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 Mathematical & Statistical Methods


New Books and Resources for DSC Members

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We are in the process of writing and adding new material (compact eBooks) exclusively available to our members, and written in simple English, by world leading experts in AI, data science, and machine learning. We invite you to sign up here to not miss these free books. This book is intended for busy professionals working with data of any kind: engineers, BI analysts, statisticians, operations research, AI and machine learning professionals, economists, data scientists, biologists, and quants, ranging from beginners to executives. In about 300 pages and 28 chapters it covers many new topics, offering a fresh perspective on the subject, including rules of thumb and recipes that are easy to automate or integrate in black-box systems, as well as new model-free, data-driven foundations to statistical science and predictive analytics. The approach focuses on robust techniques; it is bottom-up (from applications to theory), in contrast to the traditional top-down approach. The material is accessible to practitioners with a one-year college-level exposure to statistics and probability.


Machine Learning for Signal Processing: Data Science, Algorithms, and Computational Statistics: Max A. Little: 9780198714934: Amazon.com: Books

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This book provides an excellent pathway for gaining first-class expertise in machine learning. It provides both the technical background that explains why certain approaches, but not others, are best practice in real world problems, and a framework for how to think about and approach new problems. I highly recommend it for people with a signal processing background who are seeking to become an expert in machine learning.


A Quasi-Newton Method Based Vertical Federated Learning Framework for Logistic Regression

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Data privacy and security becomes a major concern in building machine learning models from different data providers. Federated learning shows promise by leaving data at providers locally and exchanging encrypted information. This paper studies the vertical federated learning structure for logistic regression where the data sets at two parties have the same sample IDs but own disjoint subsets of features. Existing frameworks adopt the first-order stochastic gradient descent algorithm, which requires large number of communication rounds. To address the communication challenge, we propose a quasi-Newton method based vertical federated learning framework for logistic regression under the additively homomorphic encryption scheme.


Stochastic Newton and Cubic Newton Methods with Simple Local Linear-Quadratic Rates

arXiv.org Machine Learning

We present two new remarkably simple stochastic second-order methods for minimizing the average of a very large number of sufficiently smooth and strongly convex functions. The first is a stochastic variant of Newton's method (SN), and the second is a stochastic variant of cubically regularized Newton's method (SCN). We establish local linear-quadratic convergence results. Unlike existing stochastic variants of second order methods, which require the evaluation of a large number of gradients and/or Hessians in each iteration to guarantee convergence, our methods do not have this shortcoming. For instance, the simplest variants of our methods in each iteration need to compute the gradient and Hessian of a {\em single} randomly selected function only. In contrast to most existing stochastic Newton and quasi-Newton methods, our approach guarantees local convergence faster than with first-order oracle and adapts to the problem's curvature. Interestingly, our method is not unbiased, so our theory provides new intuition for designing new stochastic methods.


Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

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This book is used as the textbook for the course EE103 (Stanford) and EE133A (UCLA), where you will find additional related material. If you find an error not listed in our errata list, please do let us know about it. You're welcome to use the lecture slides posted below, but we'd appreciate it if you acknowledge the source.


Linear Algebra and Learning from Data

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Also included is an essay from SIAM News'The Functions of Deep Learning' (December 2018) A second distributor for SIAM members is siam.org We will confirm orders for this new book by email.


Gilbert Strang: Linear Algebra, Deep Learning, Teaching, and MIT OpenCourseWare AI Podcast

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Gilbert Strang is a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times. This conversation is part of the Artificial Intelligence podcast.


A Bootstrap-based Inference Framework for Testing Similarity of Paired Networks

arXiv.org Machine Learning

We live in an interconnected world where network valued data arises in many domains, and, fittingly, statistical network analysis has emerged as an active area in the literature. However, the topic of inference in networks has received relatively less attention. In this work, we consider the paired network inference problem where one is given two networks on the same set of nodes, and the goal is to test whether the given networks are stochastically similar in terms of some notion of similarity. We develop a general inferential framework based on parametric bootstrap to address this problem. Under this setting, we address two specific and important problems: the equality problem, i.e., whether the two networks are generated from the same random graph model, and the scaling problem, i.e., whether the underlying probability matrices of the two random graph models are scaled versions of each other.


Chi-Squared For Feature Selection using SelectKBest

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In this video, I'll show you how SelectKBest uses Chi-squared test for feature selection for categorical features & target columns. We calculate Chi-square between each feature & the target & select the desired number of features with best Chi-square scores or the lowest p-values. The Chi-squared (χ2) test is used in statistics to test the independence of two events. More specifically in feature selection we use it to test whether the occurrence of a specific feature & the target are independent or not. For each feature & target combination, a corresponding high χ2 chi-square score or a low p-value indicates that the target column is dependent on the feature column.


Machine learning for causal inference in Biostatistics

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General inference problems and quantifying uncertainty have long been the cornerstone of statistical science. While machine learning advances have permeated many disciplines, inference for these procedures, and in particular, causal inference, has not been widespread. However, this is rapidly changing. As different scientific fields begin to converge on machine learning for causal inference, we thought now would be an excellent time to have a public discussion. In our roles as editors of Biostatistics, we decided to organize a series of commentaries on the topic from scholars with expertise in statistics, computer science, epidemiology, health economics, policy, and law.