An official website of the United States government
We present a new search method for mathematical formulas based on Operator Trees (OPTs) representing the application of operators to operands. Our method provides (1) a simple indexing scheme using OPT leaf-root paths, (2) practical matching of the K largest common subexpressions, and (3) scoring matched OPT subtrees by counting nodes corresponding to visible symbols, weighting operators lower than operands. Using the largest common subexpression (K=1), we outperform existing formula search engines for non-wildcard queries on the NTCIR-12 Wikipedia Formula Browsing Task. Stronger results are obtained when using additional subexpressions for scoring. Without parallelization or pruning, our system has practical execution times with low variance when compared to other state-of-the-art formula search engines.
more »
« less
Tangent-CFT: An Embedding Model for Mathematical Formulas
When searching for mathematical content, accurate measures of formula similarity can help with tasks such as document ranking, query recommendation, and result set clustering. While there have been many attempts at embedding words and graphs, formula embedding is in its early stages. We introduce a new formula em- bedding model that we use with two hierarchical representations, (1) Symbol Layout Trees (SLTs) for appearance, and (2) Operator Trees (OPTs) for mathematical content. Following the approach of graph embeddings such as DeepWalk, we generate tuples represent- ing paths between pairs of symbols depth-first, embed tuples using the fastText n-gram embedding model, and then represent an SLT or OPT by its average tuple embedding vector. We then combine SLT and OPT embeddings, leading to state-of-the-art results for the NTCIR-12 formula retrieval task. Our fine-grained holistic vector representations allow us to retrieve many more partially similar for- mulas than methods using structural matching in trees. Combining our embedding model with structural matching in the Approach0 formula search engine produces state-of-the-art results for both fully and partially relevant results on the NTCIR-12 benchmark. Source code for our system is publicly available.
more »
« less
- Award ID(s):
- 1717997
- PAR ID:
- 10124325
- Date Published:
- Journal Name:
- ICTIR '19 Proceedings of the 2019 ACM SIGIR International Conference on Theory of Information Retrieval
- Page Range / eLocation ID:
- 11 to 18
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Representational Learning in the form of high dimensional embeddings have been used for multiple pattern recognition applications. There has been a significant interest in building embedding based systems for learning representations in the mathematical domain. At the same time, retrieval of structured information such as mathematical expressions is an important need for modern IR systems. In this work, our motivation is to introduce a robust framework for learning representations for similarity based retrieval of mathematical expressions. Given a query by example, the embedding can find the closest matching expression as a function of euclidean distance between them. We leverage recent advancements in image-based and graph-based deep learning algorithms to learn our similarity embeddings. We do this first, by using unimodal encoders in graph space and image space and then, a multi-modal combination of the same. To overcome the lack of training data, we force the networks to learn a deep metric using triplets generated with a heuristic scoring function. We also adopt a custom strategy for mining hard samples to train our neural networks. Our system produces rankings similar to those generated by the original scoring function, but using only a fraction of the time. Our results establish the viability of using such a multi-modal embedding for this task.more » « less
-
Formula retrieval systems using substructure matching are effective, but suffer from slow retrieval times caused by the complexity of structure matching. We present a specialized inverted index and rank-safe dynamic pruning algorithm for faster substructure retrieval. Formulas are indexed from their Operator Tree (OPT) representations. Our model is evaluated using the NTCIR-12 Wikipedia Formula Browsing Task and a new formula corpus produced from Math StackExchange posts. Our approach preserves the effectiveness of structure matching while allowing queries to be executed in real-time.more » « less
-
A wide variety of machine learning tasks such as knowledge base completion, ontology alignment, and multi-label classification can benefit from incorporating into learning differentiable representations of graphs or taxonomies. While vectors in Euclidean space can theoretically represent any graph, much recent work shows that alternatives such as complex, hyperbolic, order, or box embeddings have geometric properties better suited to modeling real-world graphs. Experimentally these gains are seen only in lower dimensions, however, with performance benefits diminishing in higher dimensions. In this work, we introduce a novel variant of box embeddings that uses a learned smoothing parameter to achieve better representational capacity than vector models in low dimensions, while also avoiding performance saturation common to other geometric models in high dimensions. Further, we present theoretical results that prove box embeddings can represent any DAG. We perform rigorous empirical evaluations of vector, hyperbolic, and region-based geometric representations on several families of synthetic and real-world directed graphs. Analysis of these results exposes correlations between different families of graphs, graph characteristics, model size, and embedding geometry, providing useful insights into the inductive biases of various differentiable graph representations.more » « less
-
Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than 100x on the benchmark datasets while maintaining an error rate below 0.1%.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3341981.3344235
