Solution Path Clustering with Adaptive Concave Penalty

Fast accumulation of large amounts of complex data has created a need for more sophisticated statistical methodologies to discover interesting patterns and better extract information from these data. The large scale of the data often results in challenging high-dimensional estimation problems where only a minority of the data shows specific grouping patterns. To address these emerging challenges, we develop a new clustering methodology that introduces the idea of a regularization path into unsupervised learning. A regularization path for a clustering problem is created by varying the degree of sparsity constraint that is imposed on the differences between objects via the minimax concave penalty with adaptive tuning parameters. Instead of providing a single solution represented by a cluster assignment for each object, the method produces a short sequence of solutions that determines not only the cluster assignment but also a corresponding number of clusters for each solution. The optimization of the penalized loss function is carried out through an MM algorithm with block coordinate descent. The advantages of this clustering algorithm compared to other existing methods are as follows: it does not require the input of the number of clusters; it is capable of simultaneously separating irrelevant or noisy observations that show no grouping pattern, which can greatly improve data interpretation; it is a general methodology that can be applied to many clustering problems. We test this method on various simulated datasets and on gene expression data, where it shows better or competitive performance compared against several clustering methods.Comment: 36 page

Training Gaussian Mixture Models at Scale via Coresets

How can we train a statistical mixture model on a massive data set? In this work we show how to construct coresets for mixtures of Gaussians. A coreset is a weighted subset of the data, which guarantees that models fitting the coreset also provide a good fit for the original data set. We show that, perhaps surprisingly, Gaussian mixtures admit coresets of size polynomial in dimension and the number of mixture components, while being independent of the data set size. Hence, one can harness computationally intensive algorithms to compute a good approximation on a significantly smaller data set. More importantly, such coresets can be efficiently constructed both in distributed and streaming settings and do not impose restrictions on the data generating process. Our results rely on a novel reduction of statistical estimation to problems in computational geometry and new combinatorial complexity results for mixtures of Gaussians. Empirical evaluation on several real-world datasets suggests that our coreset-based approach enables significant reduction in training-time with negligible approximation error

Efficient algorithm for the k-means problem with Must-Link and Cannot-Link constraints

Constrained clustering, such as k -means with instance-level Must-Link (ML) and Cannot-Link (CL) auxiliary information as the constraints, has been extensively studied recently, due to its broad applications in data science and AI. Despite some heuristic approaches, there has not been any algorithm providing a non-trivial approximation ratio to the constrained k -means problem. To address this issue, we propose an algorithm with a provable approximation ratio of O(logk) when only ML constraints are considered. We also empirically evaluate the performance of our algorithm on real-world datasets having artificial ML and disjoint CL constraints. The experimental results show that our algorithm outperforms the existing greedy-based heuristic methods in clustering accuracy

Doctor of Philosophy

dissertationStatistical learning theory has garnered attention during the last decade because it provides the theoretical and mathematical framework for solving pattern recognition problems, such as dimensionality reduction, clustering, and shape analysis. In statis

Computational methods for large-scale single-cell RNA-seq and multimodal data

Emerging single cell genomics technologies such as single cell RNA-seq (scRNA-seq) and single cell ATAC-seq provide new opportunities for discovery of previously unknown cell types, facilitating the study of biological processes such as tumor progression, and delineating molecular mechanism differences between species. Due to the high dimensionality of the data produced by the technologies, computation and mathematics have been the cornerstone in decoding meaningful information from the data. Computational models have been challenged by the exponential growth of the data thanks to the continuing decrease in sequencing costs and growth of large-scale genomic projects such as the Human Cell Atlas. In addition, recent single-cell technologies have enabled us to measure multiple modalities such as transcriptome, protome, and epigenome in the same cell. This requires us to establish new computational methods which can cope with multiple layers of the data. To address these challenges, the main goal of this thesis was to develop computational methods and mathematical models for analyzing large-scale scRNA-seq and multimodal omics data. In particular, I have focused on fundamental single-cell analysis such as clustering and visualization. The most common task in scRNA-seq data analysis is the identification of cell types. Numerous methods have been proposed for this problem with a current focus on methods for the analysis of large scale scRNA-seq data. I developed Specter, a computational method that utilizes recent algorithmic advances in fast spectral clustering and ensemble learning. Specter achieves a substantial improvement in accuracy over existing methods and identifies rare cell types with high sensitivity. Specter allows us to process a dataset comprising 2 million cells in just 26 minutes. Moreover, the analysis of CITE-seq data, that simultaneously provides gene expression and protein levels, showed that Specter is able to incorporate multimodal omics measurements to resolve subtle transcriptomic differences between subpopulations of cells. We have effectively handled big data for clustering analysis using Specter. The question is how to cope with the big data for other downstream analyses such as trajectory inference and data integration. The most simple scheme is to shrink the data by selecting a subset of cells (the sketch) that best represents the full data set. Therefore I developed an algorithm called Sphetcher that makes use of the thresholding technique to efficiently pick representative cells that evenly cover the transcriptomic space occupied by the original data set. I showed that the sketch computed by Sphetcher constitutes a more accurate presentation of the original transcriptomic landscape than existing methods, which leads to a more balanced composition of cell types and a large fraction of rare cell types in the sketch. Sphetcher bridges the gap between the scalability of computational methods and the volume of the data. Moreover, I demonstrated that Sphetcher can incorporate prior information (e.g. cell labels) to inform the inference of the trajectory of human skeletal muscle myoblast differentiation. The biological processes such as development, differentiation, and cell cycle can be monitored by performing single cell sequencing at different time points, each corresponding to a snapshot of the process. A class of computational methods called trajectory inference aims to reconstruct the developmental trajectories from these snapshots. Trajectory inference (TI) methods such as Monocle, can computationally infer a pseudotime variable which serves as a proxy for developmental time. In order to compare two trajectories inferred by TI methods, we need to align the pseudotime between two trajectories. Current methods for aligning trajectories are based on the concept of dynamic time warping, which is limited to simple linear trajectories. Since complex trajectories are common in developmental processes, I adopted arboreal matchings to compare and align complex trajectories with multiple branch points diverting cells into alternative fates. Arboreal matchings were originally proposed in the context of phylogenetic trees and I theoretically linked them to dynamic time warping. A suite of exact and heuristic algorithms for aligning complex trajectories was implemented in a software Trajan. When aligning single-cell trajectories describing human muscle differentiation and myogenic reprogramming, Trajan automatically identifies the core paths from which we are able to reproduce recently reported barriers to reprogramming. In a perturbation experiment, I showed that Trajan correctly maps identical cells in a global view of trajectories, as opposed to a pairwise application of dynamic time warping. Visualization using dimensionality reduction techniques such as t-SNE and UMAP is a fundamental step in the analysis of high-dimensional data. Visualization has played a pivotal role in discovering the dynamic trends in single cell genomics data. I developed j-SNE and j-UMAP as their generalizations to the joint visualization of multimodal omics data, e.g., CITE-seq data. The approach automatically learns the relative importance of each modality in order to obtain a concise representation of the data. When comparing with the conventional approaches, I demonstrated that j-SNE and j-UMAP produce unified embeddings that better agree with known cell types and that harmonize RNA and protein velocity landscapes

A New Generation of Mixture-Model Cluster Analysis with Information Complexity and the Genetic EM Algorithm

In this dissertation, we extend several relatively new developments in statistical model selection and data mining in order to improve one of the workhorse statistical tools - mixture modeling (Pearson, 1894). The traditional mixture model assumes data comes from several populations of Gaussian distributions. Thus, what remains is to determine how many distributions, their population parameters, and the mixing proportions. However, real data often do not fit the restrictions of normality very well. It is likely that data from a single population exhibiting either asymmetrical or nonnormal tail behavior could be erroneously modeled as two populations, resulting in suboptimal decisions. To avoid these pitfalls, we develop the mixture model under a broader distributional assumption by fitting a group of multivariate elliptically-contoured distributions (Anderson and Fang, 1990; Fang et al., 1990). Special cases include the multivariate Gaussian and power exponential distributions, as well as the multivariate generalization of the Student’s T. This gives us the flexibility to model nonnormal tail and peak behavior, though the symmetry restriction still exists. The literature has many examples of research generalizing the Gaussian mixture model to other distributions (Farrell and Mersereau, 2004; Hasselblad, 1966; John, 1970a), but our effort is more general. Further, we generalize the mixture model to be non-parametric, by developing two types of kernel mixture model. First, we generalize the mixture model to use the truly multivariate kernel density estimators (Wand and Jones, 1995). Additionally, we develop the power exponential product kernel mixture model, which allows the density to adjust to the shape of each dimension independently. Because kernel density estimators enforce no functional form, both of these methods can adapt to nonnormal asymmetric, kurtotic, and tail characteristics. Over the past two decades or so, evolutionary algorithms have grown in popularity, as they have provided encouraging results in a variety of optimization problems. Several authors have applied the genetic algorithm - a subset of evolutionary algorithms - to mixture modeling, including Bhuyan et al. (1991), Krishna and Murty (1999), and Wicker (2006). These procedures have the benefit that they bypass computational issues that plague the traditional methods. We extend these initialization and optimization methods by combining them with our updated mixture models. Additionally, we “borrow” results from robust estimation theory (Ledoit and Wolf, 2003; Shurygin, 1983; Thomaz, 2004) in order to data-adaptively regularize population covariance matrices. Numerical instability of the covariance matrix can be a significant problem for mixture modeling, since estimation is typically done on a relatively small subset of the observations. We likewise extend various information criteria (Akaike, 1973; Bozdogan, 1994b; Schwarz, 1978) to the elliptically-contoured and kernel mixture models. Information criteria guide model selection and estimation based on various approximations to the Kullback-Liebler divergence. Following Bozdogan (1994a), we use these tools to sequentially select the best mixture model, select the best subset of variables, and detect influential observations - all without making any subjective decisions. Over the course of this research, we developed a full-featured Matlab toolbox (M3) which implements all the new developments in mixture modeling presented in this dissertation. We show results on both simulated and real world datasets. Keywords: mixture modeling, nonparametric estimation, subset selection, influence detection, evidence-based medical diagnostics, unsupervised classification, robust estimation