Visual reasoning is crucial for multimodal large language models (MLLMs) to address complex chart queries, yet high-quality rationale data remains scarce. Existing methods leveraged (M)LLMs for data generation, but direct prompting often yields limited precision and diversity. In this paper, we propose Chain of Functions (CoF), a novel programmatic reasoning data generation pipeline that utilizes freely-explored reasoning paths as supervision to ensure data precision and diversity. Specifically, it starts with human-free exploration among the atomic functions (e.g., maximum data and arithmetic operations) to generate diverse function chains, which are then translated into linguistic rationales and questions with only a moderate open-sourced LLM. CoF provides multiple benefits: 1) Precision: function-governed generation reduces hallucinations compared to freeform generation; 2) Diversity: enumerating function chains enables varied question taxonomies; 3) Explainability: function chains serve as built-in rationales, allowing fine-grained evaluation beyond overall accuracy; 4) Practicality: it eliminates reliance on extremely large models. Employing CoF, we construct the ChartCoF dataset, with 1.4k complex reasoning Q&A for fine-grained analysis and 50k Q&A for reasoning enhancement.Experiments show that ChartCoF improves performance for MLLMs on widely used benchmarks, and the fine-grained evaluation on ChartCoF reveals varying performance across question taxonomies and step numbers for each MLLM. Furthermore, the novel paradigm of function-governed rationale generation in CoF could inspire broader applications beyond charts. The code and data have been publicly available at https://github.com/microsoft/Chain-of-Functions.

Retrieval augmented generation has revolutionized large language model (LLM) outputs by providing factual supports. Nevertheless, it struggles to capture all the necessary knowledge for complex reasoning questions. Existing retrieval methods typically divide reference documents into passages, treating them in isolation. These passages, however, are often interrelated, such as passages that are contiguous or share the same keywords. Therefore, it is crucial to recognize such relatedness for enhancing the retrieval process. In this paper, we propose a novel retrieval method, called GNN-Ret, which leverages graph neural networks (GNNs) to enhance retrieval by exploiting the relatedness between passages. Specifically, we first construct a graph of passages by connecting passages that are structure-related or keyword-related. A graph neural network (GNN) is then leveraged to exploit the relationships between passages and improve the retrieval of supporting passages. Furthermore, we extend our method to handle multi-hop reasoning questions using a recurrent graph neural network (RGNN), named RGNN-Ret. At each step, RGNN-Ret integrates the graphs of passages from previous steps, thereby enhancing the retrieval of supporting passages. Extensive experiments on benchmark datasets demonstrate that GNN-Ret achieves higher accuracy for question answering with a single query of LLMs than strong baselines that require multiple queries, and RGNN-Ret further improves accuracy and achieves state-of-the-art performance, with up to 10.4 accuracy improvement on the 2WikiMQA dataset.