Collision-free path planning in cluttered, real-world environments relies on a representation of the collision-free space, and existing representations broadly fall into two categories. Explicit representations, such as unions of convex sets, can be plugged into optimization-based planners as hard collision-free constraints, but their parameters scale poorly with configuration-space dimension. Implicit representations, by contrast, are flexible and scale well to complex geometries, yet typically lack such guarantees. We bridge this gap with ILD (Invertible Latent Decomposition), a framework that jointly learns an invertible mapping and a union of explicit convex polytopes in the resulting latent space. Planning is carried out over these latent convex sets, and the invertible mapping decodes the resulting paths back to the original configuration space while preserving feasibility with respect to the refined explicit safe regions. We further propose Visibility-Guided Sampling (VGS) to keep the convex sets connected for path planning. Across 2D navigation, 6-DoF, and 14-DoF manipulation environments, ILD achieves broader coverage, better inter-set connectivity, and higher path-planning success rates than prior baselines, with zero observed false positives after test-time refinement. On a 14-DoF bimanual manipulator, we further demonstrate real-time collision-free planning, with test-time refinement adapting to scene-geometry changes during real-world deployment on a single 6-DoF arm.