Kernel dimensionality reduction (KDR) algorithms find a low dimensional representation of the original data by optimizing kernel dependency measures that are capable of capturing nonlinear relationships. The standard strategy is to first map the data into a high dimensional feature space using kernels prior to a projection onto a low dimensional space. While KDR methods can be easily solved by keeping the most dominant eigenvectors of the kernel matrix, its features are no longer easy to interpret. Alternatively, Interpretable KDR (IKDR) is different in that it projects onto a subspace \textit{before} the kernel feature mapping, therefore, the projection matrix can indicate how the original features linearly combine to form the new features. Unfortunately, the IKDR objective requires a non-convex manifold optimization that is difficult to solve and can no longer be solved by eigendecomposition. Recently, an efficient iterative spectral (eigendecomposition) method (ISM) has been proposed for this objective in the context of alternative clustering. However, ISM only provides theoretical guarantees for the Gaussian kernel. This greatly constrains ISM's usage since any kernel method using ISM is now limited to a single kernel. This work extends the theoretical guarantees of ISM to an entire family of kernels, thereby empowering ISM to solve any kernel method of the same objective. In identifying this family, we prove that each kernel within the family has a surrogate $\Phi$ matrix and the optimal projection is formed by its most dominant eigenvectors. With this extension, we establish how a wide range of IKDR applications across different learning paradigms can be solved by ISM. To support reproducible results, the source code is made publicly available on \url{https://github.com/ANONYMIZED}.

Methods and experimental results are presented for interpreting 3D spatial language descriptions used for human to robot communication in a fetch task. The work is based on human subject experiments in which spatial language descriptions were logged from younger and older adult participants. A spatial language model is proposed, and methods are presented for translating natural spatial language descriptions into robot commands that allow the robot to find the requested object. Robot command representation and robot behavior are also discussed. Experimental results compare path metrics of the robot system and human subjects in a common simulation environment. The overall success rate of the robot trials is 85%.

This paper outlines a new study that investigates spatial language for use in human-robot communication. The scenario studied is a home setting in which the elderly resident has misplaced an object, such as eyeglasses, and the robot will help the resident find the object. We present results from phase I of the study in which we investigate spatial language generated to a human addressee or a robot addressee in a virtual environment.