Math Search for the Masses: Multimodal Search Interfaces and Appearance-Based Retrieval

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.Comment: Paper for Invited Talk at 2015 Conference on Intelligent Computer Mathematics (July, Washington DC

Math expression retrieval using symbol pairs in layout trees

We have developed a layout-based math retrieval system by indexing on pairs of symbols in mathematical expressions. Existing approaches to layout-based retrieval include tree edit distance-based matching on MathML trees (Kamali and Tompa, 2013) and longest common subsequence matching in LATEX strings (Kumar et al., 2012). In our work, we compare our new layout-based retrieval method with a math retrieval system built using the conventional text-based retrieval system Lucene (Zanibbi and Yuan, 2011), as such systems are commonly used for math search. We show that the search results returned by our system are scored by participants in a study as significantly more similar than those of the comparison system and that our system is fast enough to be used in real time

Symbolic and Visual Retrieval of Mathematical Notation using Formula Graph Symbol Pair Matching and Structural Alignment

Large data collections containing millions of math formulae in different formats are available on-line. Retrieving math expressions from these collections is challenging. We propose a framework for retrieval of mathematical notation using symbol pairs extracted from visual and semantic representations of mathematical expressions on the symbolic domain for retrieval of text documents. We further adapt our model for retrieval of mathematical notation on images and lecture videos. Graph-based representations are used on each modality to describe math formulas. For symbolic formula retrieval, where the structure is known, we use symbol layout trees and operator trees. For image-based formula retrieval, since the structure is unknown we use a more general Line of Sight graph representation. Paths of these graphs define symbol pairs tuples that are used as the entries for our inverted index of mathematical notation. Our retrieval framework uses a three-stage approach with a fast selection of candidates as the first layer, a more detailed matching algorithm with similarity metric computation in the second stage, and finally when relevance assessments are available, we use an optional third layer with linear regression for estimation of relevance using multiple similarity scores for final re-ranking. Our model has been evaluated using large collections of documents, and preliminary results are presented for videos and cross-modal search. The proposed framework can be adapted for other domains like chemistry or technical diagrams where two visually similar elements from a collection are usually related to each other

A Survey on Retrieval of Mathematical Knowledge

We present a short survey of the literature on indexing and retrieval of mathematical knowledge, with pointers to 72 papers and tentative taxonomies of both retrieval problems and recurring techniques.Comment: CICM 2015, 20 page

VMEXT: A Visualization Tool for Mathematical Expression Trees

Mathematical expressions can be represented as a tree consisting of terminal symbols, such as identifiers or numbers (leaf nodes), and functions or operators (non-leaf nodes). Expression trees are an important mechanism for storing and processing mathematical expressions as well as the most frequently used visualization of the structure of mathematical expressions. Typically, researchers and practitioners manually visualize expression trees using general-purpose tools. This approach is laborious, redundant, and error-prone. Manual visualizations represent a user's notion of what the markup of an expression should be, but not necessarily what the actual markup is. This paper presents VMEXT - a free and open source tool to directly visualize expression trees from parallel MathML. VMEXT simultaneously visualizes the presentation elements and the semantic structure of mathematical expressions to enable users to quickly spot deficiencies in the Content MathML markup that does not affect the presentation of the expression. Identifying such discrepancies previously required reading the verbose and complex MathML markup. VMEXT also allows one to visualize similar and identical elements of two expressions. Visualizing expression similarity can support support developers in designing retrieval approaches and enable improved interaction concepts for users of mathematical information retrieval systems. We demonstrate VMEXT's visualizations in two web-based applications. The first application presents the visualizations alone. The second application shows a possible integration of the visualizations in systems for mathematical knowledge management and mathematical information retrieval. The application converts LaTeX input to parallel MathML, computes basic similarity measures for mathematical expressions, and visualizes the results using VMEXT.Comment: 15 pages, 4 figures, Intelligent Computer Mathematics - 10th International Conference CICM 2017, Edinburgh, UK, July 17-21, 2017, Proceeding

Variable typing: Assigning meaning to variables in mathematical text

Information about the meaning of mathematical variables in text is useful in NLP/IR tasks such as symbol disambiguation, topic modeling and mathematical information retrieval (MIR). We introduce variable typing, the task of assigning one mathematical type (multi-word technical terms referring to mathematical concepts) to each variable in a sentence of mathematical text. As part of this work, we also introduce a new annotated data set composed of 33,524 data points extracted from scientific documents published on arXiv. Our intrinsic evaluation demonstrates that our data set is sufficient to successfully train and evaluate current classifiers from three different model architectures. The best performing model is evaluated on an extrinsic task: MIR, by producing a typed formula index. Our results show that the best performing MIR models make use of our typed index, compared to a formula index only containing raw symbols, thereby demonstrating the usefulness of variable typing

Effective Math-Aware Ad-Hoc Retrieval based on Structure Search and Semantic Similarities

Despite the prevalence of digital scientific and educational contents on the Internet, only a few search engines are capable to retrieve them efficiently and effectively. The main challenge in freely searching scientific literature arises from the presence of structured math formulas and their heterogeneous and contextually important surrounding words. This thesis introduces an effective math-aware, ad-hoc retrieval model that incorporates structure search and semantic similarities. Transformer-based neural retrievers have been adopted to capture additional semantics using domain-adapted supervised retrieval. To enable structure search, I suggest an unsupervised retrieval model that can filter potential mathematical formulas based on structure similarity. This similarity is determined by measuring the largest common substructure(s) in a formula tree representation, known as the Operator Tree (OPT). The structure matching is approximated by employing maximum matching of path-based structure features. The proposed structure similarity measurement can be tailored based on the desired effectiveness and efficiency trade-offs. It may consider various node types, such as operators and operands, and accommodate different numbers of common subtrees with varying weights. In addition to structure similarity, this unsupervised model also captures symbol substitutions through a greedy matching algorithm applied to the matched substructure(s). To achieve efficient structure search, I introduce a dynamic pruning algorithm to the problem of structure retrieval. The proposed retrieval algorithm efficiently identifies the maximum common subtree among formula candidates and safely eliminates potential structure matches that exceed a dynamic threshold. To accomplish this, three rank-safe pruning strategies are suggested and compared against exhaustive search baselines. Additionally, more aggressive thresholding policies are proposed to balance effectiveness with further speed improvements. A novel hierarchical inverted index has been implemented. This index is designed to be compatible with traditional information retrieval (IR) infrastructure and optimization techniques. To capture other semantic similarities, I have incorporated neural retrievers into a hybrid setting with structure search. This approach has achieved the state-of-the-art effectiveness in recent math information retrieval tasks. In comparison to strict and unsupervised matching, I have found that supervised neural retrievers are able to capture additional semantic similarities in a highly complementary manner. In order to learn effective representations in heterogeneous math contents, I have proposed a novel pretraining architecture that can improve the contextual awareness between math and its surrounding texts. This pretraining scheme generates effective downstream single-vector representations, eliminating the efficiency bottleneck from using multi-vector dense representations. In the end, the thesis examines future directions, specifically the integration of recent advancements in language modeling. This includes incorporating ongoing exciting developments of large language models for improved math information retrieval. A preliminary evaluation has been conducted to assess the impact of these advancements