Sparse variational Gaussian processes (SVGPs) are a widely used, principled approach to scalable Gaussian process modelling. Their success stems from their theoretically interpretable structure: they provide a lower bound on the marginal likelihood of the full GP while allowing scal- able stochastic optimization. Recent work has explored modifications of the posterior conditional distribution and has shown that such extensions can improve upon this well-established framework. In particular, introducing a block-diagonal structured scal- ing matrix leads to a tighter lower bound on the GP marginal likelihood. We investigate whether selecting blocks that tighten the variational objective improves test RMSE and NLL at a fixed inducing-point budget. Since the scaling matrix contributes to the variational lower bound through an upper bound on the log-determinant of the full residual covariance matrix, our work makes the following contributions. First, we introduce a simple, scalable, geometry-aware algorithm, with explicit computational complexity and low computational overhead, that recovers a large fraction of this log- determinant contribution by selecting an appropriate partition of the data. Second, we show how to train these models using a doubly stochastic optimization framework. Finally, we investigate how sparse GP methods based on alternative conditional posterior structures can be used to learn better hyperparameters and inducing points for SVGP.

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