Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, 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.

Computational Optimal Transport

We present a novel representation of maps between pairs of shapes that allows for efficient inference and manipulation. Key to our approach is a generalization of the notion of map that puts in correspondence real-valued functions rather than points on the shapes. By choosing a multi-scale basis for the function space on each shape, such as the eigenfunctions of its Laplace-Beltrami operator, we obtain a representation of a map that is very compact, yet fully suitable for global inference. Perhaps more remarkably, most natural constraints on a map, such as descriptor preservation, landmark correspondences, part preservation and operator commutativity become linear in this formulation. Moreover, the representation naturally supports certain algebraic operations such as map sum, difference and composition, and enables a number of applications, such as function or annotation transfer without establishing point-to-point correspondences. We exploit these properties to devise an efficient shape matching method, at the core of which is a single linear solve. The new method achieves state-of-the-art results on an isometric shape matching benchmark. We also show how this representation can be used to improve the quality of maps produced by existing shape matching methods, and illustrate its usefulness in segmentation transfer and joint analysis of shape collections.

https://d2f99xq7vri1nk.cloudfront.net/legacy_app_files/pdf/pub-GeoPro-FMaps-web.pdf

Functional maps: a flexible representation of maps between shapes

Knowledge tracing—where a machine models the knowledge of a student as they interact with coursework—is a well established problem in computer supported education. Though effectively modeling student knowledge would have high educational impact, the task has many inherent challenges. In this paper we explore the utility of using Recurrent Neural Networks (RNNs) to model student learning. The RNN family of models have important advantages over previous methods in that they do not require the explicit encoding of human domain knowledge, and can capture more complex representations of student knowledge. Using neural networks results in substantial improvements in prediction performance on a range of knowledge tracing datasets. Moreover the learned model can be used for intelligent curriculum design and allows straightforward interpretation and discovery of structure in student tasks. These results suggest a promising new line of research for knowledge tracing and an exemplary application task for RNNs.

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Deep knowledge tracing

Programming language processing (similar to natural language processing) is a hot research topic in the field of software engineering; it has also aroused growing interest in the artificial intelligence community. However, different from a natural language sentence, a program contains rich, explicit, and complicated structural information. Hence, traditional NLP models may be inappropriate for programs. In this paper, we propose a novel tree-based convolutional neural network (TBCNN) for programming language processing, in which a convolution kernel is designed over programs' abstract syntax trees to capture structural information. TBCNN is a generic architecture for programming language processing; our experiments show its effectiveness in two different program analysis tasks: classifying programs according to functionality, and detecting code snippets of certain patterns. TBCNN outperforms baseline methods, including several neural models for NLP.

/pdf/convolutional-neural-networks-over-tree-structures-for-1mxr3ybv9x.pdf

Convolutional neural networks over tree structures for programming language processing

Research at the intersection of machine learning, programming languages, and software engineering has recently taken important steps in proposing learnable probabilistic models of source code that exploit code's abundance of patterns. In this article, we survey this work. We contrast programming languages against natural languages and discuss how these similarities and differences drive the design of probabilistic models. We present a taxonomy based on the underlying design principles of each model and use it to navigate the literature. Then, we review how researchers have adapted these models to application areas and discuss cross-cutting and application-specific challenges and opportunities.

A Survey of Machine Learning for Big Code and Naturalness

This paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

/pdf/convolutional-wasserstein-distances-efficient-optimal-1nmt1ddhao.pdf

Convolutional wasserstein distances: efficient optimal transportation on geometric domains

Providing feedback, both assessing final work and giving hints to stuck students, is difficult for open-ended assignments in massive online classes which can range from thousands to millions of students. We introduce a neural network method to encode programs as a linear mapping from an embedded precondition space to an embedded postcondition space and propose an algorithm for feedback at scale using these linear maps as features. We apply our algorithm to assessments from the Code.org Hour of Code and Stanford University's CS1 course, where we propagate human comments on student assignments to orders of magnitude more submissions.

/pdf/learning-program-embeddings-to-propagate-feedback-on-student-4dn8fje8jh.pdf

Learning Program Embeddings to Propagate Feedback on Student Code

Finding an informative, structure-preserving map between two shapes has been a long-standing problem in geometry processing, involving a variety of solution approaches and applications. However, in many cases, we are given not only two related shapes, but a collection of them, and considering each pairwise map independently does not take full advantage of all existing information. For example, a notorious problem with computing shape maps is the ambiguity introduced by the symmetry problem — for two similar shapes which have reflectional symmetry there exist two maps which are equally favorable, and no intrinsic mapping algorithm can distinguish between them based on these two shapes alone. Another prominent issue with shape mapping algorithms is their relative sensitivity to how “similar” two shapes are — good maps are much easier to obtain when shapes are very similar. Given the context of additional shape maps connecting our collection, we propose to add the constraint of global map consistency, requiring that any composition of maps between two shapes should be independent of the path chosen in the network. This requirement can help us choose among the equally good symmetric alternatives, or help us replace a “bad” pairwise map with the composition of a few “good” maps between shapes that in some sense interpolate the original ones. We show how, given a collection of pairwise shape maps, to define an optimization problem whose output is a set of alternative maps, compositions of those given, which are consistent, and individually at times much better than the original. Our method is general, and can work on any collection of shapes, as long as a seed set of good pairwise maps is provided. We demonstrate the effectiveness of our method for improving maps generated by state-of-the-art mapping methods on various shape databases.

/pdf/an-optimization-approach-to-improving-collections-of-shape-3r7dhaenk2.pdf

An Optimization Approach to Improving Collections of Shape Maps

Mild traumatic brain injury (mTBI, also referred to as concussion) accounts for the majority of all TBIs. The consequences of repetitive mTBI have become of particular concern for individuals engaged in certain sports or in military operations. Many mTBI patients suffer long-lasting neurobehavioral impairments. In order to expedite pre-clinical research and therapy development, there is a need for animal models that reflect the long-term cognitive and pathological features seen in patients. In the present study, we developed and characterized a mouse model of repetitive mTBI, induced onto the closed head over the left frontal hemisphere with an electromagnetic stereotaxic impact device. Using GFAP-luciferase bioluminescence reporter mice that provide a readout of astrocyte activation, we observed an increase in bioluminescence relative to the force delivered by the impactor after single impact and cumulative effects of repetitive mTBI. Using the injury parameters established in the reporter mice, we induced a repetitive mTBI in wild-type C57BL/6J mice and characterized the long-term outcome. Animals received repetitive mTBI showed a significant impairment in spatial learning and memory when tested at 2 and 6 months after injury. A robust astrogliosis and increased p-Tau immunoreactivity was observed upon post-mortem pathological examinations. These findings are consistent with the deficits and pathology associated with mTBI in humans and support the use of this model to evaluate potential therapeutic approaches.

/pdf/long-term-cognitive-impairments-and-pathological-alterations-1oip2gd7hq.pdf

Long-term cognitive impairments and pathological alterations in a mouse model of repetitive mild traumatic brain injury

Massive open online courses (MOOCs), one of the latest internet revolutions have engendered hope that constant iterative improvement and economies of scale may cure the ``cost disease" of higher education. While scalable in many ways, providing feedback for homework submissions (particularly open-ended ones) remains a challenge in the online classroom. In courses where the student-teacher ratio can be ten thousand to one or worse, it is impossible for instructors to personally give feedback to students or to understand the multitude of student approaches and pitfalls. Organizing and making sense of massive collections of homework solutions is thus a critical web problem. Despite the challenges, the dense solution space sampling in highly structured homeworks for some MOOCs suggests an elegant solution to providing quality feedback to students on a massive scale. We outline a method for decomposing online homework submissions into a vocabulary of "code phrases", and based on this vocabulary, we architect a queryable index that allows for fast searches into the massive dataset of student homework submissions. To demonstrate the utility of our homework search engine we index over a million code submissions from users worldwide in Stanford's Machine Learning MOOC and (a) semi-automatically learn shared structure amongst homework submissions and (b) generate specific feedback for student mistakes. Codewebs is a tool that leverages the redundancy of densely sampled, highly structured homeworks in order to force-multiply teacher effort. Giving articulate, instant feedback is a crucial component of the online learning process and thus by building a homework search engine we hope to take a step towards higher quality free education.

/pdf/codewebs-scalable-homework-search-for-massive-open-online-zazrhw6ib4.pdf

Andy Nguyen

Papers

Convolutional wasserstein distances: efficient optimal transportation on geometric domains

Learning Program Embeddings to Propagate Feedback on Student Code

An Optimization Approach to Improving Collections of Shape Maps

Long-term cognitive impairments and pathological alterations in a mouse model of repetitive mild traumatic brain injury

Codewebs: scalable homework search for massive open online programming courses