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Some hyperbolic systems are known to include implicit preservation of differential constraints: these are for example the time conservation of the curl or the divergence of a vector that appear as an implicit constraint. In this article, we show that this kind of constraint can be easily conserved at the discrete level with the classical discontinuous Galerkin method, provided the right approximation space is used for the vectorial space, and under some mild assumption on the numerical flux. For this, we develop a discrete differential geometry framework for some well chosen piece-wise polynomial vector approximation space. More precisely, we define the discrete Hodge star operator, the exterior derivative, and their adjoints. The discrete adjoint divergence and curl are proven to be exactly preserved by the discontinuous Galerkin method under a small assumption on the numerical flux. Numerical tests are performed on the wave system, the two dimensional Maxwell system and the induction equation, and confirm that the differential constraints are preserved at machine precision while keeping the high order of accuracy.

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

Operating effectively in complex environments while complying with specified constraints is crucial for the safe and successful deployment of robots that interact with and operate around people. In this work, we focus on generating long-horizon trajectories that adhere to novel static and temporally-extended constraints/instructions at test time. We propose a data-driven diffusion-based framework, LTLDoG, that modifies the inference steps of the reverse process given an instruction specified using finite linear temporal logic ($\text{LTL}_f$). LTLDoG leverages a satisfaction value function on $\text{LTL}_f$ and guides the sampling steps using its gradient field. This value function can also be trained to generalize to new instructions not observed during training, enabling flexible test-time adaptability. Experiments in robot navigation and manipulation illustrate that the method is able to generate trajectories that satisfy formulae that specify obstacle avoidance and visitation sequences.

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy).

Classical sequential recommendation models generally adopt ID embeddings to store knowledge learned from user historical behaviors and represent items. However, these unique IDs are challenging to be transferred to new domains. With the thriving of pre-trained language model (PLM), some pioneer works adopt PLM for pre-trained recommendation, where modality information (e.g., text) is considered universal across domains via PLM. Unfortunately, the behavioral information in ID embeddings is still verified to be dominating in PLM-based recommendation models compared to modality information and thus limits these models' performance. In this work, we propose a novel ID-centric recommendation pre-training paradigm (IDP), which directly transfers informative ID embeddings learned in pre-training domains to item representations in new domains. Specifically, in pre-training stage, besides the ID-based sequential model for recommendation, we also build a Cross-domain ID-matcher (CDIM) learned by both behavioral and modality information. In the tuning stage, modality information of new domain items is regarded as a cross-domain bridge built by CDIM. We first leverage the textual information of downstream domain items to retrieve behaviorally and semantically similar items from pre-training domains using CDIM. Next, these retrieved pre-trained ID embeddings, rather than certain textual embeddings, are directly adopted to generate downstream new items' embeddings. Through extensive experiments on real-world datasets, both in cold and warm settings, we demonstrate that our proposed model significantly outperforms all baselines. Codes will be released upon acceptance.

With the development of laws and regulations related to privacy preservation, it has become difficult to collect personal data to perform machine learning. In this context, federated learning, which is distributed learning without sharing personal data, has been proposed. In this paper, we focus on federated learning for user authentication. We show that it is difficult to achieve both privacy preservation and high accuracy with existing methods. To address these challenges, we propose IPFed which is privacy-preserving federated learning using random projection for class embedding. Furthermore, we prove that IPFed is capable of learning equivalent to the state-of-the-art method. Experiments on face image datasets show that IPFed can protect the privacy of personal data while maintaining the accuracy of the state-of-the-art method.

The exploration and understanding of Executable and Linkable Format (ELF) objects underpin various critical activities in computer systems, from debugging to reverse engineering. Traditional UNIX tooling like readelf, nm, and objdump have served the community reliably over the years. However, as the complexity and scale of software projects has grown, there arises a need for more intuitive, flexible, and powerful methods to investigate ELF objects. In this paper, we introduce sqlelf, an innovative tool that empowers users to probe ELF objects through the expressive power of SQL. By modeling ELF objects as relational databases, sqlelf offers the following advantages over conventional methods. Our evaluations demonstrate that sqlelf not only provides more nuanced and comprehensive insights into ELF objects but also significantly reduces the effort and time traditionally required for ELF exploration tasks

We introduce Score identity Distillation (SiD), an innovative data-free method that distills the generative capabilities of pretrained diffusion models into a single-step generator. SiD not only facilitates an exponentially fast reduction in Fr\'echet inception distance (FID) during distillation but also approaches or even exceeds the FID performance of the original teacher diffusion models. By reformulating forward diffusion processes as semi-implicit distributions, we leverage three score-related identities to create an innovative loss mechanism. This mechanism achieves rapid FID reduction by training the generator using its own synthesized images, eliminating the need for real data or reverse-diffusion-based generation, all accomplished within significantly shortened generation time. Upon evaluation across four benchmark datasets, the SiD algorithm demonstrates high iteration efficiency during distillation and surpasses competing distillation approaches, whether they are one-step or few-step, data-free, or dependent on training data, in terms of generation quality. This achievement not only redefines the benchmarks for efficiency and effectiveness in diffusion distillation but also in the broader field of diffusion-based generation. The PyTorch implementation is available at //github.com/mingyuanzhou/SiD

This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.

Artificial neural networks thrive in solving the classification problem for a particular rigid task, acquiring knowledge through generalized learning behaviour from a distinct training phase. The resulting network resembles a static entity of knowledge, with endeavours to extend this knowledge without targeting the original task resulting in a catastrophic forgetting. Continual learning shifts this paradigm towards networks that can continually accumulate knowledge over different tasks without the need to retrain from scratch. We focus on task incremental classification, where tasks arrive sequentially and are delineated by clear boundaries. Our main contributions concern 1) a taxonomy and extensive overview of the state-of-the-art, 2) a novel framework to continually determine the stability-plasticity trade-off of the continual learner, 3) a comprehensive experimental comparison of 11 state-of-the-art continual learning methods and 4 baselines. We empirically scrutinize method strengths and weaknesses on three benchmarks, considering Tiny Imagenet and large-scale unbalanced iNaturalist and a sequence of recognition datasets. We study the influence of model capacity, weight decay and dropout regularization, and the order in which the tasks are presented, and qualitatively compare methods in terms of required memory, computation time, and storage.