Re-finding electronic documents from a personal computer is a frequent demand to users. In a simple re-finding task, people can use many methods to retrieve a document, such as navigating directly to the document's folder, searching with a desktop search engine, or checking the Recent Files List. However, when encountering a difficult re-finding task, people usually cannot remember the attributes used by conventional re-finding methods, such as file path, file name, keywords etc., the re-finding would fail. We propose a new method to support difficult re-finding tasks. When a user is reading a document, we collect all kinds of possible memory pieces of the user about the document, such as number of pages, number of images, number of math formulas, cumulative reading time, reading frequency, printing experiences etc. If the user wants to re-find a document later, we use these collected attributes to filter out the target document. To alleviate the user's cognitive burden, we use a question and answer wizard interface and provide recommendations to the answers for the user, the recommendations are generated by analyzing the collected attributes of each document and the user's experiences about them.
We introduce the entropic measure transform (EMT) problem for a general process and prove the existence of a unique optimal measure characterizing the solution. The density process of the optimal measure is characterized using a semimartingale BSDE under general conditions. The EMT is used to reinterpret the conditional entropic risk-measure and to obtain a convenient formula for the conditional expectation of a process which admits an affine representation under a related measure. The entropic measure transform is then used provide a new characterization of defaultable bond prices, forward prices, and futures prices when the asset is driven by a jump diffusion. The characterization of these pricing problems in terms of the EMT provides economic interpretations as a maximization of returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy. The EMT is shown to extend the optimal stochastic control characterization of default-free bond prices of Gombani and Runggaldier (Math. Financ. 23(4):659-686, 2013). These methods are illustrated numerically with an example in the defaultable bond setting.
We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force in the system. The local agents derive their stability only from the central agent. In the mean field limit of a large number of local agents we show that the systemic risk decreases when the strength of the interaction of the local agents with the central agent increases. This means that the probability of transition from one of the two stable quasi-equilibria to the other one decreases. We also show that the systemic risk increases when the strength of the interaction of the central agent with the mean field of the local agents increases. Following the financial interpretation of such models and their behavior given in our previous paper (Garnier, Papanicolaou and Yang, SIAM J. Fin. Math. 4, 2013, 151-184), we may interpret the results of this paper in the following way. From the point of view of systemic risk, and while keeping the perceived risk of the local agents approximately constant, it is better to strengthen the interaction of the local agents with the central agent than the other way around.
With the ever-increasing quantity and variety of data worldwide, the Web has become a rich repository of mathematical formulae. This necessitates the creation of robust and scalable systems for Mathematical Information Retrieval, where users search for mathematical information using individual formulae (query-by-expression) or a combination of keywords and formulae. Often, the pages that best satisfy users' information needs contain expressions that only approximately match the query formulae. For users trying to locate or re-find a specific expression, browse for similar formulae, or who are mathematical non-experts, the similarity of formulae depends more on the relative positions of symbols than on deep mathematical semantics. We propose the Maximum Subtree Similarity (MSS) metric for query-by-expression that produces intuitive rankings of formulae based on their appearance, as represented by the types and relative positions of symbols. Because it is too expensive to apply the metric against all formulae in large collections, we first retrieve expressions using an inverted index over tuples that encode relationships between pairs of symbols, ranking hits using the Dice coefficient. The top-k formulae are then re-ranked using MSS. Our approach obtains state-of-the-art performance on the NTCIR-11 Wikipedia formula retrieval benchmark and is efficient in terms of both index space and overall retrieval time. Retrieval systems for other graphical forms, including chemical diagrams, flowcharts, figures, and tables, may also benefit from adopting our approach.
Similarity search in math is to find mathematical expressions that are similar to a user's query. We conceptualized the similarity factors between mathematical expressions, and proposed an approach to math similarity search (MSS) by defining metrics based on those similarity factors [11]. Our preliminary implementation indicated the advantage of MSS compared to non-similarity based search. In order to more effectively and efficiently search similar math expressions, MSS is further optimized. This paper focuses on performance evaluation and optimization of MSS. Our results show that the proposed optimization process significantly improved the performance of MSS with respect to both relevance ranking and recall.
We summarize math search engines and search interfaces produced by the Document and Pattern Recognition Lab in recent years, and in particular the min math search interface and the Tangent search engine. Source code for both systems are publicly available. "The Masses" refers to our emphasis on creating systems for mathematical non-experts, who may be looking to define unfamiliar notation, or browse documents based on the visual appearance of formulae rather than their mathematical semantics.
We consider the Moran process in a graph called the "star" and obtain the asymptotic expression for the fixation probability of a single mutant when the size of the graph is large. The expression obtained corrects the previously known expression announced in reference [E Lieberman, C Hauert, and MA Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312-316, 2005] and further studied in [M. Broom and J. Rychtar. An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. R. Soc. A-Math. Phys. Eng. Sci., 464(2098):2609-2627, 2008]. We also show that the star graph is an accelerator of evolution, if the graph is large enough.
Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay. Commun.~Math.~Sci., 18(1):109-121, 2020](arXiv:1808.02341) proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method's efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.
A practical automatic textual math word problems (MWPs) solver should be able to solve various textual MWPs while most existing works only focused on one-unknown linear MWPs. Herein, we propose a simple but efficient method called Universal Expression Tree (UET) to make the first attempt to represent the equations of various MWPs uniformly. Then a semantically-aligned universal tree-structured solver (SAU-Solver) based on an encoder-decoder framework is proposed to resolve multiple types of MWPs in a unified model, benefiting from our UET representation. Our SAU-Solver generates a universal expression tree explicitly by deciding which symbol to generate according to the generated symbols' semantic meanings like human solving MWPs. Besides, our SAU-Solver also includes a novel subtree-level semanticallyaligned regularization to further enforce the semantic constraints and rationality of the generated expression tree by aligning with the contextual information. Finally, to validate the universality of our solver and extend the research boundary of MWPs, we introduce a new challenging Hybrid Math Word Problems dataset (HMWP), consisting of three types of MWPs. Experimental results on several MWPs datasets show that our model can solve universal types of MWPs and outperforms several state-of-the-art models.
This work characterizes the range of the single-quadrant approximate discrete Radon transform (ADRT) of square images. The characterization follows from a set of linear constraints on the codomain. We show that for data satisfying these constraints, the exact and fast inversion formula [Rim, Appl. Math. Lett. 102 106159, 2020] yields a square image in a stable manner. The range characterization is obtained by first showing that the ADRT is a bijection between images supported on infinite half-strips, then identifying the linear subspaces that stay finitely supported under the inversion formula.
We have implemented a machine translation system, the PolyMath Translator, for LaTeX documents containing mathematical text. The current implementation translates English LaTeX to French LaTeX, attaining a BLEU score of 53.5 on a held-out test corpus of mathematical sentences. It produces LaTeX documents that can be compiled to PDF without further editing. The system first converts the body of an input LaTeX document into English sentences containing math tokens, using the pandoc universal document converter to parse LaTeX input. We have trained a Transformer-based translator model, using OpenNMT, on a combined corpus containing a small proportion of domain-specific sentences. Our full system uses both this Transformer model and Google Translate, the latter being used as a backup to better handle linguistic features that do not appear in our training dataset. If the Transformer model does not have confidence in its translation, as determined by a high perplexity score, then we use Google Translate with a custom glossary. This backup was used 26% of the time on our test corpus of mathematical sentences. The PolyMath Translator is available as a web service at www.polymathtrans.ai.
Automatically solving math word problems is a critical task in the field of natural language processing. Recent models have reached their performance bottleneck and require more high-quality data for training. We propose a novel data augmentation method that reverses the mathematical logic of math word problems to produce new high-quality math problems and introduce new knowledge points that can benefit learning the mathematical reasoning logic. We apply the augmented data on two SOTA math word problem solving models and compare our results with a strong data augmentation baseline. Experimental results show the effectiveness of our approach. We release our code and data at https://github.com/yiyunya/RODA.
Automatic math word problem solving has attracted growing attention in recent years. The evaluation datasets used by previous works have serious limitations in terms of scale and diversity. In this paper, we release a new large-scale and template-rich math word problem dataset named Ape210K. It consists of 210K Chinese elementary school-level math problems, which is 9 times the size of the largest public dataset Math23K. Each problem contains both the gold answer and the equations needed to derive the answer. Ape210K is also of greater diversity with 56K templates, which is 25 times more than Math23K. Our analysis shows that solving Ape210K requires not only natural language understanding but also commonsense knowledge. We expect Ape210K to be a benchmark for math word problem solving systems. Experiments indicate that state-of-the-art models on the Math23K dataset perform poorly on Ape210K. We propose a copy-augmented and feature-enriched sequence to sequence (seq2seq) model, which outperforms existing models by 3.2% on the Math23K dataset and serves as a strong baseline of the Ape210K dataset. The gap is still significant between human and our baseline model, calling for further research efforts. We make Ape210K dataset publicly available at https://github.com/yuantiku/ape210k
The ability of deep learning (DL) to improve the practice of medicine and its clinical outcomes faces a looming obstacle: model interpretation. Without description of how outputs are generated, a collaborating physician can neither resolve when the model's conclusions are in conflict with his or her own, nor learn to anticipate model behavior. Current research aims to interpret networks that diagnose ECG recordings, which has great potential impact as recordings become more personalized and widely deployed. A generalizable impact beyond ECGs lies in the ability to provide a rich test-bed for the development of interpretive techniques in medicine. Interpretive techniques for Deep Neural Networks (DNNs), however, tend to be heuristic and observational in nature, lacking the mathematical rigor one might expect in the analysis of math equations. The motivation of this paper is to offer a third option, a scientific approach. We treat the model output itself as a phenomenon to be explained through component parts and equations governing their behavior. We argue that these component parts should also be "black boxes" --additional targets to interpret heuristically with clear functional connection to the original. We show how to rigorously factor a DNN into a hierarchical equation consisting of black box variables. This is not a subdivision into physical parts, like an organism into its cells; it is but one choice of an equation into a collection of abstract functions. Yet, for DNNs trained to identify normal ECG waveforms on PhysioNet 2017 Challenge data, we demonstrate this choice yields interpretable component models identified with visual composite sketches of ECG samples in corresponding input regions. Moreover, the recursion distills this interpretation: additional factorization of component black boxes corresponds to ECG partitions that are more morphologically pure.
Deep neural networks have achieved human-level accuracy on almost all perceptual benchmarks. It is interesting that these advances were made using two ideas that are decades old: (a) an artificial neuron based on a linear summator and (b) SGD training. However, there are important metrics beyond accuracy: computational efficiency and stability against adversarial perturbations. In this paper, we propose two closely connected methods to improve these metrics on contour recognition tasks: (a) a novel model of an artificial neuron, a "strong neuron," with low hardware requirements and inherent robustness against adversarial perturbations and (b) a novel constructive training algorithm that generates sparse networks with $O(1)$ connections per neuron. We demonstrate the feasibility of our approach through experiments on SVHN and GTSRB benchmarks. We achieved an impressive 10x-100x reduction in operations count (10x when compared with other sparsification approaches, 100x when compared with dense networks) and a substantial reduction in hardware requirements (8-bit fixed-point math was used) with no reduction in model accuracy. Superior stability against adversarial perturbations (exceeding that of adversarial training) was achieved without any counteradversarial measures, relying on the robustness of strong neurons alone. We also proved that constituent blocks of our strong neuron are the only activation functions with perfect stability against adversarial attacks.
Authors of research papers in the fields of mathematics, and other math-heavy disciplines commonly employ the Mathematics Subject Classification (MSC) scheme to search for relevant literature. The MSC is a hierarchical alphanumerical classification scheme that allows librarians to specify one or multiple codes for publications. Digital Libraries in Mathematics, as well as reviewing services, such as zbMATH and Mathematical Reviews (MR) rely on these MSC labels in their workflows to organize the abstracting and reviewing process. Especially, the coarse-grained classification determines the subject editor who is responsible for the actual reviewing process. In this paper, we investigate the feasibility of automatically assigning a coarse-grained primary classification using the MSC scheme, by regarding the problem as a multi-class classification machine learning task. We find that our method achieves an (F_1)-score of over 77%, which is remarkably close to the agreement of zbMATH and MR ((F_1)-score of 81%). Moreover, we find that the method's confidence score allows for reducing the effort by 86% compared to the manual coarse-grained classification effort while maintaining a precision of 81% for automatically classified articles.
We describe two systems for supporting beginner students in acquiring basic skills in expressing statements in the formalism of first-order predicate logic; the first, called "math dictations", presents users with the task of formalizing a given natural-language sentence, while the second, called "Game of Def", challenges users to give a formal description of a set of a geometric pattern displayed to them. In both cases, an automatic checking takes place.
In this paper, we show how selecting and combining encodings of natural and mathematical language affect classification and clustering of documents with mathematical content. We demonstrate this by using sets of documents, sections, and abstracts from the arXiv preprint server that are labeled by their subject class (mathematics, computer science, physics, etc.) to compare different encodings of text and formulae and evaluate the performance and runtimes of selected classification and clustering algorithms. Our encodings achieve classification accuracies up to $82.8\%$ and cluster purities up to $69.4\%$ (number of clusters equals number of classes), and $99.9\%$ (unspecified number of clusters) respectively. We observe a relatively low correlation between text and math similarity, which indicates the independence of text and formulae and motivates treating them as separate features of a document. The classification and clustering can be employed, e.g., for document search and recommendation. Furthermore, we show that the computer outperforms a human expert when classifying documents. Finally, we evaluate and discuss multi-label classification and formula semantification.
ML decision-aid systems are increasingly common on the web, but their successful integration relies on people trusting them appropriately: they should use the system to fill in gaps in their ability, but recognize signals that the system might be incorrect. We measured how people's trust in ML recommendations differs by expertise and with more system information through a task-based study of 175 adults. We used two tasks that are difficult for humans: comparing large crowd sizes and identifying similar-looking animals. Our results provide three key insights: (1) People trust incorrect ML recommendations for tasks that they perform correctly the majority of the time, even if they have high prior knowledge about ML or are given information indicating the system is not confident in its prediction; (2) Four different types of system information all increased people's trust in recommendations; and (3) Math and logic skills may be as important as ML for decision-makers working with ML recommendations.
With the success of deep learning in various fields and the advent of numerous Internet of Things (IoT) devices, it is essential to lighten models suitable for low-power devices. In keeping with this trend, MicroNet Challenge, which is the challenge to build efficient models from the view of both storage and computation, was hosted at NeurIPS 2019. To develop efficient models through this challenge, we propose a framework, coined as SIPA, consisting of four stages: Searching, Improving, Pruning, and Accelerating. With the proposed framework, our team, OSI AI, compressed 334x the parameter storage and 357x the math operation compared to WideResNet-28-10 and took 4th place in the CIFAR-100 track at MicroNet Challenge 2019 with the top 10% highly efficient computation. Our source code is available from https://github.com/Lee-Gihun/MicroNet_OSI-AI.