Inclusion of Forbidden Minors in Random Representable Matroids

In 1984, Kelly and Oxley introduced the model of a random representable matroid $M[A_n]$ corresponding to a random matrix $A_n \in \mathbb{F}_q^{m(n) \times n}$, whose entries are drawn independently and uniformly from $\mathbb{F}_q$. Whereas properties such as rank, connectivity, and circuit size have been well-studied, forbidden minors have not yet been analyzed. Here, we investigate the asymptotic probability as $n \to \infty$ that a fixed $\mathbb{F}_q$-representable matroid $M$ is a minor of $M[A_n]$. (We always assume $m(n) \geq \text{rank}(M)$ for all sufficiently large $n$, otherwise $M$ can never be a minor of the corresponding $M[A_n]$.) When $M$ is free, we show that $M$ is asymptotically almost surely (a.a.s.) a minor of $M[A_n]$. When $M$ is not free, we show a phase transition: $M$ is a.a.s. a minor if $n - m(n) \to \infty$, but is a.a.s. not if $m(n) - n \to \infty$. In the more general settings of $m \leq n$ and $m > n$, we give lower and upper bounds, respectively, on both the asymptotic and non-asymptotic probability that $M$ is a minor of $M[A_n]$. The tools we develop to analyze matroid operations and minors of random matroids may be of independent interest. Our results directly imply that $M[A_n]$ is a.a.s. not contained in any proper, minor-closed class $\mathcal{M}$ of $\mathbb{F}_q$-representable matroids, provided: (i) $n - m(n) \to \infty$, and (ii) $m(n)$ is at least the minimum rank of any $\mathbb{F}_q$-representable forbidden minor of $\mathcal{M}$, for all sufficiently large $n$. As an application, this shows that graphic matroids are a vanishing subset of linear matroids, in a sense made precise in the paper. Our results provide an approach for applying the rich theory around matroid minors to the less-studied field of random matroids.Comment: to appear in Discrete Mathematic

Quit Using Pseudorapidity, Transverse Energy, and Massless Constituents

Use a massive jet's true-rapidity instead of its pseudorapidity, even for event displays and for determining if a jet is near a calorimeter edge. Use transverse momentum instead of transverse energy since only the former is conserved. Use massive constituents because using massless constituents reduces the jet mass by an amount proportional to the square of the number of hadrons in the jet, and can amount to several GeV. These three recommendations are important for precision measurements when jets are constructed by adding constituent 4-vectors.Comment: 3 pages, 4 figure

A posteriori error estimator for adaptive local basis functions to solve Kohn-Sham density functional theory

Kohn-Sham density functional theory is one of the most widely used electronic structure theories. The recently developed adaptive local basis functions form an accurate and systematically improvable basis set for solving Kohn-Sham density functional theory using discontinuous Galerkin methods, requiring a small number of basis functions per atom. In this paper we develop residual-based a posteriori error estimates for the adaptive local basis approach, which can be used to guide non-uniform basis refinement for highly inhomogeneous systems such as surfaces and large molecules. The adaptive local basis functions are non-polynomial basis functions, and standard a posteriori error estimates for $hp$-refinement using polynomial basis functions do not directly apply. We generalize the error estimates for $hp$-refinement to non-polynomial basis functions. We demonstrate the practical use of the a posteriori error estimator in performing three-dimensional Kohn-Sham density functional theory calculations for quasi-2D aluminum surfaces and a single-layer graphene oxide system in water.Comment: 34 pages, 12 figure

Co-Emulation of Scan-Chain Based Designs Utilizing SCE-MI Infrastructure

As the complexity of the scan algorithm is dependent on the number of design registers, large SoC scan designs can no longer be verified in RTL simulation unless partitioned into smaller sub-blocks. This paper proposes a methodology to decrease scan-chain verification time utilizing SCE-MI, a widely used communication protocol for emulation, and an FPGA-based emulation platform. A high-level (SystemC) testbench and FPGA synthesizable hardware transactor models are developed for the scan-chain ISCAS89 S400 benchmark circuit for high-speed communication between the host CPU workstation and the FPGA emulator. The emulation results are compared to other verification methodologies (RTL Simulation, Simulation Acceleration, and Transaction-based emulation), and found to be 82% faster than regular RTL simulation. In addition, the emulation runs in the MHz speed range, allowing the incorporation of software applications, drivers, and operating systems, as opposed to the Hz range in RTL simulation or sub-megahertz range as accomplished in transaction-based emulation. In addition, the integration of scan testing and acceleration/emulation platforms allows more complex DFT methods to be developed and tested on a large scale system, decreasing the time to market for products

Communication-Efficient Distributed Statistical Inference

We present a Communication-efficient Surrogate Likelihood (CSL) framework for solving distributed statistical inference problems. CSL provides a communication-efficient surrogate to the global likelihood that can be used for low-dimensional estimation, high-dimensional regularized estimation and Bayesian inference. For low-dimensional estimation, CSL provably improves upon naive averaging schemes and facilitates the construction of confidence intervals. For high-dimensional regularized estimation, CSL leads to a minimax-optimal estimator with controlled communication cost. For Bayesian inference, CSL can be used to form a communication-efficient quasi-posterior distribution that converges to the true posterior. This quasi-posterior procedure significantly improves the computational efficiency of MCMC algorithms even in a non-distributed setting. We present both theoretical analysis and experiments to explore the properties of the CSL approximation

Estimating the Coefficients of a Mixture of Two Linear Regressions by Expectation Maximization

We give convergence guarantees for estimating the coefficients of a symmetric mixture of two linear regressions by expectation maximization (EM). In particular, we show that the empirical EM iterates converge to the target parameter vector at the parametric rate, provided the algorithm is initialized in an unbounded cone. In particular, if the initial guess has a sufficiently large cosine angle with the target parameter vector, a sample-splitting version of the EM algorithm converges to the true coefficient vector with high probability. Interestingly, our analysis borrows from tools used in the problem of estimating the centers of a symmetric mixture of two Gaussians by EM. We also show that the population EM operator for mixtures of two regressions is anti-contractive from the target parameter vector if the cosine angle between the input vector and the target parameter vector is too small, thereby establishing the necessity of our conic condition. Finally, we give empirical evidence supporting this theoretical observation, which suggests that the sample based EM algorithm performs poorly when initial guesses are drawn accordingly. Our simulation study also suggests that the EM algorithm performs well even under model misspecification (i.e., when the covariate and error distributions violate the model assumptions)

Estimation of convex supports from noisy measurements

A popular class of problem in statistics deals with estimating the support of a density from $n$ observations drawn at random from a $d$-dimensional distribution. The one-dimensional case reduces to estimating the end points of a univariate density. In practice, an experimenter may only have access to a noisy version of the original data. Therefore, a more realistic model allows for the observations to be contaminated with additive noise. In this paper, we consider estimation of convex bodies when the additive noise is distributed according to a multivariate Gaussian distribution, even though our techniques could easily be adapted to other noise distributions. Unlike standard methods in deconvolution that are implemented by thresholding a kernel density estimate, our method avoids tuning parameters and Fourier transforms altogether. We show that our estimator, computable in $(O(\ln n))^{(d-1)/2}$ time, converges at a rate of $O_d(\log\log n/\sqrt{\log n})$ in Hausdorff distance, in accordance with the polylogarithmic rates encountered in Gaussian deconvolution problems. Part of our analysis also involves the optimality of the proposed estimator. We provide a lower bound for the minimax rate of estimation in Hausdorff distance that is $\Omega_d(1/\log^2 n)$

Neural Temporal-Difference and Q-Learning Provably Converge to Global Optima

Temporal-difference learning (TD), coupled with neural networks, is among the most fundamental building blocks of deep reinforcement learning. However, due to the nonlinearity in value function approximation, such a coupling leads to nonconvexity and even divergence in optimization. As a result, the global convergence of neural TD remains unclear. In this paper, we prove for the first time that neural TD converges at a sublinear rate to the global optimum of the mean-squared projected Bellman error for policy evaluation. In particular, we show how such global convergence is enabled by the overparametrization of neural networks, which also plays a vital role in the empirical success of neural TD. Beyond policy evaluation, we establish the global convergence of neural (soft) Q-learning, which is further connected to that of policy gradient algorithms

Distributed Stochastic Variance Reduced Gradient Methods and A Lower Bound for Communication Complexity

We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.Comment: significant addition to both theory and experimental result

A Toolkit of the Stop Search via the Chargino Decay

The top squark (stop) may dominantly decay to a bottom quark and a chargino if the mass difference between the stop and the lightest neutralino is comparable or less than the top quark mass. Such a moderately compressed spectrum is a challenging scenario for the stop search at the Large Hadron Collider, because it is difficult to separate the signals from the top and anti-top background. In this paper we focus on the di-leptonic decay channel, and consider many kinematic variables as possible discriminators. These include several MT2 variables and new "compatible-masses" variables which fully utilize all kinematic information of the background. We use several sample spectra with different characteristics to study the efficiencies of these variables in distinguishing the signal from the background. The finding is that different combinations of variables or strategies should be used for different spectra to maximally enhance the signal significance and expand the reach of the stop search in this scenario. The new variables that we proposed in this paper are also useful for other new physics searches with di-leptonic top and anti-top events as the dominant background.Comment: 32 pages, 14 figure