Figure S2: Number of iterations at each grid point for the Newton and gradient descent methods applying to the ℓ2-regularized logistic regression over simulated data generated in Example 2. We summarize the results in Figure S1-S3. Figure S1 presents the results for ridge regression. In this case, the number of iterations by gradient method first increases and then stays flat as tk grows. Newton method, however, only takes one 1.51.5 iteration at each grid point. Moreover, the level of approximation (i.e., ϵ) seems to have no impact onthe number of iterations at each grid point, which is highly desirable.

Detecting 3D keypoints with semantic consistency is widely used in many scenarios such as pose estimation, shape registration and robotics. Currently, most unsupervised 3D keypoint detection methods focus on the rigid-body objects. However, when faced with deformable objects, the keypoints they identify do not preserve semantic consistency well. In this paper, we introduce an innovative unsupervised keypoint detector Key-Grid for both the rigid-body and deformable objects, which is an autoencoder framework. The encoder predicts keypoints and the decoder utilizes the generated keypoints to reconstruct the objects. Unlike previous work, we leverage the identified keypoint in formation to form a 3D grid feature heatmap called grid heatmap, which is used in the decoder section.

We study the problem of change-point detection and localisation for functional data sequentially observed on a generald-dimensional space, where we allow thefunctional curvestobeeither sparsely ordensely sampled.

We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which areordinary differential equations subject toboundary conditions.