  This paper develops a novel Bayesian approach for nonlinear regression with
symmetric matrix predictors, often used to encode connectivity of different
nodes. Unlike methods that vectorize matrices as predictors that result in a
large number of model parameters and unstable estimation, we propose a Bayesian
multi-index regression method, resulting in a projection-pursuit-type estimator
that leverages the structure of matrix-valued predictors. We establish the
model identifiability conditions and impose a sparsity-inducing prior on the
projection directions for sparse sampling to prevent overfitting and enhance
interpretability of the parameter estimates. Posterior inference is conducted
through Bayesian backfitting. The performance of the proposed method is
evaluated through simulation studies and a case study investigating the
relationship between brain connectivity features and cognitive scores.
