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Graph Transformers (GTs) have significantly advanced the field of graph representation learning by overcoming the limitations of message-passing graph neural networks (GNNs) and demonstrating promising performance and expressive power. However, the quadratic complexity of self-attention mechanism in GTs has limited their scalability, and previous approaches to address this issue often suffer from expressiveness degradation or lack of versatility. To address this issue, we propose AnchorGT, a novel attention architecture for GTs with global receptive field and almost linear complexity, which serves as a flexible building block to improve the scalability of a wide range of GT models. Inspired by anchor-based GNNs, we employ structurally important $k$-dominating node set as anchors and design an attention mechanism that focuses on the relationship between individual nodes and anchors, while retaining the global receptive field for all nodes. With its intuitive design, AnchorGT can easily replace the attention module in various GT models with different network architectures and structural encodings, resulting in reduced computational overhead without sacrificing performance. In addition, we theoretically prove that AnchorGT attention can be strictly more expressive than Weisfeiler-Lehman test, showing its superiority in representing graph structures. Our experiments on three state-of-the-art GT models demonstrate that their AnchorGT variants can achieve better results while being faster and significantly more memory efficient.

This study explores the concept of functional equivalence within the framework of the Non-Axiomatic Reasoning System (NARS), specifically through OpenNARS for Applications (ONA). Functional equivalence allows organisms to categorize and respond to varied stimuli based on their utility rather than perceptual similarity, thus enhancing cognitive efficiency and adaptability. In this study, ONA was modified to allow the derivation of functional equivalence. This paper provides practical examples of the capability of ONA to apply learned knowledge across different functional situations, demonstrating its utility in complex problem-solving and decision-making. An extended example is included, where training of ONA aimed to learn basic human-like language abilities, using a systematic procedure in relating spoken words, objects and written words. The research carried out as part of this study extends the understanding of functional equivalence in AGI systems, and argues for its necessity for level of flexibility in learning and adapting necessary for human-level AGI.

We present a new algorithm for imitation learning in infinite horizon linear MDPs dubbed ILARL which greatly improves the bound on the number of trajectories that the learner needs to sample from the environment. In particular, we remove exploration assumptions required in previous works and we improve the dependence on the desired accuracy $\epsilon$ from $\mathcal{O}\br{\epsilon^{-5}}$ to $\mathcal{O}\br{\epsilon^{-4}}$. Our result relies on a connection between imitation learning and online learning in MDPs with adversarial losses. For the latter setting, we present the first result for infinite horizon linear MDP which may be of independent interest. Moreover, we are able to provide a strengthen result for the finite horizon case where we achieve $\mathcal{O}\br{\epsilon^{-2}}$. Numerical experiments with linear function approximation shows that ILARL outperforms other commonly used algorithms.

The current state-of-the-art theoretical analysis of Actor-Critic (AC) algorithms significantly lags in addressing the practical aspects of AC implementations. This crucial gap needs bridging to bring the analysis in line with practical implementations of AC. To address this, we advocate for considering the MMCLG criteria: \textbf{M}ulti-layer neural network parametrization for actor/critic, \textbf{M}arkovian sampling, \textbf{C}ontinuous state-action spaces, the performance of the \textbf{L}ast iterate, and \textbf{G}lobal optimality. These aspects are practically significant and have been largely overlooked in existing theoretical analyses of AC algorithms. In this work, we address these gaps by providing the first comprehensive theoretical analysis of AC algorithms that encompasses all five crucial practical aspects (covers MMCLG criteria). We establish global convergence sample complexity bounds of $\tilde{\mathcal{O}}\left({\epsilon^{-3}}\right)$. We achieve this result through our novel use of the weak gradient domination property of MDP's and our unique analysis of the error in critic estimation.

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

We generalize McDiarmid's inequality for functions with bounded differences on a high probability set, using an extension argument. Those functions concentrate around their conditional expectations. We further extend the results to concentration in general metric spaces.

This paper presents a novel latent 3D diffusion model for the generation of neural voxel fields, aiming to achieve accurate part-aware structures. Compared to existing methods, there are two key designs to ensure high-quality and accurate part-aware generation. On one hand, we introduce a latent 3D diffusion process for neural voxel fields, enabling generation at significantly higher resolutions that can accurately capture rich textural and geometric details. On the other hand, a part-aware shape decoder is introduced to integrate the part codes into the neural voxel fields, guiding the accurate part decomposition and producing high-quality rendering results. Through extensive experimentation and comparisons with state-of-the-art methods, we evaluate our approach across four different classes of data. The results demonstrate the superior generative capabilities of our proposed method in part-aware shape generation, outperforming existing state-of-the-art methods.

We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering the evolution of observables, Koopman operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis. While powerful, traditional Dynamic Mode Decomposition (DMD) techniques often struggle with continuous spectra. Rigged DMD addresses these challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions using snapshot data from the system's evolution. At its core, Rigged DMD builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition with high-order kernels for smoothing. This provides a robust decomposition encompassing both discrete and continuous spectral elements. We derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures. Additionally, we propose a novel framework for constructing rigged Hilbert spaces using time-delay embedding, significantly extending the algorithm's applicability. We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity, demonstrating Rigged DMD's convergence, efficiency, and versatility. This work paves the way for future research and applications of decompositions with continuous spectra.

State-of-the-art Convolutional Neural Network (CNN) benefits a lot from multi-task learning (MTL), which learns multiple related tasks simultaneously to obtain shared or mutually related representations for different tasks. The most widely-used MTL CNN structure is based on an empirical or heuristic split on a specific layer (e.g., the last convolutional layer) to minimize different task-specific losses. However, this heuristic sharing/splitting strategy may be harmful to the final performance of one or multiple tasks. In this paper, we propose a novel CNN structure for MTL, which enables automatic feature fusing at every layer. Specifically, we first concatenate features from different tasks according to their channel dimension, and then formulate the feature fusing problem as discriminative dimensionality reduction. We show that this discriminative dimensionality reduction can be done by 1x1 Convolution, Batch Normalization, and Weight Decay in one CNN, which we refer to as Neural Discriminative Dimensionality Reduction (NDDR). We perform ablation analysis in details for different configurations in training the network. The experiments carried out on different network structures and different task sets demonstrate the promising performance and desirable generalizability of our proposed method.