing a differentially private synthetic data distribution with Wasserstein-$1$ error $\tilde{O}(1/n)$ based on a dataset of $n$ points in $[-1,1]$. This bound is optimal up to log factors and matches a recent breakthrough of Boedihardjo, Strohmer, and Vershynin [Probab. Theory. Rel., 2024], which uses a more complex “superregular random walk” method to beat an $O(1/\sqrt{n})$ accuracy barrier inherent to earlier approaches. (2) We give an $\tilde{O}(n^2/\epsilon)$ time algorithm for the linear algebraic problem of estimating the spectral density of an $n\times n$ symmetric matrix up to $\epsilon$ error in the Wasserstein distance. Our result accelerates prior methods from Chen et al. [ICML 2021] and Braverman et al. [STOC 2022]. (3) We tighten an analysis of Vinayak, Kong, Valiant, and Kakade [ICML 2019] on the maximum likelihood estimator for the statistical problem of “Learning Populations of Parameters”, extending the parameter regime in which sample optimal results can be obtained. Beyond these main results, we provide an extension of our bound to estimating distributions in $d > 1$ dimensions. We hope that these bounds will find applications more broadly to problems involving distribution recovery from noisy moment information.

We consider the problem of estimating the spectral density of a normalized graph adjacency matrix. Concretely, given an undirected graph $G = (V, E, w)$ with $n$ nodes and positive edge weights $w \in \mathbb{R}^{E}_{> 0}$, the goal is to return eigenvalue estimates $\widehat{\lambda}_1 \le \cdots\le \widehat{\lambda}_n$ such that \begin{align*} \frac{1}{n} \sum_{i\in\{1,\ldots, n\}}|\widehat{\lambda}_i-\lambda_i(N_G)|\le \varepsilon, \end{align*} where ${\lambda}_1(N_G)\le \cdots\le{\lambda}_n(N_G)$ are the eigenvalues of $G$’s normalized adjacency matrix, $N_G$. This goal is equivalent to requiring that the Wasserstein-1 distance between the uniform distribution on $\lambda_1, \ldots, \lambda_n$ and the uniform distribution on $\widehat{\lambda}_1, \ldots, \widehat{\lambda}_n$ is less than $\varepsilon$. We provide a randomized algorithm that achieves the guarantee above with $O(n\varepsilon^{-2})$ queries to a degree and neighbor oracle and in $O(n\varepsilon^{-3})$ time. This improves on previous state-of-the-art methods, including an $O(n\varepsilon^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $\varepsilon$, a $2^{O(\varepsilon^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call \emph{nuclear sparsification}. We provide an $O(n\varepsilon^{-2})$-query and $O(n\varepsilon^{-2})$-time algorithm for computing $O(n\varepsilon^{-2})$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first \emph{deterministic} algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph).

We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $\epsilon$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $(1\pm2^{-\Omega(1/\epsilon)})$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $2^{O(1/\epsilon)}$ random walks initiated at uniformly random nodes in the graph.As a strengthening of our main result, we show that improving the dependence on $1/\epsilon$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $\epsilon$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $2^{\Omega(1/\epsilon)}$ random walks of length $2^{\Omega(1/\epsilon)}$ started at random nodes.

We study the problem of k-way clustering in signed graphs. Considerable attention in recent years has been devoted to analyzing and modeling signed graphs, where the affinity measure between nodes takes either positive or negative values. Recently, Cucuringu et al. (2019) proposed a spectral method, namely SPONGE (Signed Positive over Negative Generalized Eigenproblem), which casts the clustering task as a generalized eigenvalue problem optimizing a suitably defined objective function. This approach is motivated by social balance theory, where the clustering task aims to decompose a given network into disjoint groups, such that individuals within the same group are connected by as many positive edges as possible, while individuals from different groups are mainly connected by negative edges. Through extensive numerical experiments, SPONGE was shown to achieve state-of-the-art empirical performance. On the theoretical front, Cucuringu et al. (2019) analyzed SPONGE, as well as the popular Signed Laplacian based spectral method under the setting of a Signed Stochastic Block Model, for k=2 equal-sized clusters, in the regime where the graph is moderately dense. In this work, we build on the results in Cucuringu et al. (2019) on two fronts for the normalized versions of SPONGE and the Signed Laplacian. Firstly, for both algorithms, we extend the theoretical analysis in Cucuringu et al. (2019) to the general setting of k >= 2 unequal-sized clusters in the moderately dense regime. Secondly, we introduce regularized versions of both methods to handle sparse graphs -- a regime where standard spectral methods are known to underperform -- and provide theoretical guarantees under the same setting of a Signed Stochastic Block Model. To the best of our knowledge, regularized spectral methods have so far not been considered in the setting of clustering signed graphs. We complement our theoretical results with an extensive set of numerical experiments on synthetic data, and three real world data sets standard in the signed networks literature.

$k$-means clustering is NP-hard in the worst case but previous work has shown efficient algorithms assuming the optimal $k$-means clusters are \emph{stable} under additive or multiplicative perturbation of data. This has two caveats. First, we do not know how to efficiently verify this property of optimal solutions that are NP-hard to compute in the first place. Second, the stability assumptions required for polynomial time $k$-means algorithms are often unreasonable when compared to the ground-truth clusters in real-world data. A consequence of multiplicative perturbation resilience is \emph{center proximity}, that is, every point is closer to the center of its own cluster than the center of any other cluster, by some multiplicative factor $\alpha > 1$. We study the problem of minimizing the Euclidean $k$-means objective only over clusterings that satisfy $\alpha$-center proximity. We give a simple algorithm to find the optimal $\alpha$-center-proximal $k$-means clustering in running time exponential in $k$ and $1/(\alpha - 1)$ but linear in the number of points and the dimension. We define an analogous $\alpha$-center proximity condition for outliers, and give similar algorithmic guarantees for $k$-means with outliers and $\alpha$-center proximity. On the hardness side we show that for any $\alpha’ > 1$, there exists an $\alpha \leq \alpha’$, $(\alpha >1)$, and an $\e_0 > 0$ such that minimizing the $k$-means objective over clusterings that satisfy $\alpha$-center proximity is NP-hard to approximate within a multiplicative $(1+\e_0)$ factor.