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.
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.
- Award ID(s):
- Publication Date:
- NSF-PAR ID:
- Journal Name:
- ICTIR '19 Proceedings of the 2019 ACM SIGIR International Conference on Theory of Information Retrieval
- Page Range or eLocation-ID:
- 11 to 18
- Sponsoring Org:
- National Science Foundation
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