The Scale–Shape Effect on Sliced-Gaussian Regularizer-based Latent Prediction Methods

Self-supervised regularizers like SIGReg match embedding distributions to isotropic Gaussians by independently penalizing deviations in scale and geometric configuration. However, the substitution rate between these regularization penalties, defined as the scale–shape curvature ($\tau$), is fixed exclusively by the dimensionality of the latent space (called width here). Neither of the theoretical regularizer-objective's kernel (over infinite data and random slice directions), the bandwidth, the slice count, nor the slice dimension can alter this property, and no stationary kernel improves upon the conditioning achieved by the standard baseline. In this work, we propose $\eta$, a single weight that rescales the shape component of the objective’s kernel, and with it $\tau$, without adding a new penalty term. Consequently, the minimizer remains invariant, the kernel preserves positive definiteness for all $\eta >0$, and the configuration $\eta =1$ exactly recovers the standard objective (for e.g. SIGReg). Every $\eta$ result reported here is obtained on the analytic infinite-slice objective, the closed-form $M\to \infty$ limit of SIGReg; $\eta$ has not been applied to the stochastic $M$-slice estimator itself. Modulating $\eta$ adjusts the rank of the regularized layer and we show empirical results across two modalities, time-series and image data. This rank modulation is subject to two explicit operational assumptions: sufficient headroom below the rank ceiling, and a non-monotonic layer response, such as an un-regularized backbone preceding a projection head where the regularizer is applied. In downstream evaluations, $\eta$ shifts performance bi-directionally depending on the specific task and readout mechanism. We therefore present this parameterized axis of variation rather than prescribing an optimal hyperparameter configuration.