Optimal Transport (OT) is a mathematical gem at the interface between probability, analysis and optimization. The goal of that theory is to define geometric tools that are useful to compare probability distributions. Earlier contributions originated from Monge's work in the 18th century, to be later rediscovered under a different formalism by Tolstoi in the 1920's, Kantorovich, Hitchcock and Koopmans in the 1940's. The problem was solved numerically by Dantzig in 1949 and others in the 1950's within the framework of linear programming, paving the way for major industrial applications in the second half of the 20th century. OT was later rediscovered under a different light by analysts in the 90's, following important work by Brenier and others, as well as in the computer vision/graphics fields under the name of earth mover's distances. Recent years have witnessed yet another revolution in the spread of OT, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression,classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

We discuss general perturbation inequalities when the perturbation is random. As applications, we obtain several new results concerning two important problems: matrix sparsification and matrix completion.

Training deep neural networks (DNNs) efficiently is a challenge due to the associated highly nonconvex optimization. Recently, the efficiency of the block coordinate descent (BCD) type methods has been empirically illustrated for DNN training. The main idea of BCD is to decompose the highly composite and nonconvex DNN training problem into several almost separable simple subproblems. However, their convergence property has not been thoroughly studied. In this paper, we establish some unified global convergence guarantees of BCD type methods for a wide range of DNN training models, including but not limited to multilayer perceptrons (MLPs), convolutional neural networks (CNNs) and residual networks (ResNets). This paper nontrivially extends the existing convergence results of nonconvex BCD from the smooth case to the nonsmooth case. Our convergence analysis is built upon the powerful Kurdyka-{\L}ojasiewicz (KL) framework but some new techniques are introduced, including the establishment of the KL property of the objective functions of many commonly used DNNs, where the loss function can be taken as squared, hinge and logistic losses, and the activation function can be taken as rectified linear units (ReLUs), sigmoid and linear link functions. The efficiency of BCD method is also demonstrated by a series of exploratory numerical experiments.

The problem of machine learning with missing values is common in many areas. A simple approach is to first construct a dataset without missing values simply by discarding instances with missing entries or by imputing a fixed value for each missing entry, and then train a prediction model with the new dataset. A drawback of this naive approach is that the uncertainty in the missing entries is not properly incorporated in the prediction. In order to evaluate prediction uncertainty, the multiple imputation (MI) approach has been studied, but the performance of MI is sensitive to the choice of the probabilistic model of the true values in the missing entries, and the computational cost of MI is high because multiple models must be trained. In this paper, we propose an alternative approach called the Interval-based Prediction Uncertainty Bounding (IPUB) method. The IPUB method represents the uncertainties due to missing entries as intervals, and efficiently computes the lower and upper bounds of the prediction results when all possible training sets constructed by imputing arbitrary values in the intervals are considered. The IPUB method can be applied to a wide class of convex learning algorithms including penalized least-squares regression, support vector machine (SVM), and logistic regression. We demonstrate the advantages of the IPUB method by comparing it with an existing method in numerical experiment with benchmark datasets.

We study the problem of detecting change points (CPs) that are characterized by a subset of dimensions in a multi-dimensional sequence. A method for detecting those CPs can be formulated as a two-stage method: one for selecting relevant dimensions, and another for selecting CPs. It has been difficult to properly control the false detection probability of these CP detection methods because selection bias in each stage must be properly corrected. Our main contribution in this paper is to formulate a CP detection problem as a selective inference problem, and show that exact (non-asymptotic) inference is possible for a class of CP detection methods. We demonstrate the performances of the proposed selective inference framework through numerical simulations and its application to our motivating medical data analysis problem.

In a budget announcement, Hong Kong's financial secretary Paul Chan is setting aside $50 billion for Innovation and Technology development. "To shine in the fierce I&T race amidst keen competition, Hong Kong must optimize its resources by focusing on developing its areas of strength, namely biotechnology, artificial intelligence, smart city and financial technologies," he said. An Innovation and Technology Fund will get a $10 billion injection, earmarked for applied research and development work. There will also be additional tax deductions for domestic expenditure on R&D incurred by enterprises. Enterprises will enjoy a 300% tax deduction for the first $2 million qualifying R&D expenditure, and a 200% deduction for the remainder.

Chinese tech giant Alibaba on Wednesday said it set up a joint research institute in Singapore, together with a local university. The research institute will focus on developing artificial intelligence applications in a variety of areas including health care, smart homes and urban transportation. Alibaba said that the institute, launched with Singapore's Nanyang Technological University (NTU), was the firm's first joint research center outside China. It will be housed on the university's campus. Initially, the center would start with 50 researchers from Alibaba as well as the university.

Michelle Obama took an apparent swipe at Donald Trump's social media habits, saying she uses social media "like a grown-up". "How many kids do you know that the first thing that comes off the top of their head is the first thing they should express? It's like, 'Take a minute. Talk to your crew before you put that [out there] and then spell check and check the grammar,'" the former First Lady said during a panel in New York, according to People. While Ms Obama did not mention the President by name, Mr Trump is known for stream-of-consciousness bursts of tweets that periodically contain grammatical and spelling errors.

Count Bill Gates as part of the growing camp of Hyperloop skeptics. Billionaire tech mogul Elon Musk has gotten closer to making his radical, pod-based transportation system a reality. But the Microsoft co-founder has cast some doubt on whether or not Hyperloop, which promises to ferry passengers hundreds of miles in a matter of minutes, will actually work. 'I am not sure the Hyperloop concept makes sense,' Gates said Tuesday during a question and answer session on Reddit. 'Making it safe is hard,' he added.

Artificial intelligence software developed by an Israeli startup has proved in an international study to be quicker and more accurate at analyzing legal documents than experienced lawyers. The software developed by Tel Aviv based LawGeex was able to analyze nondisclosure agreements with more accuracy and speed than 20 experienced lawyers, the results of a collaborative study between leading US institutions and the company show. Get The Start-Up Israel's Daily Start-Up by email and never miss our top stories Free Sign Up As part of the study the researchers compared the work of the experienced lawyers, some with decades of contract experience, to LawGeex's AI software program, and found that the software was able to achieve nearly 10 percent higher accuracy and complete the task in significantly less time. This study marks the first time that AI technology has been tested with a typical task, such as reviewing a nondisclosure agreement, undertaken by lawyers on a daily basis, the company said in a statement. Both the lawyers and LawGeex's AI software were given five previously unseen contracts, which contained 153 paragraphs of technical legal language that were modeled after common nondisclosure agreements.