Correlation Clustering with Low-Rank Matrices

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data

Predicted Residual Error Sum of Squares of Mixed Models: An Application for Genomic Prediction.

Genomic prediction is a statistical method to predict phenotypes of polygenic traits using high-throughput genomic data. Most diseases and behaviors in humans and animals are polygenic traits. The majority of agronomic traits in crops are also polygenic. Accurate prediction of these traits can help medical professionals diagnose acute diseases and breeders to increase food products, and therefore significantly contribute to human health and global food security. The best linear unbiased prediction (BLUP) is an important tool to analyze high-throughput genomic data for prediction. However, to judge the efficacy of the BLUP model with a particular set of predictors for a given trait, one has to provide an unbiased mechanism to evaluate the predictability. Cross-validation (CV) is an essential tool to achieve this goal, where a sample is partitioned into K parts of roughly equal size, one part is predicted using parameters estimated from the remaining K - 1 parts, and eventually every part is predicted using a sample excluding that part. Such a CV is called the K-fold CV. Unfortunately, CV presents a substantial increase in computational burden. We developed an alternative method, the HAT method, to replace CV. The new method corrects the estimated residual errors from the whole sample analysis using the leverage values of a hat matrix of the random effects to achieve the predicted residual errors. Properties of the HAT method were investigated using seven agronomic and 1000 metabolomic traits of an inbred rice population. Results showed that the HAT method is a very good approximation of the CV method. The method was also applied to 10 traits in 1495 hybrid rice with 1.6 million SNPs, and to human height of 6161 subjects with roughly 0.5 million SNPs of the Framingham heart study data. Predictabilities of the HAT and CV methods were all similar. The HAT method allows us to easily evaluate the predictabilities of genomic prediction for large numbers of traits in very large populations

Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths

We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on "Network Analysis and Modelling" (available at https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB

Active Semi-Supervised Learning Using Sampling Theory for Graph Signals

We consider the problem of offline, pool-based active semi-supervised learning on graphs. This problem is important when the labeled data is scarce and expensive whereas unlabeled data is easily available. The data points are represented by the vertices of an undirected graph with the similarity between them captured by the edge weights. Given a target number of nodes to label, the goal is to choose those nodes that are most informative and then predict the unknown labels. We propose a novel framework for this problem based on our recent results on sampling theory for graph signals. A graph signal is a real-valued function defined on each node of the graph. A notion of frequency for such signals can be defined using the spectrum of the graph Laplacian matrix. The sampling theory for graph signals aims to extend the traditional Nyquist-Shannon sampling theory by allowing us to identify the class of graph signals that can be reconstructed from their values on a subset of vertices. This approach allows us to define a criterion for active learning based on sampling set selection which aims at maximizing the frequency of the signals that can be reconstructed from their samples on the set. Experiments show the effectiveness of our method.Comment: 10 pages, 6 figures, To appear in KDD'1

Bandit-Aided Boosting

In this paper we apply multi-armed bandits (MABs) to accelerate ADABOOST. ADABOOST constructs a strong classifier in a stepwise fashion by selecting simple base classifiers and using their weighted "vote" to determine the final classification. We model this stepwise base classifier selection as a sequential decision problem, and optimize it with MABs. Each arm represent a subset of the base classifier set. The MAB gradually learns the "utility" of the subsets, and selects one of the subsets in each iteration. ADABOOST then searches only this subset instead of optimizing the base classifier over the whole space. The reward is defined as a function of the accuracy of the base classifier. We investigate how the MAB algorithms (UCB, UCT) can be applied in the case of boosted stumps, trees, and products of base classifiers. On benchmark datasets, our bandit-based approach achieves only slightly worse test errors than the standard boosted learners for a computational cost that is an order of magnitude smaller than with standard ADABOOST