Kernel k-Groups via Hartigan's Method

Energy statistics was proposed by Sz\'{e}kely in the 80's inspired by Newton's gravitational potential in classical mechanics, and it provides a model-free hypothesis test for equality of distributions. In its original form, energy statistics was formulated in Euclidean spaces. More recently, it was generalized to metric spaces of negative type. In this paper, we consider a formulation for the clustering problem using a weighted version of energy statistics in spaces of negative type. We show that this approach leads to a quadratically constrained quadratic program in the associated kernel space, establishing connections with graph partitioning problems and kernel methods in unsupervised machine learning. To find local solutions of such an optimization problem, we propose an extension of Hartigan's method to kernel spaces. Our method has the same computational cost as kernel k-means algorithm, which is based on Lloyd's heuristic, but our numerical results show an improved performance, especially in high dimensions.

The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing

Distance-based methods, also called "energy statistics", are leading methods for two-sample and independence tests from the statistics community. Kernel methods, developed from "kernel mean embeddings", are leading methods for two-sample and independence tests from the machine learning community. Previous works demonstrated the equivalence of distance and kernel methods only at the population level, for each kind of test, requiring an embedding theory of kernels. We propose a simple, bijective transformation between semimetrics and nondegenerate kernels. We prove that for finite samples, two-sample tests are special cases of independence tests, and the distance-based statistic is equivalent to the kernel-based statistic, including the biased, unbiased, and normalized versions. In other words, upon setting the kernel or metric to be bijective of each other, running any of the four algorithms will yield the exact same answer up to numerical precision. This deepens and unifies our understanding of interpoint comparison based methods.

Linear Optimal Low Rank Projection for High-Dimensional Multi-Class Data

Classifying samples into categories becomes intractable when a single sample can have millions to billions of features, such as in genetics or imaging data. Principal Components Analysis (PCA) is widely used to identify a low-dimensional representation of such features for further analysis. However, PCA ignores class labels, such as whether or not a subject has cancer, thereby discarding information that could substantially improve downstream classification performance. We describe an approach, "Linear Optimal Low-rank" projection (LOL), which extends PCA by incorporating the class labels in a fashion that is advantageous over existing supervised dimensionality reduction techniques. We prove, and substantiate with synthetic experiments, that LOL leads to a better representation of the data for subsequent classification than other linear approaches, while adding negligible computational cost. We then demonstrate that LOL substantially outperforms PCA in differentiating cancer patients from healthy controls using genetic data, and in differentiating gender using magnetic resonance imaging data with $>$500 million features and 400 gigabytes of data. LOL therefore allows the solution of previous intractable problems, yet requires only a few minutes to run on a desktop computer.

Connectome Smoothing via Low-rank Approximations

In statistical connectomics, the quantitative study of brain networks, estimating the mean of a population of graphs based on a sample is a core problem. Often, this problem is especially difficult because the sample or cohort size is relatively small, sometimes even a single subject. While using the element-wise sample mean of the adjacency matrices is a common approach, this method does not exploit any underlying structural properties of the graphs. We propose using a low-rank method which incorporates tools for dimension selection and diagonal augmentation to smooth the estimates and improve performance over the naive methodology for small sample sizes. Theoretical results for the stochastic blockmodel show that this method offers major improvements when there are many vertices. Similarly, we demonstrate that the low-rank methods outperform the standard sample mean for a variety of independent edge distributions as well as human connectome data derived from magnetic resonance imaging, especially when sample sizes are small. Moreover, the low-rank methods yield "eigen-connectomes", which correlate with the lobe-structure of the human brain and superstructures of the mouse brain. These results indicate that low-rank methods are an important part of the tool box for researchers studying populations of graphs in general, and statistical connectomics in particular.

Statistical inference on random dot product graphs: a survey

The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describe a comprehensive paradigm for statistical inference on random dot product graphs, a paradigm centered on spectral embeddings of adjacency and Laplacian matrices. We examine the analogues, in graph inference, of several canonical tenets of classical Euclidean inference: in particular, we summarize a body of existing results on the consistency and asymptotic normality of the adjacency and Laplacian spectral embeddings, and the role these spectral embeddings can play in the construction of single- and multi-sample hypothesis tests for graph data. We investigate several real-world applications, including community detection and classification in large social networks and the determination of functional and biologically relevant network properties from an exploratory data analysis of the Drosophila connectome. We outline requisite background and current open problems in spectral graph inference.

A Large Deformation Diffeomorphic Approach to Registration of CLARITY Images via Mutual Information

CLARITY is a method for converting biological tissues into translucent and porous hydrogel-tissue hybrids. This facilitates interrogation with light sheet microscopy and penetration of molecular probes while avoiding physical slicing. In this work, we develop a pipeline for registering CLARIfied mouse brains to an annotated brain atlas. Due to the novelty of this microscopy technique it is impractical to use absolute intensity values to align these images to existing standard atlases. Thus we adopt a large deformation diffeomorphic approach for registering images via mutual information matching. Furthermore we show how a cascaded multi-resolution approach can improve registration quality while reducing algorithm run time. As acquired image volumes were over a terabyte in size, they were far too large for work on personal computers. Therefore the NeuroData computational infrastructure was deployed for multi-resolution storage and visualization of these images and aligned annotations on the web.

Semiparametric spectral modeling of the Drosophila connectome

We present semiparametric spectral modeling of the complete larval Drosophila mushroom body connectome. Motivated by a thorough exploratory data analysis of the network via Gaussian mixture modeling (GMM) in the adjacency spectral embedding (ASE) representation space, we introduce the latent structure model (LSM) for network modeling and inference. LSM is a generalization of the stochastic block model (SBM) and a special case of the random dot product graph (RDPG) latent position model, and is amenable to semiparametric GMM in the ASE representation space. The resulting connectome code derived via semiparametric GMM composed with ASE captures latent connectome structure and elucidates biologically relevant neuronal properties.

Manifold Matching using Shortest-Path Distance and Joint Neighborhood Selection

Matching datasets of multiple modalities has become an important task in data analysis. Existing methods often rely on the embedding and transformation of each single modality without utilizing any correspondence information, which often results in sub-optimal matching performance. In this paper, we propose a nonlinear manifold matching algorithm using shortest-path distance and joint neighborhood selection. Specifically, a joint nearest-neighbor graph is built for all modalities. Then the shortest-path distance within each modality is calculated from the joint neighborhood graph, followed by embedding into and matching in a common low-dimensional Euclidean space. Compared to existing algorithms, our approach exhibits superior performance for matching disparate datasets of multiple modalities.

Joint Embedding of Graphs

Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a method to jointly embed multiple undirected graphs. Given a set of graphs, the joint embedding method identifies a linear subspace spanned by rank one symmetric matrices and projects adjacency matrices of graphs into this subspace. The projection coefficients can be treated as features of the graphs. We also propose a random graph model which generalizes classical random graph model and can be used to model multiple graphs. We show through theory and numerical experiments that under the model, the joint embedding method produces estimates of parameters with small errors. Via simulation experiments, we demonstrate that the joint embedding method produces features which lead to state of the art performance in classifying graphs. Applying the joint embedding method to human brain graphs, we find it extract interpretable features that can be used to predict individual composite creativity index.

Probabilistic Fluorescence-Based Synapse Detection

Brain function results from communication between neurons connected by complex synaptic networks. Synapses are themselves highly complex and diverse signaling machines, containing protein products of hundreds of different genes, some in hundreds of copies, arranged in precise lattice at each individual synapse. Synapses are fundamental not only to synaptic network function but also to network development, adaptation, and memory. In addition, abnormalities of synapse numbers or molecular components are implicated in most mental and neurological disorders. Despite their obvious importance, mammalian synapse populations have so far resisted detailed quantitative study. In human brains and most animal nervous systems, synapses are very small and very densely packed: there are approximately 1 billion synapses per cubic millimeter of human cortex. This volumetric density poses very substantial challenges to proteometric analysis at the critical level of the individual synapse. The present work describes new probabilistic image analysis methods for single-synapse analysis of synapse populations in both animal and human brains.