The mathematical language for talking about connections, which usually depends on networks--vertices (dots) and edges (lines connecting them)--has been an invaluable way to model real-world phenomena since at least the 18th century. But a few decades ago, the emergence of giant data sets forced researchers to expand their toolboxes and, at the same time, gave them sprawling sandboxes in which to apply new mathematical insights. Since then, said Josh Grochow, a computer scientist at the University of Colorado, Boulder, there's been an exciting period of rapid growth as researchers have developed new kinds of network models that can find complex structures and signals in the noise of big data. Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research develop ments and trends in mathe matics and the physical and life sciences. Grochow is among a growing chorus of researchers who point out that when it comes to finding connections in big data, graph theory has its limits.

This monograph develops a comprehensive statistical learning framework that is robust to (distributional) perturbations in the data using Distributionally Robust Optimization (DRO) under the Wasserstein metric. Beginning with fundamental properties of the Wasserstein metric and the DRO formulation, we explore duality to arrive at tractable formulations and develop finite-sample, as well as asymptotic, performance guarantees. We consider a series of learning problems, including (i) distributionally robust linear regression; (ii) distributionally robust regression with group structure in the predictors; (iii) distributionally robust multi-output regression and multiclass classification, (iv) optimal decision making that combines distributionally robust regression with nearest-neighbor estimation; (v) distributionally robust semi-supervised learning, and (vi) distributionally robust reinforcement learning. A tractable DRO relaxation for each problem is being derived, establishing a connection between robustness and regularization, and obtaining bounds on the prediction and estimation errors of the solution. Beyond theory, we include numerical experiments and case studies using synthetic and real data. The real data experiments are all associated with various health informatics problems, an application area which provided the initial impetus for this work.

Most data is automatically collected and only ever "seen" by algorithms. Yet, data compressors preserve perceptual fidelity rather than just the information needed by algorithms performing downstream tasks. In this paper, we characterize the bit-rate required to ensure high performance on all predictive tasks that are invariant under a set of transformations, such as data augmentations. Based on our theory, we design unsupervised objectives for training neural compressors. Using these objectives, we train a generic image compressor that achieves substantial rate savings (more than $1000\times$ on ImageNet) compared to JPEG on 8 datasets, without decreasing downstream classification performance.

This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we provide some details and proofs of some results from (Strang, 1993). We then describe the fundamental theorem of linear algebra from different views and find the properties and relationships behind the views. The fundamental theorem of linear algebra is essential in many fields, such as electrical engineering, computer science, machine learning, and deep learning.

This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we provide some details and proofs of some results from (Strang, 1993). We then describe the fundamental theorem of linear algebra from different views and find the properties and relationships behind the views. The fundamental theorem of linear algebra is essential in many fields, such as electrical engineering, computer science, machine learning, and deep learning. This survey is primarily a summary of purpose, significance of important theories behind it. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in theory behind the fundamental theorem of linear algebra and rigorous analysis in order to seamlessly introduce its properties in four subspaces in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results and given the paucity of scope to present this discussion, e.g., the separated analysis of the (orthogonal) projection matrices. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields. Some excellent examples include (Rose, 1982; Strang, 2009; Trefethen and Bau III, 1997; Strang, 2019, 2021).

Building thriving internet or web products requires optimizing for multiple business metrics. In many applications, these business metrics can move in conflicting directions, and delicately balancing these tradeoffs can be vital for the success of a product. For example, emails and notifications are foundational components of many companies' growth strategies today. At LinkedIn, when deciding which emails and notifications to send, we focus on the engagement generated, while remaining mindful of the volume sent . Similarly, when providing edge recommendations, total edges formed, and the engagement on those edges can be at odds: optimizing for total connections might lead to a large number of edges that have no engagement, bringing no additional value to the members.

Risk-bounded motion planning is an important yet difficult problem for safety-critical tasks. While existing mathematical programming methods offer theoretical guarantees in the context of constrained Markov decision processes, they either lack scalability in solving larger problems or produce conservative plans. Recent advances in deep reinforcement learning improve scalability by learning policy networks as function approximators. In this paper, we propose an extension of soft actor critic model to estimate the execution risk of a plan through a risk critic and produce risk-bounded policies efficiently by adding an extra risk term in the loss function of the policy network. We define the execution risk in an accurate form, as opposed to approximating it through a summation of immediate risks at each time step that leads to conservative plans. Our proposed model is conditioned on a continuous spectrum of risk bounds, allowing the user to adjust the risk-averse level of the agent on the fly. Through a set of experiments, we show the advantage of our model in terms of both computational time and plan quality, compared to a state-of-the-art mathematical programming baseline, and validate its performance in more complicated scenarios, including nonlinear dynamics and larger state space.

Highly populated cities face several challenges, one of them being the intense traffic congestion. In recent years, the concept of Urban Air Mobility has been put forward by large companies and organizations as a way to address this problem, and this approach has been rapidly gaining ground. This disruptive technology involves aerial vehicles (AVs) for hire than can be utilized by customers to travel between locations within large cities. This concept has the potential to drastically decrease traffic congestion and reduce air pollution, since these vehicles typically use electric motors powered by batteries. This work studies the problem of scheduling the assignment of AVs to customers, having as a goal to maximize the serviced customers and minimize the energy consumption of the AVs by forcing them to fly at the lowest possible altitude. Initially, an Integer Linear Program (ILP) formulation is presented, that is solved offline and optimally, followed by a near-optimal algorithm, that solves the problem incrementally, one AV at a time, to address scalability issues, allowing scheduling in problems involving large numbers of locations, AVs, and customer requests.

Detection of change points constitutes a canonical statistical problem due to numerous applications in diverse areas, including economics and finance (Basseville et al. [1993], Frisén [2008]), quality process control (Qiu [2013]), functional genomics and neuroscience (Koepcke et al. [2016]). The offline version of the problem, wherein one examines the data retrospectively and aims to detect the presence and/or location of change points has been studied extensively for a variety of statistical models, including signal plus noise, regression, graphical, random graph, factor and time series models and various algorithms have been developed to accomplish this task -dynamic programming, regularized cost functions, binary segmentation, multiscale methods, etc., see, e.g. the review article Niu et al. [2016]. In the presence of multiple change points, consistency of the estimated location of the change points under certain regularity assumptions on the temporal spacing between change points and on the magnitude of the changes in the underlying model parameters have been established, see, e.g. Fryzlewicz [2014], Frick et al. [2014] and Wang and Samworth [2018] amongst several others, here the former two are under a fixed p framework and the latter under a high dimensional framework. Further, when a single change point has been assumed, the asymptotic distribution of the change point estimator has been established for various statistical models, see, e.g., [Bai, 1994, 1997], Csorgo and Horváth [1997], under fixed p setting, and [Bhattacharjee et al., 2017, 2019], [Kaul et al., 2020, 2021], under diverging dimensionality, where the last two articles allow potential high dimensionality.

Mixed-integer linear programs (MILPs) are widely used in artificial intelligence and operations research to model complex decision problems like scheduling and routing. Designing such programs however requires both domain and modelling expertise. In this paper, we study the problem of acquiring MILPs from contextual examples, a novel and realistic setting in which examples capture solutions and non-solutions within a specific context. The resulting learning problem involves acquiring continuous parameters -- namely, a cost vector and a feasibility polytope -- but has a distinctly combinatorial flavor. To solve this complex problem, we also contribute MISSLE, an algorithm for learning MILPs from contextual examples. MISSLE uses a variant of stochastic local search that is guided by the gradient of a continuous surrogate loss function. Our empirical evaluation on synthetic data shows that MISSLE acquires better MILPs faster than alternatives based on stochastic local search and gradient descent.