We propose a data-driven Fourier-trained neural-network method for estimating fixed-horizon probability densities from empirical characteristic-function (CF) information. The estimator is a positive Gaussian--Laplace mixture with closed-form CF, so training can be performed directly in Fourier space while preserving nonnegativity and unit mass. We consider two sampling settings. In the direct i.i.d. sampling setting, the method is trained against an empirical CF constructed from i.i.d. samples. In the resampling-based pseudo-sampling setting, it is trained against an empirical pseudo-CF constructed from dependent data by resampling. For the direct i.i.d. case, we derive an expected $L_2$ error bound that separates Fourier truncation, empirical training error, discretization, and CF sampling error. For the pseudo-sampling case, we obtain a conditional analogue with two additional pseudo-law discrepancy terms. We develop a multidimensional extension of the framework and analyze its computational complexity. Numerical experiments show competitive performance relative to Expectation--Maximization on Gaussian-mixture benchmarks, clear gains on heavy-tailed targets, $L_2$ error decay consistent with the theory in a well-specified setting, and effective estimation of one-year Australian equity return law from resampled dependent data.

Despite enormous successful applications of graph neural networks (GNNs), theoretical understanding of their generalization ability, especially for node-level tasks where data are not independent and identically-distributed (IID), has been sparse. The theoretical investigation of the generalization performance is beneficial for understanding fundamental issues (such as fairness) of GNN models and designing better learning methods. In this paper, we present a novel PAC-Bayesian analysis for GNNs under a non-IID semi-supervised learning setup. Moreover, we analyze the generalization performances on different subgroups of unlabeled nodes, which allows us to further study an accuracy-(dis)parity-style (un)fairness of GNNs from a theoretical perspective. Under reasonable assumptions, we demonstrate that the distance between a test subgroup and the training set can be a key factor affecting the GNN performance on that subgroup, which calls special attention to the training node selection for fair learning. Experiments across multiple GNN models and datasets support our theoretical results4.

Deep neural networks are ubiquitously adopted in many applications, such as computer vision, natural language processing, and graph analytics. However, well-trained neural networks can make prediction errors after deployment as the world changes.

Both models consist of 2 layers and the hidden dimension is fixed to 64. We add a weight decay of 5e-4 for Cora, Citeseer, and Pubmed,and0fortherest. The optimizer configuration and the training schedule are the same as Section A.2. Kh(c ˆci) (7) where i N V denotes the evaluated node, andh is the bandwidth of the kernel function. The classwise-ECEs are summarized in Table 3, and the KDE-ECEs are collected in Table 4. Weadopt a heuristic which proportionally rescales the non top-1 output probabilities so that the calibrated probabilistic output sums up to one. While the ECEs ofCaGCN inits original paper are promising [23], we observethat the ECEs of CaGCN are often unstable and sometimes even worse than that of the uncalibrated model in our experiments.

Based on this analysis, our second contribution is the discovering of a type of unfairness that arises from theoretically predictable accuracy disparity across some subgroups of test nodes.

Abstract--Received signal strength indicator (RSSI) is the primary representation of Wi-Fi fingerprints and serves as a crucial tool for indoor localization. However, existing RSSI-based positioning methods often suffer from reduced accuracy due to environmental complexity and challenges in processing multi-source information. T o address these issues, we propose a novel multi-graph heterogeneous GNN framework (MG-HGNN) to enhance spatial awareness and improve positioning performance. In this framework, two graph construction branches perform node and edge embedding, respectively, to generate informative graphs. Subsequently, a heterogeneous graph neural network is employed for graph representation learning, enabling accurate positioning. The MG-HGNN framework introduces the following key innovations: 1) multi-type task-directed graph construction that combines label estimation and feature encoding for richer graph information; 2) a heterogeneous GNN structure that enhances the performance of conventional GNN models. Evaluations on the UJIIndoorLoc and UTSIndoorLoc public datasets demonstrate that MG-HGNN not only achieves superior performance compared to several state-of-the-art methods, but also provides a novel perspective for enhancing GNN-based localization methods. Ablation studies further confirm the rationality and effectiveness of the proposed framework. Index T erms--Fingerprint-based localization, graph neural network, heterogeneous network, received signal strength indicator (RSSI). NDOOR localization technologies aim to estimate the position of mobile users or devices in indoor environments where satellite-based systems such as GPS are ineffective [1]. Over the past decade, a variety of wireless indoor localization techniques have been developed based on different sensing modalities, including Bluetooth Low Energy (BLE) [2], Ultra Wideband (UWB) [3], Radio Frequency Identification (RFID) [4], magnetic field sensing [5], and Wi-Fi [6], [7]. Among them, Wi-Fi based localization has attracted a lot of attention due to the ubiquity of Wi-Fi infrastructure, low deployment cost, and compatibility with existing mobile devices without requiring additional hardware [1]. This work has been submitted to the IEEE for possible publication. This work is supported by the National Key Research and Development Program of China [Grant No. 2024QY1103], the Shandong Provincial Natural Science Foundation, China [Grant No. ZR2024QF138].(Corresponding Yibu Wang, Zhaoxin Zhang, Ning Li, and Tianzi Zhao are with the School of Computer Science and Technology, Harbin Institute of Technology, China (e-mail: 24b903081@stu.hit.edu.cn; Xinlong Zhao is with the China Mineral Resources Group Big Data Co., Ltd, China (e-mail: xinlong.zhao@qq.com).

Graph Neural Networks (GNNs) have demonstrated remarkable effectiveness on graph-based tasks. However, their predictive confidence is often miscalibrated, typically exhibiting under-confidence, which harms the reliability of their decisions. Existing calibration methods for GNNs normally introduce additional calibration components, which fail to capture the intrinsic relationship between the model and the prediction confidence, resulting in limited theoretical guarantees and increased computational overhead. To address this issue, we propose a simple yet efficient graph calibration method. We establish a unified theoretical framework revealing that model confidence is jointly governed by class-centroid-level and node-level calibration at the final layer. Based on this insight, we theoretically show that reducing the weight decay of the final-layer parameters alleviates GNN under-confidence by acting on the class-centroid level, while node-level calibration acts as a finer-grained complement to class-centroid level calibration, which encourages each test node to be closer to its predicted class centroid at the final-layer representations. Extensive experiments validate the superiority of our method.