High-dimensional Sparse Inverse Covariance Estimation using Greedy Methods

In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two novel greedy approaches to solving this problem. The first estimates the non-zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps. The second estimates the neighborhood of each node in the graph separately, again using greedy forward and backward steps, and combines the intermediate neighborhoods to form an overall estimate. The principal contribution of this paper is a rigorous analysis of the sparsistency, or consistency in recovering the sparsity pattern of the inverse covariance matrix. Surprisingly, we show that both the local and global greedy methods learn the full structure of the model with high probability given just $O(d\log(p))$ samples, which is a \emph{significant} improvement over state of the art $\ell_1$-regularized Gaussian MLE (Graphical Lasso) that requires $O(d^2\log(p))$ samples. Moreover, the restricted eigenvalue and smoothness conditions imposed by our greedy methods are much weaker than the strong irrepresentable conditions required by the $\ell_1$-regularization based methods. We corroborate our results with extensive simulations and examples, comparing our local and global greedy methods to the $\ell_1$-regularized Gaussian MLE as well as the Neighborhood Greedy method to that of nodewise $\ell_1$-regularized linear regression (Neighborhood Lasso).Comment: Accepted to AI STAT 2012 for Oral Presentatio

On Gorenstein Surfaces Dominated by P^2

In this paper we prove that a normal Gorenstein surface dominated by the projective plane P^2 is isomorphic to a quotient P^2/G, where G is a finite group of automorphisms of P^2 (except possibly for one surface V_8'). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.Comment: Nagoya Mathematical Journal, to appea

Entropic Inequalities for a Class of Quantum Secret Sharing States

It is well-known that von Neumann entropy is nonmonotonic unlike Shannon entropy (which is monotonically nondecreasing). Consequently, it is difficult to relate the entropies of the subsystems of a given quantum state. In this paper, we show that if we consider quantum secret sharing states arising from a class of monotone span programs, then we can partially recover the monotonicity of entropy for the so-called unauthorized sets. Furthermore, we can show for these quantum states the entropy of the authorized sets is monotonically nonincreasing.Comment: LaTex, 5 page

Non-ketonic hyperglycemia presenting with acute hemichorea and ballism

Received: October 8, 2017 Accepted: May 10, 2018 Published: August 17, 2018Financial support: Author declares that no financial assistance was taken from any source.Potential conflicts of interest: Author declares no conflicts of interest.Non-ketotic hyperglycemia is a complication of poorly controlled diabetes mellitus. Rarely, it can present like an acute neurological syndrome with unilateral choreiform and ballistic movements. Such a presentation usually raises the suspicion of a cerebrovascular event and prompts more workup. Moreover, the neuroimaging in this condition also suggests a variety of potential possibilities. Identification of this rare presentation of non-ketotic hyperglycemia helps with the appropriate management and avoid unnecessary investigations. In this case report, we report the case of an elderly woman who presented with hemichorea-ballism due to non-ketotic hyperglycemia and discuss the literature on this presentation. We also highlighted the differential diagnosis based on neuroimaging.Pradeep C. Bollu MD (1) (Department of Neurology, University of Missouri, Columbia, Missouri)Includes bibliographical reference

A Tale of Two Data-Intensive Paradigms: Applications, Abstractions, and Architectures

Scientific problems that depend on processing large amounts of data require overcoming challenges in multiple areas: managing large-scale data distribution, co-placement and scheduling of data with compute resources, and storing and transferring large volumes of data. We analyze the ecosystems of the two prominent paradigms for data-intensive applications, hereafter referred to as the high-performance computing and the Apache-Hadoop paradigm. We propose a basis, common terminology and functional factors upon which to analyze the two approaches of both paradigms. We discuss the concept of "Big Data Ogres" and their facets as means of understanding and characterizing the most common application workloads found across the two paradigms. We then discuss the salient features of the two paradigms, and compare and contrast the two approaches. Specifically, we examine common implementation/approaches of these paradigms, shed light upon the reasons for their current "architecture" and discuss some typical workloads that utilize them. In spite of the significant software distinctions, we believe there is architectural similarity. We discuss the potential integration of different implementations, across the different levels and components. Our comparison progresses from a fully qualitative examination of the two paradigms, to a semi-quantitative methodology. We use a simple and broadly used Ogre (K-means clustering), characterize its performance on a range of representative platforms, covering several implementations from both paradigms. Our experiments provide an insight into the relative strengths of the two paradigms. We propose that the set of Ogres will serve as a benchmark to evaluate the two paradigms along different dimensions.Comment: 8 pages, 2 figure