We consider the problem of approximate $K$-means clustering with outliers and side information provided by same-cluster queries and possibly noisy answers. Our solution shows that, under some mild assumptions on the smallest cluster size, one can obtain an $(1+\epsilon)$-approximation for the optimal potential with probability at least $1-\delta$, where $\epsilon>0$ and $\delta\in(0,1)$, using an expected number of $O(\frac{K^3}{\epsilon \delta})$ noiseless same-cluster queries and comparison-based clustering of complexity $O(ndK + \frac{K^3}{\epsilon \delta})$; here, $n$ denotes the number of points and $d$ the dimension of space. Compared to a handful of other known approaches that perform importance sampling to account for small cluster sizes, the proposed query technique reduces the number of queries by a factor of roughly $O(\frac{K^6}{\epsilon^3})$, at the cost of possibly missing very small clusters. We extend this settings to the case where some queries to the oracle produce erroneous information, and where certain points, termed outliers, do not belong to any clusters. Our proof techniques differ from previous methods used for $K$-means clustering analysis, as they rely on estimating the sizes of the clusters and the number of points needed for accurate centroid estimation and subsequent nontrivial generalizations of the double Dixie cup problem. We illustrate the performance of the proposed algorithm both on synthetic and real datasets, including MNIST and CIFAR $10$.

Algorithms for clustering points in metric spaces is a long-studied area of research. Clustering has seen a multitude of work both theoretically, in understanding the approximation guarantees possible for many objective functions such as k-median and k-means clustering, and experimentally, in finding the fastest algorithms and seeding procedures for Lloyd's algorithm. The performance of a given clustering algorithm depends on the specific application at hand, and this may not be known up front. For example, a "typical instance" may vary depending on the application, and different clustering heuristics perform differently depending on the instance. In this paper, we define an infinite family of algorithms generalizing Lloyd's algorithm, with one parameter controlling the the initialization procedure, and another parameter controlling the local search procedure. This family of algorithms includes the celebrated k-means++ algorithm, as well as the classic farthest-first traversal algorithm. We design efficient learning algorithms which receive samples from an application-specific distribution over clustering instances and learn a near-optimal clustering algorithm from the class. We show the best parameters vary significantly across datasets such as MNIST, CIFAR, and mixtures of Gaussians. Our learned algorithms never perform worse than k-means++, and on some datasets we see significant improvements.

We advocate Laplacian K-modes for joint clustering and density mode finding, and propose a concave-convex relaxation of the problem, which yields a parallel algorithm that scales up to large datasets and high dimensions. We optimize a tight bound (auxiliary function) of our relaxation, which, at each iteration, amounts to computing an independent update for each cluster-assignment variable, with guar- anteed convergence. Therefore, our bound optimizer can be trivially distributed for large-scale data sets. Furthermore, we show that the density modes can be obtained as byproducts of the assignment variables via simple maximum-value operations whose additional computational cost is linear in the number of data points. Our formulation does not need storing a full affinity matrix and computing its eigenvalue decomposition, neither does it perform expensive projection steps and Lagrangian-dual inner iterates for the simplex constraints of each point. Fur- thermore, unlike mean-shift, our density-mode estimation does not require inner- loop gradient-ascent iterates. It has a complexity independent of feature-space dimension, yields modes that are valid data points in the input set and is appli- cable to discrete domains as well as arbitrary kernels. We report comprehensive experiments over various data sets, which show that our algorithm yields very competitive performances in term of optimization quality (i.e., the value of the discrete-variable objective at convergence) and clustering accuracy.

We provide the first information theoretical tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretic necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model.

We introduce the Sublevel Set (SS) method, a generic method to obtain sufficient guarantees of near-optimality and uniqueness (up to small perturbations) for a clustering. This method can be instantiated for a variety of clustering loss functions for which convex relaxations exist. Obtaining the guarantees in practice amounts to solving a convex optimization. We demonstrate the applicability of this method by obtaining distribution free guarantees for K-means clustering on realistic data sets.

Consider the situation where you are given an existing $k$-way clustering $\pi$. A challenge for explainable AI is to find a compact and distinct explanations of each cluster which in this paper is using instance-level descriptors/tags from a common dictionary. Since the descriptors/tags were not given to the clustering method, this is not a semi-supervised learning situation. We show that the \emph{feasibility} problem of just testing whether any distinct description (not the most compact) exists is generally intractable for just two clusters. This means that unless \textbf{P} = \cnp, there cannot exist an efficient algorithm for the cluster description problem. Hence, we explore ILP formulations for smaller problems and a relaxed but restricted setting that leads to a polynomial time algorithm for larger problems. We explore several extension to the basic setting such as the ability to ignore some instances and composition constraints on the descriptions of the clusters. We show our formulation's usefulness on Twitter data where the communities were found using social connectivity (i.e. \texttt{follower} relation) but the explanation of the communities is based on behavioral properties of the nodes (i.e. hashtag usage) not available to the clustering method.

We design new differentially private algorithms for the Euclidean k-means problem, both in the centralized model and in the local model of differential privacy. In both models, our algorithms achieve significantly improved error guarantees than the previous state-of-the-art. In addition, in the local model, our algorithm significantly reduces the number of interaction rounds. Although the problem has been widely studied in the context of differential privacy, all of the existing constructions achieve only super constant approximation factors. We present, for the first time, efficient private algorithms for the problem with constant multiplicative error. Furthermore, we show how to modify our algorithms so they compute private coresets for k-means clustering in both models.

The inference of the causal relationship between a pair of observed variables is a fundamental problem in science, and most existing approaches are based on one single causal model. In practice, however, observations are often collected from multiple sources with heterogeneous causal models due to certain uncontrollable factors, which renders causal analysis results obtained by a single model skeptical. In this paper, we generalize the Additive Noise Model (ANM) to a mixture model, which consists of a finite number of ANMs, and provide the condition of its causal identifiability. To conduct model estimation, we propose Gaussian Process Partially Observable Model (GPPOM), and incorporate independence enforcement into it to learn latent parameter associated with each observation. Causal inference and clustering according to the underlying generating mechanisms of the mixture model are addressed in this work. Experiments on synthetic and real data demonstrate the effectiveness of our proposed approach.

We introduce a framework to transfer knowledge acquired from a repository of (heterogeneous) supervised datasets to new unsupervised datasets. Our perspective avoids the subjectivity inherent in unsupervised learning by reducing it to supervised learning, and provides a principled way to evaluate unsupervised algorithms. We demonstrate the versatility of our framework via rigorous agnostic bounds on a variety of unsupervised problems. In the context of clustering, our approach helps choose the number of clusters and the clustering algorithm, remove the outliers, and provably circumvent Kleinberg's impossibility result. Experiments across hundreds of problems demonstrate improvements in performance on unsupervised data with simple algorithms despite the fact our problems come from heterogeneous domains. Additionally, our framework lets us leverage deep networks to learn common features across many small datasets, and perform zero shot learning.

We study the problem of finding clusters in random bipartite graphs. We present a simple two-step algorithm which provably finds even tiny clusters of size $O(n^\epsilon)$, where $n$ is the number of vertices in the graph and $\epsilon > 0$. Previous algorithms were only able to identify clusters of size $\Omega(\sqrt{n})$. We evaluate the algorithm on synthetic and on real-world data; the experiments show that the algorithm can find extremely small clusters even in presence of high destructive noise.