Jun Fan

Linhao Song, Jun Fan, Di-Rong Chen et al.

In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties are largely unknown beyond universality of approximation or the existing analysis does not apply to the rectified linear unit (ReLU) activation function. To fill in this void, we investigate here the approximation power of functional deep neural networks associated with the ReLU activation function by constructing a continuous piecewise linear interpolation under a simple triangulation. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions. Finally, our study may also shed some light on the understanding of functional data learning algorithms.

Yan Wu, Yang Yang, Jun Fan et al.

Radiated noise in unmanned underwater vehicles (UUVs) is an important indicator for characterizing acoustic signatures and evaluating platform performance. To address the strong dependence of traditional physics-based modeling and numerical simulation methods on target structural information and environmental boundary conditions, and their inability to achieve continuous spatial spectrum-response modeling in three-dimensional scenes, this paper proposes a neural radiated-noise field (NRNF). An NRNF represents the UUV radiated-noise spectrum as a continuous function of the three-dimensional UUV position, the three-dimensional hydrophone position, the UUV yaw angle, and the frequency, enabling query-based prediction at arbitrary spatial locations. The proposed method employs sinusoidal encoding for position and frequency, and introduces a learnable three-dimensional scene feature grid to explicitly represent environmental structure and propagation effects. A spectrum-prediction dataset is constructed from lake trials, and the proposed model is evaluated under three settings: horizontal extrapolation, depth extrapolation, and cross-run generalization. Results show that the NRNF achieves an average prediction error of 3.5 dB in the 50 to 5000 Hz band. Horizontal extrapolation is easiest, depth extrapolation is the most challenging, and cross-run generalization is of intermediate difficulty. Further ablation results demonstrate that the scene feature grid significantly improves the prediction stability and spatial generalization of the model.

Yunfei Yang, Jun Fan

This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate $\mathcal{O}((WL)^{-2\tilde{s}})$ for anisotropic Besov space $\mathcal{B}^{\boldsymbol{s}}_{q,r}([0,1]^d)$ with anisotropic smoothness $\boldsymbol{s}=(s_1,\dots,s_d)$ under the embedding condition $\tilde{s} > 1/q-1/p$, where the mean smoothness $\tilde{s} = (\sum_{i=1}^d s_i^{-1})^{-1}$. For mixed smooth Besov space $\mathcal{MB}^s_{q,r}([0,1]^d)$ with mixed smoothness $s>1/q-1/p$, we show that the approximation rate $\mathcal{O}((WL)^{-2s})$ holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

Yuanzhi Xu, Qian Gao, Jun Fan et al.

The generation of factually incorrect objects, commonly known as object hallucination, remains a persistent challenge in Large Vision-Language Models (LVLMs). Current approaches to address this issue - ranging from expensive data-driven fine-tuning and high-latency contrastive decoding to rigid attention head truncation - frequently compromise either computational efficiency or the continuity of the model's feature space. To overcome these limitations, we introduce a novel, training-free inference strategy that operates as a region-aware adaptive weighting mechanism to dynamically correct semantic drift without relying on abrupt heuristic truncations. By computing an outlier-resistant statistical midpoint across various attention heads, we establish a stable anchor for reliable visual representations. We then utilize the inter-head disagreement mapped across regions to dynamically determine intervention budgets, gently suppressing hallucination-inducing attention paths through a continuous penalty modulation. This recalibration process effectively rectifies visual-semantic misalignments while fully preserving generative fluency and language priors. Comprehensive evaluations on standard multimodal benchmarks, including CHAIR, POPE, and MME, reveal that our strategy substantially curtails both instance- and sentence-level hallucinations. The results demonstrate state-of-the-art performance against contemporary baselines, confirming our method's efficiency and algorithmic robustness. Our code will be public.

Zhiyu Liu, Zhi Han, Yandong Tang et al.

The problem of low-tubal-rank tensor estimation is a fundamental task with wide applications across high-dimensional signal processing, machine learning, and image science. Traditional approaches tackle such a problem by performing tensor singular value decomposition, which is computationally expensive and becomes infeasible for large-scale tensors. Recent approaches address this issue by factorizing the tensor into two smaller factor tensors and solving the resulting problem using gradient descent. However, this kind of approach requires an accurate estimate of the tensor rank, and when the rank is overestimated, the convergence of gradient descent and its variants slows down significantly or even diverges. To address this problem, we propose an Alternating Preconditioned Gradient Descent (APGD) algorithm, which accelerates convergence in the over-parameterized setting by adding a preconditioning term to the original gradient and updating these two factors alternately. Based on certain geometric assumptions on the objective function, we establish linear convergence guarantees for more general low-tubal-rank tensor estimation problems. Then we further analyze the specific cases of low-tubal-rank tensor factorization and low-tubal-rank tensor recovery. Our theoretical results show that APGD achieves linear convergence even under over-parameterization, and the convergence rate is independent of the tensor condition number. Extensive simulations on synthetic data are carried out to validate our theoretical assertions.

Jun Fan, Jingyu Yang, Xinyu Zhang et al.

The recovery of unknown signals from quadratic measurements finds extensive applications in fields such as phase retrieval, power system state estimation, and unlabeled distance geometry. This paper investigates the finite sample properties of weakly convex--concave regularized estimators in high-dimensional quadratic measurements models. By employing a weakly convex--concave penalized least squares approach, we establish support recovery and $\ell_2$-error bounds for the local minimizer. To solve the corresponding optimization problem, we adopt two proximal gradient strategies, where the proximal step is computed either in closed form or via a weighted $\ell_1$ approximation, depending on the regularization function. Numerical examples demonstrate the efficacy of the proposed method.

Yiqun Chen, Jiaxin Mao, Yi Zhang et al.

Search result diversification (SRD), which aims to ensure that documents in a ranking list cover a broad range of subtopics, is a significant and widely studied problem in Information Retrieval and Web Search. Existing methods primarily utilize a paradigm of "greedy selection", i.e., selecting one document with the highest diversity score at a time or optimize an approximation of the objective function. These approaches tend to be inefficient and are easily trapped in a suboptimal state. To address these challenges, we introduce Multi-Agent reinforcement learning (MARL) for search result DIVersity, which called MA4DIV. In this approach, each document is an agent and the search result diversification is modeled as a cooperative task among multiple agents. By modeling the SRD ranking problem as a cooperative MARL problem, this approach allows for directly optimizing the diversity metrics, such as $α$-NDCG, while achieving high training efficiency. We conducted experiments on public TREC datasets and a larger scale dataset in the industrial setting. The experiemnts show that MA4DIV achieves substantial improvements in both effectiveness and efficiency than existing baselines, especially on the industrial dataset. The code of MA4DIV can be seen on https://github.com/chenyiqun/MA4DIV.

Ziyang Zeng, Heming Jing, Jindong Chen et al.

Ranking relevance is a fundamental task in search engines, aiming to identify the items most relevant to a given user query. Traditional relevance models typically produce scalar scores or directly predict relevance labels, limiting both interpretability and the modeling of complex relevance signals. Inspired by recent advances in Chain-of-Thought (CoT) reasoning for complex tasks, we investigate whether explicit reasoning can enhance both interpretability and performance in relevance modeling. However, existing reasoning-based Generative Relevance Models (GRMs) primarily rely on supervised fine-tuning on large amounts of human-annotated or synthetic CoT data, which often leads to limited generalization. Moreover, domain-agnostic, free-form reasoning tends to be overly generic and insufficiently grounded, limiting its potential to handle the diverse and ambiguous cases prevalent in open-domain search. In this work, we formulate relevance modeling in Xiaohongshu search as a reasoning task and introduce a Reinforcement Learning (RL)-based training framework to enhance the grounded reasoning capabilities of GRMs. Specifically, we incorporate practical business-specific relevance criteria into the multi-step reasoning prompt design and propose Stepwise Advantage Masking (SAM), a lightweight process-supervision strategy which facilitates effective learning of these criteria through improved credit assignment. To enable industrial deployment, we further distill the large-scale RL-tuned model to a lightweight version suitable for real-world search systems. Extensive experiments on industrial datasets, along with online A/B tests, demonstrate the effectiveness of our approach.

Xiaotong Liu, Shao-Bo Lin, Jun Fan et al.

Synthetic data have gained increasing attention across various domains, with a growing emphasis on their performance in downstream prediction tasks. However, most existing synthesis strategies focus on maintaining statistical information. Although some studies address prediction performance guarantees, their single-stage synthesis designs make it challenging to balance the privacy requirements that necessitate significant perturbations and the prediction performance that is sensitive to such perturbations. We propose a two-stage synthesis strategy. In the first stage, we introduce a synthesis-then-hybrid strategy, which involves a synthesis operation to generate pure synthetic data, followed by a hybrid operation that fuses the synthetic data with the original data. In the second stage, we present a kernel ridge regression (KRR)-based synthesis strategy, where a KRR model is first trained on the original data and then used to generate synthetic outputs based on the synthetic inputs produced in the first stage. By leveraging the theoretical strengths of KRR and the covariant distribution retention achieved in the first stage, our proposed two-stage synthesis strategy enables a statistics-driven restricted privacy--prediction trade-off and guarantee optimal prediction performance. We validate our approach and demonstrate its characteristics of being statistics-driven and restricted in achieving the privacy--prediction trade-off both theoretically and numerically. Additionally, we showcase its generalizability through applications to a marketing problem and five real-world datasets.

Zhongjie Shi, Jun Fan, Linhao Song et al.

Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional learning tasks, particularly in nonlinear functional regression. In this paper, we introduce a functional deep neural network with an adaptive and discretization-invariant dimension reduction method. Our functional network architecture consists of three parts: first, a kernel embedding step that features an integral transformation with an adaptive smooth kernel; next, a projection step that utilizes eigenfunction bases based on a projection Mercer kernel for the dimension reduction; and finally, a deep ReLU neural network is employed for the prediction. Explicit rates of approximating nonlinear smooth functionals across various input function spaces by our proposed functional network are derived. Additionally, we conduct a generalization analysis for the empirical risk minimization (ERM) algorithm applied to our functional net, by employing a novel two-stage oracle inequality and the established functional approximation results. Ultimately, we conduct numerical experiments on both simulated and real datasets to demonstrate the effectiveness and benefits of our functional net.

Jun Fan, Zheng-Chu Guo, Lei Shi

Spectral algorithms leverage spectral regularization techniques to analyze and process data, providing a flexible framework for addressing supervised learning problems. To deepen our understanding of their performance in real-world scenarios where the distributions of training and test data may differ, we conduct a rigorous investigation into the convergence behavior of spectral algorithms under covariate shift. In this setting, the marginal distributions of the input data differ between the training and test datasets, while the conditional distribution of the output given the input remains unchanged. Within a non-parametric regression framework over a reproducing kernel Hilbert space, we analyze the convergence rates of spectral algorithms under covariate shift and show that they achieve minimax optimality when the density ratios between the training and test distributions are uniformly bounded. However, when these density ratios are unbounded, the spectral algorithms may become suboptimal. To address this issue, we propose a novel weighted spectral algorithm with normalized weights that incorporates density ratio information into the learning process. Our theoretical analysis shows that this normalized weighted approach achieves optimal capacity-independent convergence rates, but the rates will suffer from the saturation phenomenon. Furthermore, by introducing a weight clipping technique, we demonstrate that the convergence rates of the weighted spectral algorithm with clipped weights can approach the optimal capacity-dependent convergence rates arbitrarily closely. This improvement resolves the suboptimality issue in unbounded density ratio scenarios and advances the state-of-the-art by refining existing theoretical results.

Ren-Rui Liu, Jun Fan, Lei Shi et al.

This paper focuses on the problem of unbounded density ratio estimation -- an understudied yet critical challenge in statistical learning -- and its application to covariate shift adaptation. Much of the existing literature assumes that the density ratio is either uniformly bounded or unbounded but known exactly. These conditions are often violated in practice, creating a gap between theoretical guarantees and real-world applicability. In contrast, this work directly addresses unbounded density ratios and integrates them into importance weighting for effective covariate shift adaptation. We propose a three-step estimation method that leverages unlabeled data from both the source and target distributions: (1) estimating a relative density ratio; (2) applying a truncation operation to control its unboundedness; and (3) transforming the truncated estimate back into the standard density ratio. The estimated density ratio is then employed as importance weights for regression under covariate shift. We establish rigorous, non-asymptotic convergence guarantees for both the proposed density ratio estimator and the resulting regression function estimator, demonstrating optimal or near-optimal convergence rates. Our findings offer new theoretical insights into density ratio estimation and learning under covariate shift, extending classical learning theory to more practical and challenging scenarios.

Qianxin Yi, Shao-Bo Lin, Jun Fan et al.

Reinforcement learning (RL) has been widely applied to sequential decision making, where interpretability and performance are both critical for practical adoption. Current approaches typically focus on performance and rely on post hoc explanations to account for interpretability. Different from these approaches, we focus on designing an interpretability-oriented yet performance-enhanced RL approach. Specifically, we propose a spectral based linear RL method that extends the ridge regression-based approach through a spectral filter function. The proposed method clarifies the role of regularization in controlling estimation error and further enables the design of an adaptive regularization parameter selection strategy guided by the bias-variance trade-off principle. Theoretical analysis establishes near-optimal bounds for both parameter estimation and generalization error. Extensive experiments on simulated environments and real-world datasets from Kuaishou and Taobao demonstrate that our method either outperforms or matches existing baselines in decision quality. We also conduct interpretability analyses to illustrate how the learned policies make decisions, thereby enhancing user trust. These results highlight the potential of our approach to bridge the gap between RL theory and practical decision making, providing interpretability, accuracy, and adaptability in management contexts.

Fei Zhao, Chonggang Lu, Yue Wang et al.

As a primary medium for modern information dissemination, social networking services (SNS) have experienced rapid growth, which has proposed significant challenges for platform content management and interaction quality improvement. Recently, the development of large language models (LLMs) has offered potential solutions but existing studies focus on isolated tasks, which not only encounter diminishing benefit from the data scaling within individual scenarios but also fail to flexibly adapt to diverse real-world context. To address these challenges, we introduce RedOne, a domain-specific LLM designed to break the performance bottleneck of single-task baselines and establish a comprehensive foundation for the SNS. RedOne was developed through a three-stage training strategy consisting of continue pretraining, supervised fine-tuning, and preference optimization, using a large-scale real-world dataset. Through extensive experiments, RedOne maintains strong general capabilities, and achieves an average improvement up to 14.02% across 8 major SNS tasks and 7.56% in SNS bilingual evaluation benchmark, compared with base models. Furthermore, through online testing, RedOne reduced the exposure rate in harmful content detection by 11.23% and improved the click page rate in post-view search by 14.95% compared with single-tasks finetuned baseline models. These results establish RedOne as a robust domain-specific LLM for SNS, demonstrating excellent generalization across various tasks and promising applicability in real-world scenarios.

Zhan Yu, Jun Fan, Zhongjie Shi et al.

In recent years, different types of distributed and parallel learning schemes have received increasing attention for their strong advantages in handling large-scale data information. In the information era, to face the big data challenges {that} stem from functional data analysis very recently, we propose a novel distributed gradient descent functional learning (DGDFL) algorithm to tackle functional data across numerous local machines (processors) in the framework of reproducing kernel Hilbert space. Based on integral operator approaches, we provide the first theoretical understanding of the DGDFL algorithm in many different aspects of the literature. On the way of understanding DGDFL, firstly, a data-based gradient descent functional learning (GDFL) algorithm associated with a single-machine model is proposed and comprehensively studied. Under mild conditions, confidence-based optimal learning rates of DGDFL are obtained without the saturation boundary on the regularity index suffered in previous works in functional regression. We further provide a semi-supervised DGDFL approach to weaken the restriction on the maximal number of local machines to ensure optimal rates. To our best knowledge, the DGDFL provides the first divide-and-conquer iterative training approach to functional learning based on data samples of intrinsically infinite-dimensional random functions (functional covariates) and enriches the methodologies for functional data analysis.

Keli Guo, Jun Fan, Lixing Zhu

In this paper, we establish minimax optimal rates of convergence for prediction in a semi-functional linear model that consists of a functional component and a less smooth nonparametric component. Our results reveal that the smoother functional component can be learned with the minimax rate as if the nonparametric component were known. More specifically, a double-penalized least squares method is adopted to estimate both the functional and nonparametric components within the framework of reproducing kernel Hilbert spaces. By virtue of the representer theorem, an efficient algorithm that requires no iterations is proposed to solve the corresponding optimization problem, where the regularization parameters are selected by the generalized cross validation criterion. Numerical studies are provided to demonstrate the effectiveness of the method and to verify the theoretical analysis.

Jun Fan, Dao-Hong Xiang

This paper studies binary classification problem associated with a family of loss functions called large-margin unified machines (LUM), which offers a natural bridge between distribution-based likelihood approaches and margin-based approaches. It also can overcome the so-called data piling issue of support vector machine in the high-dimension and low-sample size setting. In this paper we establish some new comparison theorems for all LUM loss functions which play a key role in the further error analysis of large-margin learning algorithms.

Zhongyang Zhang, Ling Zhang, Ze Sun et al.

Simulating the dynamic characteristics of a PN junction at the microscopic level requires solving the Poisson's equation at every time step. Solving at every time step is a necessary but time-consuming process when using the traditional finite difference (FDM) approach. Deep learning is a powerful technique to fit complex functions. In this work, deep learning is utilized to accelerate solving Poisson's equation in a PN junction. The role of the boundary condition is emphasized in the loss function to ensure a better fitting. The resulting I-V curve for the PN junction, using the deep learning solver presented in this work, shows a perfect match to the I-V curve obtained using the finite difference method, with the advantage of being 10 times faster at every time step.

Yunlong Feng, Jun Fan, Johan A. K. Suykens

This paper studies the nonparametric modal regression problem systematically from a statistical learning view. Originally motivated by pursuing a theoretical understanding of the maximum correntropy criterion based regression (MCCR), our study reveals that MCCR with a tending-to-zero scale parameter is essentially modal regression. We show that nonparametric modal regression problem can be approached via the classical empirical risk minimization. Some efforts are then made to develop a framework for analyzing and implementing modal regression. For instance, the modal regression function is described, the modal regression risk is defined explicitly and its \textit{Bayes} rule is characterized; for the sake of computational tractability, the surrogate modal regression risk, which is termed as the generalization risk in our study, is introduced. On the theoretical side, the excess modal regression risk, the excess generalization risk, the function estimation error, and the relations among the above three quantities are studied rigorously. It turns out that under mild conditions, function estimation consistency and convergence may be pursued in modal regression as in vanilla regression protocols, such as mean regression, median regression, and quantile regression. However, it outperforms these regression models in terms of robustness as shown in our study from a re-descending M-estimation view. This coincides with and in return explains the merits of MCCR on robustness. On the practical side, the implementation issues of modal regression including the computational algorithm and the tuning parameters selection are discussed. Numerical assessments on modal regression are also conducted to verify our findings empirically.

Jun Fan, Ting Hu, Qiang Wu et al.

In this paper we study the consistency of an empirical minimum error entropy (MEE) algorithm in a regression setting. We introduce two types of consistency. The error entropy consistency, which requires the error entropy of the learned function to approximate the minimum error entropy, is shown to be always true if the bandwidth parameter tends to 0 at an appropriate rate. The regression consistency, which requires the learned function to approximate the regression function, however, is a complicated issue. We prove that the error entropy consistency implies the regression consistency for homoskedastic models where the noise is independent of the input variable. But for heteroskedastic models, a counterexample is used to show that the two types of consistency do not coincide. A surprising result is that the regression consistency is always true, provided that the bandwidth parameter tends to infinity at an appropriate rate. Regression consistency of two classes of special models is shown to hold with fixed bandwidth parameter, which further illustrates the complexity of regression consistency of MEE. Fourier transform plays crucial roles in our analysis.

Ting Hu, Jun Fan, Qiang Wu et al.

We consider the minimum error entropy (MEE) criterion and an empirical risk minimization learning algorithm in a regression setting. A learning theory approach is presented for this MEE algorithm and explicit error bounds are provided in terms of the approximation ability and capacity of the involved hypothesis space when the MEE scaling parameter is large. Novel asymptotic analysis is conducted for the generalization error associated with Renyi's entropy and a Parzen window function, to overcome technical difficulties arisen from the essential differences between the classical least squares problems and the MEE setting. A semi-norm and the involved symmetrized least squares error are introduced, which is related to some ranking algorithms.