Numerical Reasoning on Knowledge Graphs with Convolutional Histograms

Numerical complex query answering methods often encode scalar values as latent vectors or learned density parameters. Although expressive, these representations do not preserve numerical order, distance, and uncertainty together in an explicit numerical space. Their intermediate numerical states can therefore be difficult to inspect in the original units. We present CHOREOGRAPH, a framework that represents each observed scalar as a localized histogram on an ordered grid defined separately for each numerical attribute. Attribute-wise quantile binning allocates finer resolution where training literals are dense. Relation projection kernels (RPKs) use one-dimensional convolution to model how numerical mass changes across relational edges and can be fitted efficiently. Distributional message passing then estimates unobserved numerical slots. CHOREOGRAPH composes numerical operations using histogram-valued intermediate states. This discrete representation supports comparison, set, and arithmetic operations without operator-specific learning. Each intermediate numerical state can therefore be traced to intervals in the original units. We evaluate CHOREOGRAPH on FB15K-237, FB15K, DB15K, and YAGO15K using benchmark queries from prior work. Across all four datasets, CHOREOGRAPH outperforms the evaluated baselines on most query structures. On FB15K, all RPKs are fitted in 9.4 seconds, and total training is $3.3\times$–$22.4\times$ faster than recent baselines.