# A Scientific Document Retrieval and Reordering Method by Incorporating HFS and LSD
## 1 Introduction  
With the continuous growth in research across science, technology, engineering, and management (STEM) disciplines, the body of scientific literature, predominantly composed of complex mathematical formulas and equations, has experienced exponential expansion, posing challenges to traditional text-based retrieval methods. The semantics conveyed by one or more mathematical expressions within a document may often encapsulate the core message of the entire document. Simultaneously, in many instances, mathematical expressions cannot be succinctly explained or searched using just a few words. To achieve more ideal retrieval results, an effective approach involves utilizing mathematical expressions for the retrieval of scientific documents [1; 2]. Due to the diversity and inherent semantic ambiguity within mathematical expressions, leveraging mathematical information for a range of research tasks--such as mathematical information retrieval and comprehension of mathematical formulas--continues to present a formidable challenge. The distinctive two-dimensional structure adds complexity to their encoding, as retrieval problems encompass not just syntactic term matching but also entail semantic reasoning. To attain a higher degree of precision in matching mathematical expressions during retrieval, it is imperative to carefully contemplate the use of appropriate data structures [3]. Sentence embeddings obtained through training provide dense vector representations. State-of-the-art sentence modeling methods are predominantly based on pretrained language models and find widespread application in various real-world scenarios, such assentence text similarity and text retrieval. Furthermore, sentence embedding methods based on pretrained language models tend to favor larger model architectures and larger training datasets to achieve enhanced performance [4; 5]. Although effective, such large models cause predicaments in terms of inference, as they are computationally expensive and time-consuming to deploy with limited computational resources. They often take on the role of 'teachers', transferring their high capabilities to smaller'student' models through knowledge distillation (KD) [6]. In summary, mathematical expressions constitute a semi-formal visual language for conveying meaning. Identifying mathematical expressions presents several challenges in the contemporary landscape. For instance, various representation methods exist for mathematical expressions and pinpointing spatial correlations can be intricate, impeding users' ability to execute uniform queries. Regarding the fusion of mathematical expressions and their contextual semantics, current approaches rely on large-scale models to obtain their embeddings, leading to substantial computational and inference expenses. In light of the aforementioned challenges, this proposes a scientific document retrieval and sorting model based on HFS (hesitant fuzzy sets) [7] and LSD (local semantic distillation). In this approach, the document undergoes preprocessing by the document preprocessor module, from which we extract formulas in Presentation MathML or latex format along with their context. Subsequently, mathematical expressions are broken down into their constituent attributes and the corresponding evaluation functions are established. After that, knowledge distillation is implemented on pretrained language models, employing the distilled model to compute contextual semantic similarity. Ultimately, this semantic similarity is harnessed to enhance the retrieval outcomes of mathematical expressions, culminating in the ultimate ranking of scientific documents. This method capitalizes on the strengths of HFS in fuzzy-decision-supporting and the benefits of KD in compressing large models while preserving their performance, thereby elevating the efficacy of scientific document retrieval systems.