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In this paper we present a general theory of $\Pi_{2}$-rules for systems of intuitionistic and modal logic. We introduce the notions of $\Pi_{2}$-rule system and of an Inductive Class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of G\"{o}del algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many $\Pi_{2}$-rule systems extending $\mathsf{LC}=\mathsf{IPC}+(p\rightarrow q)\vee (q\rightarrow p)$, and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in $\mathsf{LC}$: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all $\Pi_{2}$-rules which are admissible are derivable, and (2) show that the problem of admissibility of $\Pi_{2}$-rules over $\mathsf{LC}$ is decidable.

Dedukti is a Logical Framework based on the $\lambda$$\Pi$-Calculus Modulo Theory. We show that many theories can be expressed in Dedukti: constructive and classical predicate logic, Simple type theory, programming languages, Pure type systems, the Calculus of inductive constructions with universes, etc. and that permits to used it to check large libraries of proofs developed in other proof systems: Zenon, iProver, FoCaLiZe, HOL Light, and Matita.

Since being proposed, Neural Radiance Fields (NeRF) have achieved great success in related tasks, mainly adopting the hierarchical volume sampling (HVS) strategy for volume rendering. However, the HVS of NeRF approximates distributions using piecewise constant functions, which provides a relatively rough estimation. Based on the observation that a well-trained weight function $w(t)$ and the $L_0$ distance between points and the surface have very high similarity, we propose $L_0$-Sampler by incorporating the $L_0$ model into $w(t)$ to guide the sampling process. Specifically, we propose to use piecewise exponential functions rather than piecewise constant functions for interpolation, which can not only approximate quasi-$L_0$ weight distributions along rays quite well but also can be easily implemented with few lines of code without additional computational burden. Stable performance improvements can be achieved by applying $L_0$-Sampler to NeRF and its related tasks like 3D reconstruction. Code is available at //ustc3dv.github.io/L0-Sampler/ .

Integrating different functionalities, conventionally implemented as dedicated systems, into a single platform allows utilising the available resources more efficiently. We consider an integrated sensing and power transfer (ISAPT) system and propose the joint optimisation of the rectangular pulse-shaped transmit signal and the beamforming vector to combine sensing and wireless power transfer (WPT) functionalities efficiently. In contrast to prior works, we adopt an accurate non-linear circuit-based energy harvesting (EH) model. We formulate and solve a non-convex optimisation problem for a general number of EH receivers to maximise a weighted sum of the average harvested powers at the EH receivers while ensuring the received echo signal reflected by a sensing target (ST) has sufficient power for estimating the range to the ST with a prescribed accuracy within the considered coverage region. The average harvested power is shown to monotonically increase with the pulse duration when the average transmit power budget is sufficiently large. We discuss the trade-off between sensing performance and power transfer for the considered ISAPT system. The proposed approach significantly outperforms a heuristic baseline scheme based on a linear EH model, which linearly combines energy beamforming with the beamsteering vector in the direction to the ST as its transmit strategy.

We investigate $L_2$ boosting in the context of kernel regression. Kernel smoothers, in general, lack appealing traits like symmetry and positive definiteness, which are critical not only for understanding theoretical aspects but also for achieving good practical performance. We consider a projection-based smoother (Huang and Chen, 2008) that is symmetric, positive definite, and shrinking. Theoretical results based on the orthonormal decomposition of the smoother reveal additional insights into the boosting algorithm. In our asymptotic framework, we may replace the full-rank smoother with a low-rank approximation. We demonstrate that the smoother's low-rank ($d(n)$) is bounded above by $O(h^{-1})$, where $h$ is the bandwidth. Our numerical findings show that, in terms of prediction accuracy, low-rank smoothers may outperform full-rank smoothers. Furthermore, we show that the boosting estimator with low-rank smoother achieves the optimal convergence rate. Finally, to improve the performance of the boosting algorithm in the presence of outliers, we propose a novel robustified boosting algorithm which can be used with any smoother discussed in the study. We investigate the numerical performance of the proposed approaches using simulations and a real-world case.

This paper proposes a new notion of Markov $\alpha$-potential games to study Markov games. Two important classes of practically significant Markov games, Markov congestion games and the perturbed Markov team games, are analyzed in this framework of Markov $\alpha$-potential games, with explicit characterization of the upper bound for $\alpha$ and its relation to game parameters. Moreover, any maximizer of the $\alpha$-potential function is shown to be an $\alpha$-stationary Nash equilibrium of the game. Furthermore, two algorithms for the Nash regret analysis, namely the projected gradient-ascent algorithm and the sequential maximum improvement algorithm, are presented and corroborated by numerical experiments.

Public large-scale text-to-image diffusion models, such as Stable Diffusion, have gained significant attention from the community. These models can be easily customized for new concepts using low-rank adaptations (LoRAs). However, the utilization of multiple concept LoRAs to jointly support multiple customized concepts presents a challenge. We refer to this scenario as decentralized multi-concept customization, which involves single-client concept tuning and center-node concept fusion. In this paper, we propose a new framework called Mix-of-Show that addresses the challenges of decentralized multi-concept customization, including concept conflicts resulting from existing single-client LoRA tuning and identity loss during model fusion. Mix-of-Show adopts an embedding-decomposed LoRA (ED-LoRA) for single-client tuning and gradient fusion for the center node to preserve the in-domain essence of single concepts and support theoretically limitless concept fusion. Additionally, we introduce regionally controllable sampling, which extends spatially controllable sampling (e.g., ControlNet and T2I-Adaptor) to address attribute binding and missing object problems in multi-concept sampling. Extensive experiments demonstrate that Mix-of-Show is capable of composing multiple customized concepts with high fidelity, including characters, objects, and scenes.

Feature selection (FS) plays an important role in machine learning, which extracts important features and accelerates the learning process. In this paper, we propose a deep FS method that simultaneously conducts feature selection and differentiable $ k $-NN graph learning based on the Dirichlet Energy. The Dirichlet Energy identifies important features by measuring their smoothness on the graph structure, and facilitates the learning of a new graph that reflects the inherent structure in new feature subspace. We employ Optimal Transport theory to address the non-differentiability issue of learning $ k $-NN graphs in neural networks, which theoretically makes our method applicable to other graph neural networks for dynamic graph learning. Furthermore, the proposed framework is interpretable, since all modules are designed algorithmically. We validate the effectiveness of our model with extensive experiments on both synthetic and real-world datasets.

Temporal data, notably time series and spatio-temporal data, are prevalent in real-world applications. They capture dynamic system measurements and are produced in vast quantities by both physical and virtual sensors. Analyzing these data types is vital to harnessing the rich information they encompass and thus benefits a wide range of downstream tasks. Recent advances in large language and other foundational models have spurred increased use of these models in time series and spatio-temporal data mining. Such methodologies not only enable enhanced pattern recognition and reasoning across diverse domains but also lay the groundwork for artificial general intelligence capable of comprehending and processing common temporal data. In this survey, we offer a comprehensive and up-to-date review of large models tailored (or adapted) for time series and spatio-temporal data, spanning four key facets: data types, model categories, model scopes, and application areas/tasks. Our objective is to equip practitioners with the knowledge to develop applications and further research in this underexplored domain. We primarily categorize the existing literature into two major clusters: large models for time series analysis (LM4TS) and spatio-temporal data mining (LM4STD). On this basis, we further classify research based on model scopes (i.e., general vs. domain-specific) and application areas/tasks. We also provide a comprehensive collection of pertinent resources, including datasets, model assets, and useful tools, categorized by mainstream applications. This survey coalesces the latest strides in large model-centric research on time series and spatio-temporal data, underscoring the solid foundations, current advances, practical applications, abundant resources, and future research opportunities.

Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at //github.com/Shen-Lab/L2-GCN.