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The existing research on spectral algorithms, applied within a Reproducing Kernel Hilbert Space (RKHS), has primarily focused on general kernel functions, often neglecting the inherent structure of the input feature space. Our paper introduces a new perspective, asserting that input data are situated within a low-dimensional manifold embedded in a higher-dimensional Euclidean space. We study the convergence performance of spectral algorithms in the RKHSs, specifically those generated by the heat kernels, known as diffusion spaces. Incorporating the manifold structure of the input, we employ integral operator techniques to derive tight convergence upper bounds concerning generalized norms, which indicates that the estimators converge to the target function in strong sense, entailing the simultaneous convergence of the function itself and its derivatives. These bounds offer two significant advantages: firstly, they are exclusively contingent on the intrinsic dimension of the input manifolds, thereby providing a more focused analysis. Secondly, they enable the efficient derivation of convergence rates for derivatives of any k-th order, all of which can be accomplished within the ambit of the same spectral algorithms. Furthermore, we establish minimax lower bounds to demonstrate the asymptotic optimality of these conclusions in specific contexts. Our study confirms that the spectral algorithms are practically significant in the broader context of high-dimensional approximation.

3D articulated objects are inherently challenging for manipulation due to the varied geometries and intricate functionalities associated with articulated objects.Point-level affordance, which predicts the per-point actionable score and thus proposes the best point to interact with, has demonstrated excellent performance and generalization capabilities in articulated object manipulation. However, a significant challenge remains: while previous works use perfect point cloud generated in simulation, the models cannot directly apply to the noisy point cloud in the real-world. To tackle this challenge, we leverage the property of real-world scanned point cloud that, the point cloud becomes less noisy when the camera is closer to the object. Therefore, we propose a novel coarse-to-fine affordance learning pipeline to mitigate the effect of point cloud noise in two stages. In the first stage, we learn the affordance on the noisy far point cloud which includes the whole object to propose the approximated place to manipulate. Then, we move the camera in front of the approximated place, scan a less noisy point cloud containing precise local geometries for manipulation, and learn affordance on such point cloud to propose fine-grained final actions. The proposed method is thoroughly evaluated both using large-scale simulated noisy point clouds mimicking real-world scans, and in the real world scenarios, with superiority over existing methods, demonstrating the effectiveness in tackling the noisy real-world point cloud problem.

We discuss the inhomogeneous spiked Wigner model, a theoretical framework recently introduced to study structured noise in various learning scenarios, through the prism of random matrix theory, with a specific focus on its spectral properties. Our primary objective is to find an optimal spectral method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion -- well-known in the homogeneous case -- to our inhomogeneous, block-structured, Wigner model. We provide a thorough rigorous analysis of a transformed matrix and show that the transition for the appearance of 1) an outlier outside the bulk of the limiting spectral distribution and 2) a positive overlap between the associated eigenvector and the signal, occurs precisely at the optimal threshold, making the proposed spectral method optimal within the class of iterative methods for the inhomogeneous Wigner problem.

In narrow spaces, motion planning based on the traditional hierarchical autonomous system could cause collisions due to mapping, localization, and control noises, especially for car-like Ackermann-steering robots which suffer from non-convex and non-holonomic kinematics. To tackle these problems, we leverage deep reinforcement learning which is verified to be effective in self-decision-making, to self-explore in narrow spaces without a given map and destination while avoiding collisions. Specifically, based on our Ackermann-steering rectangular-shaped ZebraT robot and its Gazebo simulator, we propose the rectangular safety region to represent states and detect collisions for rectangular-shaped robots, and a carefully crafted reward function for reinforcement learning that does not require the waypoint guidance. For validation, the robot was first trained in a simulated narrow track. Then, the well-trained model was transferred to other simulation tracks and could outperform other traditional methods including classical and learning methods. Finally, the trained model is demonstrated in the real world with our ZebraT robot.

This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation. We provide bounds for the generalization and training error in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers. Our main result is a bound of the total error in the $H^1([0,T];L^2(\Omega))$-norm in terms of the training error and the number of training points, which can be made arbitrarily small under some assumptions. We illustrate our theoretical bounds with numerical experiments.

The performance of machine learning classification algorithms are evaluated by estimating metrics, often from the confusion matrix, using training data and cross-validation. However, these do not prove that the best possible performance has been achieved. Fundamental limits to error rates can be estimated using information distance measures. To this end, the confusion matrix has been formulated to comply with the Chernoff-Stein Lemma. This links the error rates to the Kullback-Leibler divergences between the probability density functions describing the two classes. This leads to a key result that relates Cohen's Kappa to the Resistor Average Distance which is the parallel resistor combination of the two Kullback-Leibler divergences. The Resistor Average Distance has units of bits and is estimated from the same training data used by the classification algorithm, using kNN estimates of the KullBack-Leibler divergences. The classification algorithm gives the confusion matrix and Kappa. Theory and methods are discussed in detail and then applied to Monte Carlo data and real datasets. Four very different real datasets - Breast Cancer, Coronary Heart Disease, Bankruptcy, and Particle Identification - are analysed, with both continuous and discrete values, and their classification performance compared to the expected theoretical limit. In all cases this analysis shows that the algorithms could not have performed any better due to the underlying probability density functions for the two classes. Important lessons are learnt on how to predict the performance of algorithms for imbalanced data using training datasets that are approximately balanced. Machine learning is very powerful but classification performance ultimately depends on the quality of the data and the relevance of the variables to the problem.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

Heterogeneous graph neural networks (HGNNs) as an emerging technique have shown superior capacity of dealing with heterogeneous information network (HIN). However, most HGNNs follow a semi-supervised learning manner, which notably limits their wide use in reality since labels are usually scarce in real applications. Recently, contrastive learning, a self-supervised method, becomes one of the most exciting learning paradigms and shows great potential when there are no labels. In this paper, we study the problem of self-supervised HGNNs and propose a novel co-contrastive learning mechanism for HGNNs, named HeCo. Different from traditional contrastive learning which only focuses on contrasting positive and negative samples, HeCo employs cross-viewcontrastive mechanism. Specifically, two views of a HIN (network schema and meta-path views) are proposed to learn node embeddings, so as to capture both of local and high-order structures simultaneously. Then the cross-view contrastive learning, as well as a view mask mechanism, is proposed, which is able to extract the positive and negative embeddings from two views. This enables the two views to collaboratively supervise each other and finally learn high-level node embeddings. Moreover, two extensions of HeCo are designed to generate harder negative samples with high quality, which further boosts the performance of HeCo. Extensive experiments conducted on a variety of real-world networks show the superior performance of the proposed methods over the state-of-the-arts.

Due to the significance and value in human-computer interaction and natural language processing, task-oriented dialog systems are attracting more and more attention in both academic and industrial communities. In this paper, we survey recent advances and challenges in an issue-specific manner. We discuss three critical topics for task-oriented dialog systems: (1) improving data efficiency to facilitate dialog system modeling in low-resource settings, (2) modeling multi-turn dynamics for dialog policy learning to achieve better task-completion performance, and (3) integrating domain ontology knowledge into the dialog model in both pipeline and end-to-end models. We also review the recent progresses in dialog evaluation and some widely-used corpora. We believe that this survey can shed a light on future research in task-oriented dialog systems.

Due to their inherent capability in semantic alignment of aspects and their context words, attention mechanism and Convolutional Neural Networks (CNNs) are widely applied for aspect-based sentiment classification. However, these models lack a mechanism to account for relevant syntactical constraints and long-range word dependencies, and hence may mistakenly recognize syntactically irrelevant contextual words as clues for judging aspect sentiment. To tackle this problem, we propose to build a Graph Convolutional Network (GCN) over the dependency tree of a sentence to exploit syntactical information and word dependencies. Based on it, a novel aspect-specific sentiment classification framework is raised. Experiments on three benchmarking collections illustrate that our proposed model has comparable effectiveness to a range of state-of-the-art models, and further demonstrate that both syntactical information and long-range word dependencies are properly captured by the graph convolution structure.