Learning Large-Scale MTP2_2 Gaussian Graphical Models via Bridge-Block Decomposition

This paper studies the problem of learning the large-scale Gaussian graphical models that are multivariate totally positive of order two ($\text{MTP}_2$). By introducing the concept of bridge, which commonly exists in large-scale sparse graphs, we show that the entire problem can be equivalently optimized through (1) several smaller-scaled sub-problems induced by a \emph{bridge-block decomposition} on the thresholded sample covariance graph and (2) a set of explicit solutions on entries corresponding to bridges. From practical aspect, this simple and provable discipline can be applied to break down a large problem into small tractable ones, leading to enormous reduction on the computational complexity and substantial improvements for all existing algorithms. The synthetic and real-world experiments demonstrate that our proposed method presents a significant speed-up compared to the state-of-the-art benchmarks

Direct observation of the intermediate in an ultrafast isomerization.

Using a combination of two-dimensional infrared (2D IR) and variable temperature Fourier transform infrared (FTIR) spectroscopies the rapid structural isomerization of a five-coordinate ruthenium complex is investigated. In methylene chloride, three exchanging isomers were observed: (1) square pyramidal equatorial, (1); (2) trigonal bipyramidal, (0); and (3) square pyramidal apical, (2). Exchange between 1 and 0 was found to be an endergonic process (ΔH = 0.84 (0.08) kcal mol-1, ΔS = 0.6 (0.4) eu) with an isomerization time constant of 4.3 (1.5) picoseconds (ps, 10-12 s). Exchange between 0 and 2 however was found to be exergonic (ΔH = -2.18 (0.06) kcal mol-1, ΔS = -5.3 (0.3) eu) and rate limiting with an isomerization time constant of 6.3 (1.6) ps. The trigonal bipyramidal complex was found to be an intermediate, with an activation barrier of 2.2 (0.2) kcal mol-1 and 2.4 (0.2) kcal mol-1 relative to the equatorial and apical square pyramidal isomers respectively. This study provides direct validation of the mechanism of Berry pseudorotation - the pairwise exchange of ligands in a five-coordinate complex - a process that was first described over fifty years ago. This study also clearly demonstrates that the rate of pseudorotation approaches the frequency of molecular vibrations

Photoinduced Strain‐Assisted Synthesis of a Stiff‐Stilbene Polymer by Ring‐Opening Metathesis Polymerization

Developing a novel strategy to synthesize photoresponsive polymers is of significance owing to their potential applications. We report a photoinduced strain‐assisted synthesis of main‐chain stiff‐stilbene polymers by using ring‐opening metathesis polymerization (ROMP), activating a macrocyclic π‐bond connected to a stiff‐stilbene photoswitch through a linker. Since the linker acts as an external constraint, the photoisomerization to the E‐form leads to the stiff‐stilbene being strained and thus reactive to ROMP. The photoisomerization of Z‐form to E‐form was investigated using time‐dependent NMR studies and UV/Vis spectroscopy. The DFT calculation showed that the E‐form was less stable due to a lack of planarity. By the internal strain developed due to the linker constraint through photoisomerization, the E‐form underwent ROMP by a second generation Grubbs catalyst. In contrast, Z‐form did not undergo polymerization under similar conditions. The MALDI‐TOF spectrum of E‐form after polymerization showed the presence of oligomers of >5.2 kDa

Efficient and Scalable Parametric High-Order Portfolios Design via the Skew-t Distribution

Since Markowitz's mean-variance framework, optimizing a portfolio that maximizes the profit and minimizes the risk has been ubiquitous in the financial industry. Initially, profit and risk were measured by the first two moments of the portfolio's return, a.k.a. the mean and variance, which are sufficient to characterize a Gaussian distribution. However, it is broadly believed that the first two moments are not enough to capture the characteristics of the returns' behavior, which have been recognized to be asymmetric and heavy-tailed. Although there is ample evidence that portfolio designs involving the third and fourth moments, i.e., skewness and kurtosis, will outperform the conventional mean-variance framework, they are non-trivial. Specifically, in the classical framework, the memory and computational cost of computing the skewness and kurtosis grow sharply with the number of assets. To alleviate the difficulty in high-dimensional problems, we consider an alternative expression for high-order moments based on parametric representations via a generalized hyperbolic skew-t distribution. Then, we reformulate the high-order portfolio optimization problem as a fixed-point problem and propose a robust fixed-point acceleration algorithm that solves the problem in an efficient and scalable manner. Empirical experiments also demonstrate that our proposed high-order portfolio optimization framework is of low complexity and significantly outperforms the state-of-the-art methods by 2 to 4 orders of magnitude

Does the ℓ1\ell_1-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?

We consider the problem of learning a sparse graph under Laplacian constrained Gaussian graphical models. This problem can be formulated as a penalized maximum likelihood estimation of the precision matrix under Laplacian structural constraints. Like in the classical graphical lasso problem, recent works made use of the $\ell_1$-norm regularization with the goal of promoting sparsity in Laplacian structural precision matrix estimation. However, we find that the widely used $\ell_1$-norm is not effective in imposing a sparse solution in this problem. Through empirical evidence, we observe that the number of nonzero graph weights grows with the increase of the regularization parameter. From a theoretical perspective, we prove that a large regularization parameter will surprisingly lead to a fully connected graph. To address this issue, we propose a nonconvex estimation method by solving a sequence of weighted $\ell_1$-norm penalized sub-problems and prove that the statistical error of the proposed estimator matches the minimax lower bound. To solve each sub-problem, we develop a projected gradient descent algorithm that enjoys a linear convergence rate. Numerical experiments involving synthetic and real-world data sets from the recent COVID-19 pandemic and financial stock markets demonstrate the effectiveness of the proposed method. An open source $\mathsf{R}$ package containing the code for all the experiments is available at https://github.com/mirca/sparseGraph

Adaptive Estimation of MTP2\text{MTP}_2 Graphical Models

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. Such models have received increasing attention in recent years, and have shown interesting properties, e.g., the maximum likelihood estimator exists with as little as two observations regardless of the underlying dimension. In this paper, we propose an adaptive estimation method, which consists of multiple stages: In the first stage, we solve an $\ell_1$-regularized maximum likelihood estimation problem, which leads to an initial estimate; in the subsequent stages, we iteratively refine the initial estimate by solving a sequence of weighted $\ell_1$-regularized problems. We further establish the theoretical guarantees on the estimation error, which consists of optimization error and statistical error. The optimization error decays to zero at a linear rate, indicating that the estimate is refined iteratively in subsequent stages, and the statistical error characterizes the statistical rate. The proposed method outperforms state-of-the-art methods in estimating precision matrices and identifying graph edges, as evidenced by synthetic and financial time-series data sets.Comment: 24 page

Self‐Healable and Recyclable Tactile Force Sensors with Post‐Tunable Sensitivity

It is challenging to post‐tune the sensitivity of a tactile force sensor. Herein, a facile method is reported to tailor the sensing properties of conductive polymer composites by utilizing the liquid‐like property of dynamic polymer matrix at low strain rates. The idea is demonstrated using dynamic polymer composites (CB/dPDMS) made via evaporation‐induced gelation of the suspending toluene solution of carbon black (CB) and acid‐catalyzed dynamic polydimethylsiloxane (dPDMS). The dPDMS matrices allow CB to redistribute to change the sensitivity of materials at the liquid‐like state, but exhibit typical solid‐like behavior and thus can be used as strain sensors at normal strain rates. It is shown that the gauge factor of the polymer composites can be easily post‐tuned from 1.4 to 51.5. In addition, the dynamic polymer matrices also endow the composites with interesting self‐healing ability and recyclability. Therefore, it is envisioned that this method can be useful in the design of various novel tactile sensing materials for many applications

Fast Projected Newton-like Method for Precision Matrix Estimation with Nonnegative Partial Correlations

We study the problem of estimating precision matrices in multivariate Gaussian distributions where all partial correlations are nonnegative, also known as multivariate totally positive of order two ($\mathrm{MTP}_2$). Such models have received significant attention in recent years, primarily due to interesting properties, e.g., the maximum likelihood estimator exists with as few as two observations regardless of the underlying dimension. We formulate this problem as a weighted $\ell_1$-norm regularized Gaussian maximum likelihood estimation under $\mathrm{MTP}_2$ constraints. On this direction, we propose a novel projected Newton-like algorithm that incorporates a well-designed approximate Newton direction, which results in our algorithm having the same orders of computation and memory costs as those of first-order methods. We prove that the proposed projected Newton-like algorithm converges to the minimizer of the problem. We further show, both theoretically and experimentally, that the minimizer of our formulation using the weighted $\ell_1$-norm is able to recover the support of the underlying precision matrix correctly without requiring the incoherence condition present in $\ell_1$-norm based methods. Experiments involving synthetic and real-world data demonstrate that our proposed algorithm is significantly more efficient, from a computational time perspective, than the state-of-the-art methods. Finally, we apply our method in financial time-series data, which are well-known for displaying positive dependencies, where we observe a significant performance in terms of modularity value on the learned financial networks.Comment: 43 pages; notation updated for section

New tools and methods for direct programmatic access to the dbSNP relational database

Genome-wide association studies often incorporate information from public biological databases in order to provide a biological reference for interpreting the results. The dbSNP database is an extensive source of information on single nucleotide polymorphisms (SNPs) for many different organisms, including humans. We have developed free software that will download and install a local MySQL implementation of the dbSNP relational database for a specified organism. We have also designed a system for classifying dbSNP tables in terms of common tasks we wish to accomplish using the database. For each task we have designed a small set of custom tables that facilitate task-related queries and provide entity-relationship diagrams for each task composed from the relevant dbSNP tables. In order to expose these concepts and methods to a wider audience we have developed web tools for querying the database and browsing documentation on the tables and columns to clarify the relevant relational structure. All web tools and software are freely available to the public at http://cgsmd.isi.edu/dbsnpq. Resources such as these for programmatically querying biological databases are essential for viably integrating biological information into genetic association experiments on a genome-wide scale