Multidimensional scaling is an important dimension reduction tool in statistics and machine learning. Yet few theoretical results characterizing its statistical performance exist, not to mention any in high dimensions. By considering a unified framework that includes low, moderate and high dimensions, we study multidimensional scaling in the setting of clustering noisy data. Our results suggest that, the classical multidimensional scaling can be modified to further improve the quality of embedded samples, especially when the noise level increases. To this end, we propose {\it modified multidimensional scaling} which applies a nonlinear transformation to the sample eigenvalues. The nonlinear transformation depends on the dimensionality, sample size and moment of noise. We show that modified multidimensional scaling followed by various clustering algorithms can achieve exact recovery, i.e., all the cluster labels can be recovered correctly with probability tending to one. Numerical simulations and two real data applications lend strong support to our proposed methodology.

We consider K-means clustering in networked environments (e.g., internet of things (IoT) and sensor networks) where data is inherently distributed across nodes and processing power at each node may be limited. We consider a clustering algorithm referred to as networked K-means, or N K-means, which relies only on local neighborhood information exchange. Information exchange is limited to low-dimensional statistics and not raw data at the agents. The proposed approach develops a parametric family of multi-agent clustering objectives (parameterized by ρ) and associated distributed N K-means algorithms (also parameterized by ρ). The NK-means algorithm with parameter ρ converges to a set of fixed points relative to the associated multi-agent objective (designated as generalized minima). By appropriate choice of ρ, the set of generalized minima may be brought arbitrarily close to the set of Lloyd's minima. Thus, the NK-means algorithm may be used to compute Lloyd's minima of the collective dataset up to arbitrary accuracy. Keywords: K-means clustering, Lloyd's minima, distributed algorithms, distributed machine learning, network information processing

We propose a new type of generative model for high-dimensional data that learns a manifold geometry of the data, rather than density, and can generate points evenly along this manifold. This is in contrast to existing generative models that represent data density, and are strongly affected by noise and other artifacts of data collection. We demonstrate how this approach corrects sampling biases and artifacts, thus improves several downstream data analysis tasks, such as clustering and classification. Finally, we demonstrate that this approach is especially useful in biology where, despite the advent of single-cell technologies, rare subpopulations and gene-interaction relationships are affected by biased sampling. We show that SUGAR can generate hypothetical populations, and it is able to reveal intrinsic patterns and mutual-information relationships between genes on a single-cell RNA sequencing dataset of hematopoiesis.

People belong to multiple communities, words belong to multiple topics, and books cover multiple genres; overlapping clusters are commonplace. Many existing overlapping clustering methods model each person (or word, or book) as a non-negative weighted combination of "exemplars" who belong solely to one community, with some small noise. Geometrically, each person is a point on a cone whose corners are these exemplars. This basic form encompasses the widely used Mixed Membership Stochastic Blockmodel of networks and its degree-corrected variants, as well as topic models such as LDA. We show that a simple one-class SVM yields provably consistent parameter inference for all such models, and scales to large datasets. Experimental results on several simulated and real datasets show our algorithm (called SVM-cone) is both accurate and scalable.

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 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 $10$.

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.

Clustering is a technique for the analysis of datasets obtained by empirical studies in several disciplines with a major application for biomedical research. Essentially, clustering algorithms are executed by machines aiming at finding groups of related points in a dataset. However, the result of grouping depends on both metrics for point-to-point similarity and rules for point-to-group association. Indeed, non-appropriate metrics and rules can lead to undesirable clustering artifacts. This is especially relevant for datasets, where groups with heterogeneous structures co-exist. In this work, we propose an algorithm that achieves clustering by exploring the paths between points. This allows both, to evaluate the properties of the path (such as gaps, density variations, etc.), and expressing the preference for certain paths. Moreover, our algorithm supports the integration of existing knowledge about admissible and non-admissible clusters by training a path classifier. We demonstrate the accuracy of the proposed method on challenging datasets including points from synthetic shapes in publicly available benchmarks and microscopy data.

Kleinberg [18] stated three axioms that any clustering procedure should satisfy and showed there is no clustering procedure that simultaneously satisfies all three. One of these, called the consistency axiom, requires that when the data is modified in a helpful way, i.e. if points in the same cluster are made more similar and those in different ones made less similar, the algorithm should output the same clustering. To circumvent this impossibility result, research has focused on considering clustering procedures that have a clustering quality measure (or a cost) and showing that a modification of Kleinberg's axioms that takes cost into account lead to feasible clustering procedures. In this work, we take a different approach, based on the observation that the consistency axiom fails to be satisfied when the "correct" number of clusters changes. We modify this axiom by making use of cost functions to determine the correct number of clusters, and require that consistency holds only if the number of clusters remains unchanged. We show that single linkage satisfies the modified axioms, and if the input is well-clusterable, some popular procedures such as k-means also satisfy the axioms, taking a step towards explaining the success of these objective functions for guiding the design of algorithms. 1 Introduction In a highly influential paper, Kleinberg [18] showed that clustering is impossible in the following sense: there exists no clustering function, i.e. a function that takes a point-set and a pairwise dissimilarity function

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.