Kernel dimensionality reduction (KDR) algorithms find a low dimensional representation of the original data by optimizing kernel dependency measures that are capable of capturing nonlinear relationships. The standard strategy is to first map the data into a high dimensional feature space using kernels prior to a projection onto a low dimensional space. While KDR methods can be easily solved by keeping the most dominant eigenvectors of the kernel matrix, its features are no longer easy to interpret. Alternatively, Interpretable KDR (IKDR) is different in that it projects onto a subspace \textit{before} the kernel feature mapping, therefore, the projection matrix can indicate how the original features linearly combine to form the new features. Unfortunately, the IKDR objective requires a non-convex manifold optimization that is difficult to solve and can no longer be solved by eigendecomposition. Recently, an efficient iterative spectral (eigendecomposition) method (ISM) has been proposed for this objective in the context of alternative clustering. However, ISM only provides theoretical guarantees for the Gaussian kernel. This greatly constrains ISM's usage since any kernel method using ISM is now limited to a single kernel. This work extends the theoretical guarantees of ISM to an entire family of kernels, thereby empowering ISM to solve any kernel method of the same objective. In identifying this family, we prove that each kernel within the family has a surrogate $\Phi$ matrix and the optimal projection is formed by its most dominant eigenvectors. With this extension, we establish how a wide range of IKDR applications across different learning paradigms can be solved by ISM. To support reproducible results, the source code is made publicly available on \url{https://github.com/ANONYMIZED}.

Crowdclustering is a practical way to incorporate domain knowledge into clustering, by combining opinions from multiple domain experts. Existing crowdclustering methods analyze binary pairwise similarity labels. However, in some applications, experts might provide partition labels. If we convert partition labels into pairwise similarity, then it would be difficult to understand the relationships between clustering solutions from different experts. In this paper, we propose a crowdclustering model that directly analyzes partition labels. The proposed model adopts a novel approach based on a modified multinomial logistic regression model, which simultaneously learns the number of clusters and determines hyper-planes that partition samples into clusters. The proposed model also learns a mapping between the latent clusters and expert labels, revealing the agreements and disagreements between experts. Experiments on benchmark data demonstrate that the proposed model simultaneously learns the number of clusters and discovers the clustering structure. An experiment on disease subtyping problem illustrates that the proposed model helps us understand the agreement and disagreement between experts.

Inferring causal relations from observational data is widely used for knowledge discovery in healthcare and economics. To investigate whether a treatment can affect an outcome of interest, we focus on answering counterfactual questions of this type: what would a patient’s blood pressure be had he/she received a different treatment? Nearest neighbor matching (NNM) sets the counterfactual outcome of any treatment (control) sample to be equal to the factual outcome of its nearest neighbor in the control (treatment) group. Although being simple, flexible and interpretable, most NNM approaches could be easily misled by variables that do not affect the outcome. In this paper, we address this challenge by learning subspaces that are predictive of the outcome variable for both the treatment group and control group. Applying NNM in the learned subspaces leads to more accurate estimation of the counterfactual outcomes and therefore treatment effects. We introduce an informative subspace learning algorithm by maximizing the nonlinear dependence between the candidate subspace and the outcome variable measured by the Hilbert-Schmidt Independence Criterion (HSIC). We propose a scalable estimator of HSIC, called HSIC-RFF that reduces the quadratic computational and storage complexities (with respect to the sample size) of the naive HSIC implementation to linear through constructing random Fourier features. We also prove an upper bound on the approximation error of the HSIC-RFF estimator. Experimental results on simulated datasets and real-world datasets demonstrate our proposed approach outperforms existing NNM approaches and other commonly used regression-based methods for counterfactual inference.

Utilizing expert input often improves clustering performance. However in a knowledge discovery problem, ground truth is unknown even to an expert. Thus, instead of one expert, we solicit the opinion from multiple experts. The key question motivating this work is: which experts should be assigned higher weights when there is disagreement on whether to put a pair of samples in the same group? To model the uncertainty in constraints from different experts, we build a probabilistic model for pairwise constraints through jointly modeling each expert’s accuracy and the mapping from features to latent cluster assignments. After learning our probabilistic discriminative clustering model and accuracies of different experts, 1) samples that were not annotated by any expert can be clustered using the discriminative clustering model; and 2) experts with higher accuracies are automatically assigned higher weights in determining the latent cluster assignments. Experimental results on UCI benchmark datasets and a real-world disease subtyping dataset demonstrate that our proposed approach outperforms competing alternatives, including semi-crowdsourced clustering, semi-supervised clustering with constraints from majority voting, and consensus clustering.

Expert input can improve clustering performance. In today’s collaborative environment, the availability of crowdsourced multiple expert input is becoming common. Given multiple experts’ inputs, most existing approaches can only discover one clustering structure. However, data is multi-faced by nature and can be clustered in different ways (also known as views). In an exploratory analysis problem where ground truth is not known, different experts may have diverse views on how to cluster data. In this paper, we address the problem on how to automatically discover multiple ways to cluster data given potentially diverse inputs from multiple uncertain experts. We propose a novel Bayesian probabilistic model that automatically learns the multiple expert views and the clustering structure associated with each view. The benefits of learning the experts’ views include 1) enabling the discovery of multiple diverse clustering structures, and 2) improving the quality of clustering solution in each view by assigning higher weights to experts with higher confidence. In our approach, the expert views, multiple clustering structures and expert confidences are jointly learned via variational inference. Experimental results on synthetic datasets, benchmark datasets and a real-world disease subtyping problem show that our proposed approach outperforms competing baselines, including meta clustering, semi-supervised clustering, semi-crowdsourced clustering and consensus clustering.

Clustering is a challenging problem because given the same data set, it can be grouped in multiple different ways. Which of these clustering solutions is interesting depends on its domain application. Thus, incorporating domain expert input often improves clustering performance. However, most existing semi-supervised clustering techniques can only incorporate instance-level constraints (a few labels or must-link/cannot-link constraints), which domain experts may not be comfortable providing in knowledge discovery problems because categories are not known. Fortunately, domain experts often have an idea regarding properties that clustering solutions should have in order to be useful in domain application based on domain relevant scores. In this paper, we provide a framework for jointly optimizing the usefulness and quality of a clustering solution. Experiments on a synthetic data, a benchmark data, and a real-world disease subtyping problem demonstrate the usefulness of our proposed approach.

In many applications, not all the features used to represent data samples are important. Often only a few features are relevant for the prediction task. The choice of dependence measures often affect the final result of many feature selection methods. To select features that have complex nonlinear relationships with the response variable, the dependence measure should be equitable, a concept proposed by Reshef et al. (2011); that is, the dependence measure treats linear and nonlinear relationships equally. Recently, Kinney and Atwal (2014) gave a mathematical definition of self- equitability. In this paper, we introduce a new concept of robust-equitability and identify a robust- equitable copula dependence measure, the robust copula dependence (RCD) measure. RCD is based on the $L_1$-distance of the copula density from uniform and we show that it is equitable under both equitability definitions. We also prove theoretically that RCD is much easier to estimate than mutual information. Because of these theoretical properties, the RCD measure has the following advantages compared to existing dependence measures: it is robust to different relationship forms and robust to unequal sample sizes of different features. Experiments on both synthetic and real-world data sets confirm the theoretical analysis, and illustrate the advantage of using the dependence measure RCD for feature selection.

We introduce a novel Bayesian nonparametric model that uses the concept of disease trajectories for disease subtype identification. Although our model is general, we demonstrate that by treating fractions of tissue patterns derived from medical images as compositional data, our model can be applied to study distinct progression trends between population subgroups. Specifically, we apply our algorithm to quantitative emphysema measurements obtained from chest CT scans in the COPDGene Study and show several distinct progression patterns. As emphysema is one of the major components of chronic obstructive pulmonary disease (COPD), the third leading cause of death in the United States [1], an improved definition of emphysema and COPD subtypes is of great interest. We investigate several models with our algorithm, and show that one with age , pack years (a measure of cigarette exposure), and smoking status as predictors gives the best compromise between estimated predictive performance and model complexity. This model identified nine subtypes which showed significant associations to seven single nucleotide polymorphisms (SNPs) known to associate with COPD. Additionally, this model gives better predictive accuracy than multiple, multivariate ordinary least squares regression as demonstrated in a five-fold cross validation analysis. We view our subtyping algorithm as a contribution that can be applied to bridge the gap between CT-level assessment of tissue composition to population-level analysis of compositional trends that vary between disease subtypes.

Feature selection aims to select relevant features to improve the performance of predictors. Many feature selection methods depend on the choice of dependence measures. To select features that have complex nonlinear relationships with the response variable, the dependence measure should be equitable: treating linear and nonlinear relationships equally. In this paper we introduce the concept of robust-equitability and a robust-equitable dependence measure copula correlation (Ccor). This measure has the following advantages compared to existing dependence measures: it is robust to different relationship forms and robust to unequal sample sizes of different features. In contrast, existing dependence measures cannot take these factors into account simultaneously. Experiments on synthetic and real-world datasets confirm our theoretical analysis, and illustrates its advantage in feature selection.

Clustering is a technique that is usually applied as a tool for exploratory data analysis. Because of the exploratory nature of this task, it would be beneficial if a clustering method generates interpretable results, and allows incorporating domain knowledge. This motivates us to develop a probabilistic discriminative model that learns a rectangular decision rule for each cluster, we call Discriminative Rectangle Mixture (DReaM) model. DReaM gives interpretable clustering results, because the rectangular decision rules discovered explicitly illustrate how one cluster is defined and differs from other clusters. It also facilitates us to take advantage of existing rules because we can choose informative prior distributions for the rectangular rules. Moreover, DReaM allows that the features for generating rules do not have to be the same as the features for discovering cluster structure. We approximate the distribution for the rules discovered via variational inference. Experimental results demonstrate that DReaM gives more interpretable clustering results, and yet its performance is comparable to existing clustering methods when solving traditional clustering. Furthermore, in real applications, DReaM is able to effectively take advantage of domain knowledge, and to generate reasonable clustering results.