Compressive Network Analysis

Modern data acquisition routinely produces massive amounts of network data. Though many methods and models have been proposed to analyze such data, the research of network data is largely disconnected with the classical theory of statistical learning and signal processing. In this paper, we present a new framework for modeling network data, which connects two seemingly different areas: network data analysis and compressed sensing. From a nonparametric perspective, we model an observed network using a large dictionary. In particular, we consider the network clique detection problem and show connections between our formulation with a new algebraic tool, namely Randon basis pursuit in homogeneous spaces. Such a connection allows us to identify rigorous recovery conditions for clique detection problems. Though this paper is mainly conceptual, we also develop practical approximation algorithms for solving empirical problems and demonstrate their usefulness on real-world datasets

An Investigation Of Gene Networks Influenced By Low Dose Ionizing Radiation Using Statistical And Graph Theoretical Algorithms

Increased application of radiation in health and security sectors has raised concerns about its deleterious effects. Ionizing radiation (IR) less than 10cGys is considered low dose ionizing radiation (LDIR) by the National Research Committee to assess health risks from exposure to low levels of IR. It is hard to extract the effects of mild stimulus such as LDIR on gene expression profiles using simple differential expression. We hypothesized that differential correlation instead would capture the effects of LDIR on mutual relationships between genes. We tested this hypothesis on expression profiles from five inbred strains of mice treated with LDIR. Whereas ANOVA detected little effect of LDIR on gene expression, a differential correlation graph generated by a two stage statistical filter revealed gene networks enriched with genes implicated in radiation response, DNA damage repair, apoptosis, cancer and immune system. To mimic the effects of radiation on human populations, we profiled baseline expression of recombinant inbred strains of BXD mice derived from a cross between C57BL/6J and DBA/2J standard inbred strains. To establish a threshold for extraction of gene networks from the baseline expression profiles, we compared gene enrichment in paracliques obtained at different absolute Pearson correlations (APC) using graph algorithms. Gene networks extracted at statistically significant APC (r≈0.41) exhibited even better enrichment of genes participating in common biological processes than networks extracted at higher APCs from 0.6 to 0.875. Since immune response is influenced by LDIR, we investigated the effects of genetic background on variability of immune system in a population of BXD mice. Considering immune response as a complex trait, we identified significant QTLs explaining the ratio of CD8+ and CD4+ T-cells. Multiple regression modeling of genes neighboring statistically significant QTLs identified three candidate genes (Ptprk,Acp1 and Lamb1-1) explaining 61% variance of ratio of CD4+ and CD8+ T cells. Expression profiling of parental strains of BXD mice also revealed effects of LDIR and LDIR*strain on expression of genes related to immune response. Thus using an integrated approach involving transcriptomic, SNP and immunological data, we have developed novel methods to pinpoint candidate gene networks putatively influenced by LDIR

Bayesian Model Based Tracking with Application to Cell Segmentation and Tracking

The goal of this research is to develop a model-based tracking framework with biomedical imaging applications. This is an interdisciplinary area of research with interests in machine vision, image processing, and biology. This thesis presents methods of image modeling, tracking, and data association applied to problems in multi-cellular image analysis, especially hematopoietic stem cell (HSC) images at the current stage. The focus of this research is on the development of a robust image analysis interface capable of detecting, locating, and tracking individual hematopoietic stem cells (HSCs), which proliferate and differentiate to different blood cell types continuously during their lifetime, and are of substantial interest in gene therapy, cancer, and stem-cell research. Such a system can be potentially employed in the future to track different groups of HSCs extracted from bone marrow and recognize the best candidates based on some biomedical-biological criteria. Selected candidates can further be used for bone marrow transplantation (BMT) which is a medical procedure for the treatment of various incurable diseases such as leukemia, lymphomas, aplastic anemia, immune deficiency disorders, multiple myeloma and some solid tumors. Tracking HSCs over time is a localization-based tracking problem which is one of the most challenging tracking problems to be solved. The proposed cell tracking system consists of three inter-related stages: i) Cell detection/localization, ii) The association of detected cells, iii) Background estimation/subtraction. that will be discussed in detail

Spectral methods and computational trade-offs in high-dimensional statistical inference

Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we also show that such procedures have essentially the best performance among all randomised polynomial time algorithms by exhibiting statistical and computational trade-offs in those problems. In the first chapter, we prove a useful variant of the well-known Davis{Kahan theorem, which is a spectral perturbation result that allows us to bound of the distance between population eigenspaces and their sample versions. We then propose a semi-definite programming algorithm for the sparse principal component analysis (PCA) problem, and analyse its theoretical performance using the perturbation bounds we derived earlier. It turns out that the parameter regime in which our estimator is consistent is strictly smaller than the consistency regime of a minimax optimal (yet computationally intractable) estimator. We show through reduction from a well-known hard problem in computational complexity theory that the difference in consistency regimes is unavoidable for any randomised polynomial time estimator, hence revealing subtle statistical and computational trade-offs in this problem. Such computational trade-offs also exist in the problem of restricted isometry certification. Certifiers for restricted isometry properties can be used to construct design matrices for sparse linear regression problems. Similar to the sparse PCA problem, we show that there is also an intrinsic gap between the class of matrices certifiable using unrestricted algorithms and using polynomial time algorithms. Finally, we consider the problem of high-dimensional changepoint estimation, where we estimate the time of change in the mean of a high-dimensional time series with piecewise constant mean structure. Motivated by real world applications, we assume that changes only occur in a sparse subset of all coordinates. We apply a variant of the semi-definite programming algorithm in sparse PCA to aggregate the signals across different coordinates in a near optimal way so as to estimate the changepoint location as accurately as possible. Our statistical procedure shows superior performance compared to existing methods in this problem.St John's College and Cambridge Overseas Trus

Spectral methods and computational trade-offs in high-dimensional statistical inference

Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we also show that such procedures have essentially the best performance among all randomised polynomial time algorithms by exhibiting statistical and computational trade-offs in those problems. In the first chapter, we prove a useful variant of the well-known Davis{Kahan theorem, which is a spectral perturbation result that allows us to bound of the distance between population eigenspaces and their sample versions. We then propose a semi-definite programming algorithm for the sparse principal component analysis (PCA) problem, and analyse its theoretical performance using the perturbation bounds we derived earlier. It turns out that the parameter regime in which our estimator is consistent is strictly smaller than the consistency regime of a minimax optimal (yet computationally intractable) estimator. We show through reduction from a well-known hard problem in computational complexity theory that the difference in consistency regimes is unavoidable for any randomised polynomial time estimator, hence revealing subtle statistical and computational trade-offs in this problem. Such computational trade-offs also exist in the problem of restricted isometry certification. Certifiers for restricted isometry properties can be used to construct design matrices for sparse linear regression problems. Similar to the sparse PCA problem, we show that there is also an intrinsic gap between the class of matrices certifiable using unrestricted algorithms and using polynomial time algorithms. Finally, we consider the problem of high-dimensional changepoint estimation, where we estimate the time of change in the mean of a high-dimensional time series with piecewise constant mean structure. Motivated by real world applications, we assume that changes only occur in a sparse subset of all coordinates. We apply a variant of the semi-definite programming algorithm in sparse PCA to aggregate the signals across different coordinates in a near optimal way so as to estimate the changepoint location as accurately as possible. Our statistical procedure shows superior performance compared to existing methods in this problem.St John's College and Cambridge Overseas Trus

STOCHASTIC MOBILITY MODELS IN SPACE AND TIME

An interesting fact in nature is that if we observe agents (neurons, particles, animals, humans) behaving, or more precisely moving, inside their environment, we can recognize - tough at different space or time scales - very specific patterns. The existence of those patterns is quite obvious, since not all things in nature behave totally at random, especially if we take into account thinking species like human beings. If a first phenomenon which has been deeply modeled is the gas particle motion as the template of a totally random motion, other phenomena, like foraging patterns of animals such as albatrosses, and specific instances of human mobility wear some randomness away in favor of deterministic components. Thus, while the particle motion may be satisfactorily described with a Wiener Process (also called Brownian motion), the others are better described by other kinds of stochastic processes called Levy Flights. Minding at these phenomena in a unifying way, in terms of motion of agents \u2013 either inanimate like the gas particles, or animated like the albatrosses \u2013 the point is that the latter are driven by specific interests, possibly converging into a common task, to be accomplished. The whole thesis work turns around the concept of agent intentionality at different scales, whose model may be used as key ingredient in the statistical description of complex behaviors. The two main contributions in this direction are: 1. the development of a \u201cwait and chase\u201d model of human mobility having the same two-phase pattern as animal foraging but with a greater propensity of local stays in place and therefore a less dispersed general behavior; 2. the introduction of a mobility paradigm for the neurons of a multilayer neural network and a methodology to train these new kind of networks to develop a collective behavior. The lead idea is that neurons move toward the most informative mates to better learn how to fulfill their part in the overall functionality of the network. With these specific implementations we have pursued the general goal of attributing both a cognitive and a physical meaning to the intentionality so as to be able in a near future to speak of intentionality as an additional potential in the dynamics of the masses (both at the micro and a the macro-scale), and of communication as another network in the force field. This could be intended as a step ahead in the track opened by the past century physicists with the coupling of thermodynamic and Shannon entropies in the direction of unifying cognitive and physical laws