Stochastic complexity of vectors containing cluster structure

This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and practical importance in data clustering based on Minimum Description Length (MDL) principle, such as for estimating the best number of clusters and best cluster structure for the data. Straightforward computation of the shortest code length of the vector containing cluster structure based on the NML model requires polynomial time with respect to the size of the vector and number of clusters. We show that this is a tractable problem by introducing a recursion formula for the efficient computation of normalizing constant from the NML model. The time complexity of the new formula is linear opposed to previous polynomial time with respect to the size of the vector and number of clusters.

FedSLIM: Privacy-Preserving Federated MDL-Based Descriptive Pattern Mining Across Data Silos

Federated learning has achieved considerable success for predictive modelling, yet federated descriptive analytics remains largely unexplored. Existing federated pattern mining approaches are predominantly support-based and do not optimise a principled global objective such as Minimum Description Length (MDL). We introduce FedSLIM, the first federated MDL-based framework for descriptive pattern mining. Building on the SLIM principle, FedSLIM enables collaborative optimisation of compact pattern models across distributed databases without sharing raw transactions. We propose two complementary variants that balance privacy, communication, and optimisation fidelity under different deployment assumptions. To evaluate federated MDL mining, we introduce fidelity and discovery-oriented metrics that quantify agreement with a centralised baseline and assess recovery of globally informative patterns. Experiments on multiple real-world datasets under IID and non-IID partitioning show that both variants preserve high-quality compression structure while requiring orders of magnitude less search than the centralised baseline. We further reveal a local-global discovery gap in distributed MDL mining, where globally compressive patterns may be undiscoverable through isolated local optimisation. Both variants recover globally informative patterns absent from all standalone local models, demonstrating the benefits of federated optimisation beyond independent local mining. These results establish federated MDL mining as a practical foundation for privacy-preserving descriptive analytics across distributed data silos.