  3D Gaussians, as a low-level scene representation, typically involve
thousands to millions of Gaussians. This makes it difficult to control the
scene in ways that reflect the underlying dynamic structure, where the number
of independent entities is typically much smaller. In particular, it can be
challenging to animate and move objects in the scene, which requires
coordination among many Gaussians. To address this issue, we develop a mutual
information shaping technique that enforces movement resonance between
correlated Gaussians in a motion network. Such correlations can be learned from
putative 2D object masks in different views. By approximating the mutual
information with the Jacobians of the motions, our method ensures consistent
movements of the Gaussians composing different objects under various
perturbations. In particular, we develop an efficient contrastive training
pipeline with lightweight optimization to shape the motion network, avoiding
the need for re-shaping throughout the motion sequence. Notably, our training
only touches a small fraction of all Gaussians in the scene yet attains the
desired compositional behavior according to the underlying dynamic structure.
The proposed technique is evaluated on challenging scenes and demonstrates
significant performance improvement in promoting consistent movements and 3D
object segmentation while inducing low computation and memory requirements.
