Message-Passing Algorithms for Quadratic Minimization

Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are known, but GaBP may fail to converge to the correct solution given an arbitrary positive definite quadratic function. As was observed in previous work, the GaBP algorithm fails to converge if the computation trees produced by the algorithm are not positive definite. In this work, we will show that the failure modes of the GaBP algorithm can be understood via graph covers, and we prove that a parameterized generalization of the min-sum algorithm can be used to ensure that the computation trees remain positive definite whenever the input matrix is positive definite. We demonstrate that the resulting algorithm is closely related to other iterative schemes for quadratic minimization such as the Gauss-Seidel and Jacobi algorithms. Finally, we observe, empirically, that there always exists a choice of parameters such that the above generalization of the GaBP algorithm converges

Applications of Metric Coinduction

Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One can prove a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infer by the coinduction principle that the property holds of the limit object. This can often be used to avoid complicated analytic arguments involving limits and convergence, replacing them with simpler algebraic arguments. This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets. These results point to the usefulness of coinduction as a general proof technique

Mean Shift Mask Transformer for Unseen Object Instance Segmentation

Segmenting unseen objects is a critical task in many different domains. For example, a robot may need to grasp an unseen object, which means it needs to visually separate this object from the background and/or other objects. Mean shift clustering is a common method in object segmentation tasks. However, the traditional mean shift clustering algorithm is not easily integrated into an end-to-end neural network training pipeline. In this work, we propose the Mean Shift Mask Transformer (MSMFormer), a new transformer architecture that simulates the von Mises-Fisher (vMF) mean shift clustering algorithm, allowing for the joint training and inference of both the feature extractor and the clustering. Its central component is a hypersphere attention mechanism, which updates object queries on a hypersphere. To illustrate the effectiveness of our method, we apply MSMFormer to Unseen Object Instance Segmentation, which yields a new state-of-the-art of 87.3 Boundary F-meansure on the real-world Object Clutter Indoor Dataset (OCID). Code is available at https://github.com/YoungSean/UnseenObjectsWithMeanShiftComment: 10 figure