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.

My Machine Learning Foundations curriculum provides a comprehensive overview of all of the subjects -- across mathematics, statistics, and computer science -- that underlie contemporary machine learning approaches. You can check out the full curriculum and all of the open-source Python code (featuring the NumPy, TensorFlow, and PyTorch libraries) in GitHub here. At a high level, my ML Foundations content can be broken into four subject areas: linear algebra, calculus, probability/stats, and computer science. The first quarter of the content, on linear algebra, stands alone as its own discrete course and is now available on YouTube. The playlist for my complete Linear Algebra for Machine Learning course is on YouTube here.

Enhancing existing transmission lines is a useful tool to combat transmission congestion and guarantee transmission security with increasing demand and boosting the renewable energy source. This study concerns the selection of lines whose capacity should be expanded and by how much from the perspective of independent system operator (ISO) to minimize the system cost with the consideration of transmission line constraints and electricity generation and demand balance conditions, and incorporating ramp-up and startup ramp rates, shutdown ramp rates, ramp-down rate limits and minimum up and minimum down times. For that purpose, we develop the ISO unit commitment and economic dispatch model and show it as a right-hand side uncertainty multiple parametric analysis for the mixed integer linear programming (MILP) problem. We first relax the binary variable to continuous variables and employ the Lagrange method and Karush-Kuhn-Tucker conditions to obtain optimal solutions (optimal decision variables and objective function) and critical regions associated with active and inactive constraints. Further, we extend the traditional branch and bound method for the large-scale MILP problem by determining the upper bound of the problem at each node, then comparing the difference between the upper and lower bounds and reaching the approximate optimal solution within the decision makers' tolerated error range. In additional, the objective function's first derivative on the parameters of each line is used to inform the selection of lines to ease congestion and maximize social welfare. Finally, the amount of capacity upgrade will be chosen by balancing the cost-reduction rate of the objective function on parameters and the cost of the line upgrade. Our findings are supported by numerical simulation and provide transmission line planners with decision-making guidance.

This work considers optimization of composition of functions in a nested form over Riemannian manifolds where each function contains an expectation. This type of problems is gaining popularity in applications such as policy evaluation in reinforcement learning or model customization in meta-learning. The standard Riemannian stochastic gradient methods for non-compositional optimization cannot be directly applied as stochastic approximation of inner functions create bias in the gradients of the outer functions. For two-level composition optimization, we present a Riemannian Stochastic Composition Gradient Descent (R-SCGD) method that finds an approximate stationary point, with expected squared Riemannian gradient smaller than $\epsilon$, in $O(\epsilon^{-2})$ calls to the stochastic gradient oracle of the outer function and stochastic function and gradient oracles of the inner function. Furthermore, we generalize the R-SCGD algorithms for problems with multi-level nested compositional structures, with the same complexity of $O(\epsilon^{-2})$ for the first-order stochastic oracle. Finally, the performance of the R-SCGD method is numerically evaluated over a policy evaluation problem in reinforcement learning.