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Bayesian Multi-scale Modeling of Factor Matrix without using Partition Tree

The multi-scale factor models are particularly appealing for analyzing matrix- or tensor-valued data, due to their adaptiveness to local geometry and intuitive interpretation. However, the reliance on the binary tree for recursive partitioning creates high complexity in the parameter space, making it extremely challenging to quantify its uncertainty. In this article, we discover an alternative way to generate multi-scale matrix using simple matrix operation: starting from a random matrix with each column having two unique values, its Cholesky whitening transform obeys a recursive partitioning structure. This allows us to consider a generative distribution with large prior support on common multi-scale factor models, and efficient posterior computation via Hamiltonian Monte Carlo. We demonstrate its potential in a multi-scale factor model to find broader regions of interest for human brain connectivity.