Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks

High-Dimensional Dependency Structure Learning for Physical Processes

In this paper, we consider the use of structure learning methods for probabilistic graphical models to identify statistical dependencies in high-dimensional physical processes. Such processes are often synthetically characterized using PDEs (partial differential equations) and are observed in a variety of natural phenomena, including geoscience data capturing atmospheric and hydrological phenomena. Classical structure learning approaches such as the PC algorithm and variants are challenging to apply due to their high computational and sample requirements. Modern approaches, often based on sparse regression and variants, do come with finite sample guarantees, but are usually highly sensitive to the choice of hyper-parameters, e.g., parameter $\lambda$ for sparsity inducing constraint or regularization. In this paper, we present ACLIME-ADMM, an efficient two-step algorithm for adaptive structure learning, which estimates an edge specific parameter $\lambda_{ij}$ in the first step, and uses these parameters to learn the structure in the second step. Both steps of our algorithm use (inexact) ADMM to solve suitable linear programs, and all iterations can be done in closed form in an efficient block parallel manner. We compare ACLIME-ADMM with baselines on both synthetic data simulated by partial differential equations (PDEs) that model advection-diffusion processes, and real data (50 years) of daily global geopotential heights to study information flow in the atmosphere. ACLIME-ADMM is shown to be efficient, stable, and competitive, usually better than the baselines especially on difficult problems. On real data, ACLIME-ADMM recovers the underlying structure of global atmospheric circulation, including switches in wind directions at the equator and tropics entirely from the data.Comment: 21 pages, 8 figures, International Conference on Data Mining 201