  Deep kernel processes are a recently introduced class of deep Bayesian models
that have the flexibility of neural networks, but work entirely with Gram
matrices. They operate by alternately sampling a Gram matrix from a
distribution over positive semi-definite matrices, and applying a deterministic
transformation. When the distribution is chosen to be Wishart, the model is
called a deep Wishart process (DWP). This particular model is of interest
because its prior is equivalent to a deep Gaussian process (DGP) prior, but at
the same time it is invariant to rotational symmetries, leading to a simpler
posterior distribution. Practical inference in the DWP was made possible in
recent work ("A variational approximate posterior for the deep Wishart process"
Ober and Aitchison 2021a) where the authors used a generalisation of the
Bartlett decomposition of the Wishart distribution as the variational
approximate posterior. However, predictive performance in that paper was less
impressive than one might expect, with the DWP only beating a DGP on a few of
the UCI datasets used for comparison. In this paper, we show that further
generalising their distribution to allow linear combinations of rows and
columns in the Bartlett decomposition results in better predictive performance,
while incurring negligible additional computation cost.
