### Flexible Models for Microclustering with Application to Entity Resolution

Most generative models for clustering implicitly assume that the number of data points in each cluster grows linearly with the total number of data points. Finite mixture models, Dirichlet process mixture models, and Pitman--Yor process mixture models make this assumption, as do all other infinitely exchangeable clustering models. However, for some applications, this assumption is inappropriate. For example, when performing entity resolution, the size of each cluster should be unrelated to the size of the data set, and each cluster should contain a negligible fraction of the total number of data points. These applications require models that yield clusters whose sizes grow sublinearly with the size of the data set. We address this requirement by defining the microclustering property and introducing a new class of models that can exhibit this property. We compare models within this class to two commonly used clustering models using four entity-resolution data sets.

### Breaking the Small Cluster Barrier of Graph Clustering

This paper investigates graph clustering in the planted cluster model in the presence of {\em small clusters}. Traditional results dictate that for an algorithm to provably correctly recover the clusters, {\em all} clusters must be sufficiently large (in particular, $\tilde{\Omega}(\sqrt{n})$ where $n$ is the number of nodes of the graph). We show that this is not really a restriction: by a more refined analysis of the trace-norm based recovery approach proposed in Jalali et al. (2011) and Chen et al. (2012), we prove that small clusters, under certain mild assumptions, do not hinder recovery of large ones. Based on this result, we further devise an iterative algorithm to recover {\em almost all clusters} via a "peeling strategy", i.e., recover large clusters first, leading to a reduced problem, and repeat this procedure. These results are extended to the {\em partial observation} setting, in which only a (chosen) part of the graph is observed.The peeling strategy gives rise to an active learning algorithm, in which edges adjacent to smaller clusters are queried more often as large clusters are learned (and removed). From a high level, this paper sheds novel insights on high-dimensional statistics and learning structured data, by presenting a structured matrix learning problem for which a one shot convex relaxation approach necessarily fails, but a carefully constructed sequence of convex relaxationsdoes the job.

### Identifying Linear Models in Multi-Resolution Population Data using Minimum Description Length Principle to Predict Household Income

One shirt size cannot fit everybody, while we cannot make a unique shirt that fits perfectly for everyone because of resource limitation. This analogy is true for the policy making. Policy makers cannot establish a single policy to solve all problems for all regions because each region has its own unique issue. In the other extreme, policy makers also cannot create a policy for each small village due to the resource limitation. Would it be better if we can find a set of largest regions such that the population of each region within this set has common issues and we can establish a single policy for them? In this work, we propose a framework using regression analysis and minimum description length (MDL) to find a set of largest areas that have common indicators, which can be used to predict household incomes efficiently. Given a set of household features, and a multi-resolution partition that represents administrative divisions, our framework reports a set C* of largest subdivisions that have a common model for population-income prediction. We formalize a problem of finding C* and propose the algorithm as a solution. We use both simulation datasets as well as a real-world dataset of Thailand's population household information to demonstrate our framework performance and application. The results show that our framework performance is better than the baseline methods. We show the results of our method can be used to find indicators of income prediction for many areas in Thailand. By increasing these indicator values, we expect people in these areas to gain more incomes. Hence, the policy makers can plan to establish the policies by using these indicators in our results as a guideline to solve low-income issues. Our framework can be used to support policy makers to establish policies regarding any other dependent variable beyond incomes in order to combat poverty and other issues.

### Flexible Models for Microclustering with Application to Entity Resolution

Most generative models for clustering implicitly assume that the number of data points in each cluster grows linearly with the total number of data points. Finite mixture models, Dirichlet process mixture models, and Pitman-Yor process mixture models make this assumption, as do all other infinitely exchangeable clustering models. However, for some applications, this assumption is inappropriate. For example, when performing entity resolution, the size of each cluster should be unrelated to the size of the data set, and each cluster should contain a negligible fraction of the total number of data points. These applications require models that yield clusters whose sizes grow sublinearly with the size of the data set. We address this requirement by defining the microclustering property and introducing a new class of models that can exhibit this property. We compare models within this class to two commonly used clustering models using four entity-resolution data sets.

### Feature Selection in Clustering Problems

A novel approach to combining clustering and feature selection is presented. Itimplements a wrapper strategy for feature selection, in the sense that the features are directly selected by optimizing the discriminative powerof the used partitioning algorithm. On the technical side, we present an efficient optimization algorithm with guaranteed local convergence property.The only free parameter of this method is selected by a resampling-based stability analysis. Experiments with real-world datasets demonstrate that our method is able to infer both meaningful partitions and meaningful subsets of features.