131 research outputs found
Efficient Clustering on Riemannian Manifolds: A Kernelised Random Projection Approach
Reformulating computer vision problems over Riemannian manifolds has
demonstrated superior performance in various computer vision applications. This
is because visual data often forms a special structure lying on a lower
dimensional space embedded in a higher dimensional space. However, since these
manifolds belong to non-Euclidean topological spaces, exploiting their
structures is computationally expensive, especially when one considers the
clustering analysis of massive amounts of data. To this end, we propose an
efficient framework to address the clustering problem on Riemannian manifolds.
This framework implements random projections for manifold points via kernel
space, which can preserve the geometric structure of the original space, but is
computationally efficient. Here, we introduce three methods that follow our
framework. We then validate our framework on several computer vision
applications by comparing against popular clustering methods on Riemannian
manifolds. Experimental results demonstrate that our framework maintains the
performance of the clustering whilst massively reducing computational
complexity by over two orders of magnitude in some cases
Grassmann Learning for Recognition and Classification
Computational performance associated with high-dimensional data is a common challenge for real-world classification and recognition systems. Subspace learning has received considerable attention as a means of finding an efficient low-dimensional representation that leads to better classification and efficient processing. A Grassmann manifold is a space that promotes smooth surfaces, where points represent subspaces and the relationship between points is defined by a mapping of an orthogonal matrix. Grassmann learning involves embedding high dimensional subspaces and kernelizing the embedding onto a projection space where distance computations can be effectively performed. In this dissertation, Grassmann learning and its benefits towards action classification and face recognition in terms of accuracy and performance are investigated and evaluated. Grassmannian Sparse Representation (GSR) and Grassmannian Spectral Regression (GRASP) are proposed as Grassmann inspired subspace learning algorithms. GSR is a novel subspace learning algorithm that combines the benefits of Grassmann manifolds with sparse representations using least squares loss §¤1-norm minimization for improved classification. GRASP is a novel subspace learning algorithm that leverages the benefits of Grassmann manifolds and Spectral Regression in a framework that supports high discrimination between classes and achieves computational benefits by using manifold modeling and avoiding eigen-decomposition. The effectiveness of GSR and GRASP is demonstrated for computationally intensive classification problems: (a) multi-view action classification using the IXMAS Multi-View dataset, the i3DPost Multi-View dataset, and the WVU Multi-View dataset, (b) 3D action classification using the MSRAction3D dataset and MSRGesture3D dataset, and (c) face recognition using the ATT Face Database, Labeled Faces in the Wild (LFW), and the Extended Yale Face Database B (YALE). Additional contributions include the definition of Motion History Surfaces (MHS) and Motion Depth Surfaces (MDS) as descriptors suitable for activity representations in video sequences and 3D depth sequences. An in-depth analysis of Grassmann metrics is applied on high dimensional data with different levels of noise and data distributions which reveals that standardized Grassmann kernels are favorable over geodesic metrics on a Grassmann manifold. Finally, an extensive performance analysis is made that supports Grassmann subspace learning as an effective approach for classification and recognition
Development of statistical methods for the analysis of single-cell RNA-seq data
Single-cell RNA-sequencing profiles the transcriptome of cells from diverse populations. A popular intermediate data format is a large count matrix of genes x cells. This type of data brings several analytical challenges. Here, I present three projects that I worked on during my PhD that address particular aspects of working with such datasets:
- The large number of cells in the count matrix is a challenge for fitting gamma-Poisson generalized linear models with existing tools. I developed a new R package called glmGamPoi to address this gap. I optimized the overdispersion estimation procedure to be quick and robust for datasets with many cells and small counts. I compared the performance against two popular tools (edgeR and DESeq2) and find that my inference is 6x to 13x faster and achieves a higher likelihood for a majority of the genes in four single-cell datasets.
- The variance of single-cell RNA-seq counts depends on their mean but many existing statistical tools have optimal performance when the variance is uniform. Accordingly, variance-stabilizing transformations are applied to unlock the large number of methods with such an requirement. I compared four approaches to variance-stabilize the data based on the delta method, model residuals, inferred latent expression state or count factor analysis. I describe the theoretical strength and weaknesses, and compare their empirical performance in a benchmark on simulated and real single-cell data. I find that none of the mathematically more sophisticated transformations consistently outperform the simple log(y/s+1) transformation.
- Multi-condition single-cell data offers the opportunity to find differentially expressed genes for individual cell subpopulations. However, the prevalent approach to analyze such data is to start by dividing the cells into discrete populations and then test for differential expression within each group. The results are interpretable but may miss interesting cases by (1) choosing the cluster size too small and lacking power to detect effects or (2) choosing the cluster size too large and obscuring interesting effects apparent on a smaller scale. I developed a new statistical framework for the analysis of multi-condition single-cell data that avoids the premature discretization. The approach performs regression on the latent subspaces occupied by the cells in each condition. The method is implemented as an R package called lemur
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Variational cross-validation of slow dynamical modes in molecular kinetics
Markov state models (MSMs) are a widely used method for approximating the
eigenspectrum of the molecular dynamics propagator, yielding insight into the
long-timescale statistical kinetics and slow dynamical modes of biomolecular
systems. However, the lack of a unified theoretical framework for choosing
between alternative models has hampered progress, especially for non-experts
applying these methods to novel biological systems. Here, we consider
cross-validation with a new objective function for estimators of these slow
dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures
the ability of a rank- projection operator to capture the slow subspace of
the system. It is shown that a variational theorem bounds the GMRQ from above
by the sum of the first eigenvalues of the system's propagator, but that
this bound can be violated when the requisite matrix elements are estimated
subject to statistical uncertainty. This overfitting can be detected and
avoided through cross-validation. These result make it possible to construct
Markov state models for protein dynamics in a way that appropriately captures
the tradeoff between systematic and statistical errors
- …