We demonstrate how Monte Carlo Search (MCS) algorithms, namely Nested Monte Carlo Search (NMCS) and Nested Rollout Policy Adaptation (NRPA), can be used to build graphs and find counter-examples to spectral graph theory conjectures in minutes.

Network-valued time series are currently a common form of network data. However, the study of the aggregate behavior of network sequences generated from network-valued stochastic processes is relatively rare. Most of the existing research focuses on the simple setup where the networks are independent (or conditionally independent) across time, and all edges are updated synchronously at each time step. In this paper, we study the concentration properties of the aggregated adjacency matrix and the corresponding Laplacian matrix associated with network sequences generated from lazy network-valued stochastic processes, where edges update asynchronously, and each edge follows a lazy stochastic process for its updates independent of the other edges. We demonstrate the usefulness of these concentration results in proving consistency of standard estimators in community estimation and changepoint estimation problems. We also conduct a simulation study to demonstrate the effect of the laziness parameter, which controls the extent of temporal correlation, on the accuracy of community and changepoint estimation.

We present a locally optimal tracking controller for Cable Driven Parallel Robot (CDPR) control based on a time-varying Linear Quadratic Gaussian (TV-LQG) controller. In contrast to many methods which use fixed feedback gains, our time-varying controller computes the optimal gains depending on the location in the workspace and the future trajectory. Meanwhile, we rely heavily on offline computation to reduce the burden of online implementation and feasibility checking. Following the growing popularity of probabilistic graphical models for optimal control, we use factor graphs as a tool to formulate our controller for their efficiency, intuitiveness, and modularity. The topology of a factor graph encodes the relevant structural properties of equations in a way that facilitates insight and efficient computation using sparse linear algebra solvers. We first use factor graph optimization to compute a nominal trajectory, then linearize the graph and apply variable elimination to compute the locally optimal, time varying linear feedback gains. Next, we leverage the factor graph formulation to compute the locally optimal, time-varying Kalman Filter gains, and finally combine the locally optimal linear control and estimation laws to form a TV-LQG controller. We compare the tracking accuracy of our TV-LQG controller to a state-of-the-art dual-space feed-forward controller on a 2.9m x 2.3m, 4-cable planar robot and demonstrate improved tracking accuracies of 0.8{\deg} and 11.6mm root mean square error in rotation and translation respectively.

As the name suggests, it is a collection of mathematical functions ranging from simple calculation to complex numerical analysis. Math PHP library is completely independent and works straightaway. Its features includes Algebra, Arithmetic, Finance, Functions like Map and Polynomial, Information theory (Entropy), Linear Algebra (Matrix, Vector), Numbers (Arbitrary Integer, Complex, Rational), Number Theory (Integers), Numerical Analysis (Interpolation, Numerical Differentiation, Numerical Integration, Root Finding), Probability (Combinatorics, Distributions), Sequences (Basic, Advanced, Non-Integer), Set Theory, Statistics (Anova, Averages, Circulation, Correlation, Descriptive, Distance, Divergence, Distributions, Effect Size, Experiments, Kernel Density Estimation, Multivariance, Outlier, Random Variable, Regressions, Signification testing), Trigonometry. So if you're looking for all in one library for math function, well, this is the one.

We study optimization for data-driven decision-making when we have observations of the uncertain parameters within the optimization model together with concurrent observations of covariates. Given a new covariate observation, the goal is to choose a decision that minimizes the expected cost conditioned on this observation. We investigate three data-driven frameworks that integrate a machine learning prediction model within a stochastic programming sample average approximation (SAA) for approximating the solution to this problem. Two of the SAA frameworks are new and use out-of-sample residuals of leave-one-out prediction models for scenario generation. The frameworks we investigate are flexible and accommodate parametric, nonparametric, and semiparametric regression techniques. We derive conditions on the data generation process, the prediction model, and the stochastic program under which solutions of these data-driven SAAs are consistent and asymptotically optimal, and also derive convergence rates and finite sample guarantees. Computational experiments validate our theoretical results, demonstrate the potential advantages of our data-driven formulations over existing approaches (even when the prediction model is misspecified), and illustrate the benefits of our new data-driven formulations in the limited data regime.

Linear algebra is a mathematical discipline concerned with studying vector spaces and linear mappings between them [1]. It is essential in artificial intelligence implementations because it allows for unlocking meanings in high-dimensional data, a common use case pipeline (in AI). Applications for implementation include solving problems across many use cases in AI, including machine learning, deep learning, and natural language processing. Namely, it can be utilized to predict the behavior of neural networks, and it is also being used to improve the accuracy of deep learning models. Further, linear algebra provides a way to understand and visualize high-dimensional data, often used in natural language processing tasks.

Abstract: Among many solutions to the high-dimensional approximate nearest neighbor (ANN) search problem, locality sensitive hashing (LSH) is known for its sub-linear query time and robust theoretical guarantee on query accuracy. Traditional LSH methods can generate a small number of candidates quickly from hash tables but suffer from large index sizes and hash boundary problems. Recent studies to address these issues often incur extra overhead to identify eligible candidates or remove false positives, making query time no longer sub-linear. To address this dilemma, in this paper we propose a novel LSH scheme called DB-LSH which supports efficient ANN search for large high-dimensional datasets. It organizes the projected spaces with multi-dimensional indexes rather than using fixed-width hash buckets.

Linear algebra is a mathematical discipline concerned with studying vector spaces and linear mappings between them [1]. It is essential in artificial intelligence implementations because it allows for unlocking meanings in high-dimensional data, a common use case pipeline (in AI). Applications for implementation include solving problems across many use cases in AI, including machine learning, deep learning, and natural language processing. Namely, it can be utilized to predict the behavior of neural networks, and it is also being used to improve the accuracy of deep learning models. Further, linear algebra provides a way to understand and visualize high-dimensional data, often used in natural language processing tasks.

Originally published on Towards AI the World's Leading AI and Technology News and Media Company. If you are building an AI-related product or service, we invite you to consider becoming an AI sponsor. At Towards AI, we help scale AI and technology startups. Let us help you unleash your technology to the masses. Understand the 4 reasons to apply linear algebra in deep learning and learn about the 4 use cases demonstrating applications.

Originally published on Towards AI the World's Leading AI and Technology News and Media Company. If you are building an AI-related product or service, we invite you to consider becoming an AI sponsor. At Towards AI, we help scale AI and technology startups. Let us help you unleash your technology to the masses. It's free, we don't spam, and we never share your email address.