Recently developed applications in the field of machine learning and computational physics rely on automatic differentiation techniques, that require stable and efficient linear algebra gradient computations. This technical note provides a comprehensive and detailed discussion of the derivative of the truncated singular and eigenvalue decomposition. It summarizes previous work and builds on them with an extensive description of how to derive the relevant terms. A main focus is correctly expressing the derivative in terms of the truncated part, despite lacking knowledge of the full decomposition.

Interactive perception enables robots to manipulate the environment and objects to bring them into states that benefit the perception process. Deformable objects pose challenges to this due to significant manipulation difficulty and occlusion in vision-based perception. In this work, we address such a problem with a setup involving both an active camera and an object manipulator. Our approach is based on a sequential decision-making framework and explicitly considers the motion regularity and structure in coupling the camera and manipulator. We contribute a method for constructing and computing a subspace, called Dynamic Active Vision Space (DAVS), for effectively utilizing the regularity in motion exploration. The effectiveness of the framework and approach are validated in both a simulation and a real dual-arm robot setup. Our results confirm the necessity of an active camera and coordinative motion in interactive perception for deformable objects.