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Jicong Fan

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23 papers

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AAAI Conference 2026 Conference Paper

Adaptive Riemannian Graph Neural Networks

Xudong Wang
Chris Ding
Tongxin Li
Jicong Fan

Graph data often exhibits complex geometric heterogeneity, where structures with varying local curvature, such as tree-like hierarchies and dense communities, coexist within a single network. Existing geometric GNNs, which embed graphs into single fixed-curvature manifolds or discrete product spaces, struggle to capture this diversity. We introduce Adaptive Riemannian Graph Neural Networks (ARGNN), a novel framework that learns a continuous and anisotropic Riemannian metric tensor field over the graph. It allows each node to determine its optimal local geometry, enabling the model to fluidly adapt to the graph's structural landscape. Our core innovation is an efficient parameterization of the node-wise metric tensor, specializing to a learnable diagonal form that captures directional geometric information while maintaining computational tractability. To ensure geometric regularity and stable training, we integrate a Ricci flow-inspired regularization that smooths the learned manifold. Theoretically, we establish the rigorous geometric evolution convergence guarantee for ARGNN and provide a continuous generalization that unifies prior fixed or mixed-curvature GNNs. Empirically, our method demonstrates superior performance on both homophilic and heterophilic benchmark datasets with the ability to capture diverse structures adaptively. Moreover, the learned geometries both offer interpretable insights into the underlying graph structure and empirically corroborate our theoretical analysis.

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IJCAI Conference 2025 Conference Paper

Explainable Graph Representation Learning via Graph Pattern Analysis

Xudong Wang
Ziheng Sun
Chris Ding
Jicong Fan

Explainable artificial intelligence (XAI) is an important area in the AI community, and interpretability is crucial for building robust and trustworthy AI models. While previous work has explored model-level and instance-level explainable graph learning, there has been limited investigation into explainable graph representation learning. In this paper, we focus on representation-level explainable graph learning and ask a fundamental question: What specific information about a graph is captured in graph representations? Our approach is inspired by graph kernels, which evaluate graph similarities by counting substructures within specific graph patterns. Although the pattern counting vector can serve as an explainable representation, it has limitations such as ignoring node features and being high-dimensional. To address these limitations, we introduce a framework (PXGL-GNN) for learning and explaining graph representations through graph pattern analysis. We start by sampling graph substructures of various patterns. Then, we learn the representations of these patterns and combine them using a weighted sum, where the weights indicate the importance of each graph pattern's contribution. We also provide theoretical analyses of our methods, including robustness and generalization. In our experiments, we show how to learn and explain graph representations for real-world data using pattern analysis. Additionally, we compare our method against multiple baselines in both supervised and unsupervised learning tasks to demonstrate its effectiveness.

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NeurIPS Conference 2025 Conference Paper

Fairness-aware Anomaly Detection via Fair Projection

Feng Xiao
Xiaoying Tang
Jicong Fan

Unsupervised anomaly detection is a critical task in many high-social-impact applications such as finance, healthcare, social media, and cybersecurity, where demographics involving age, gender, race, disease, etc. are used frequently. In these scenarios, possible bias from anomaly detection systems can lead to unfair treatment for different groups and even exacerbate social bias. In this work, first, we thoroughly analyze the feasibility and necessary assumptions for ensuring group fairness in unsupervised anomaly detection. Second, we propose a novel fairness-aware anomaly detection method FairAD. From the normal training data, FairAD learns a projection to map data of different demographic groups to a common target distribution that is simple and compact, and hence provides a reliable base to estimate the density of the data. The density can be directly used to identify anomalies while the common target distribution ensures fairness between different groups. Furthermore, we propose a threshold-free fairness metric that provides a global view for model's fairness, eliminating dependence on manual threshold selection. Experiments on real-world benchmarks demonstrate that our method achieves an improved trade-off between detection accuracy and fairness under both balanced and skewed data across different groups.

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AAAI Conference 2025 Conference Paper

Federated t-SNE and UMAP for Distributed Data Visualization

Dong Qiao
Xinxian Ma
Jicong Fan

High-dimensional data visualization is crucial in big data era and these techniques such as t-SNE and UMAP have been widely used in science and engineering. Big data, however, is often distributed across multiple data centers and subject to security and privacy concerns, which leads to difficulties for the standard algorithms of t-SNE and UMAP. To tackle the challenge, this work proposes Fed-tSNE and Fed-UMAP, which provide high-dimensional data visualization under the framework of federated learning, without exchanging data across clients or sending data to the central server. The main idea of Fed-tSNE and Fed-UMAP is implicitly learning the distribution information of data in a manner of federated learning and then estimating the global distance matrix for t-SNE and UMAP. To further enhance the protection of data privacy, we propose Fed-tSNE+ and Fed-UMAP+. We also extend our idea to federated spectral clustering, yielding algorithms of clustering distributed data. In addition to these new algorithms, we offer theoretical guarantees of distance and similarity estimation and analyze the property of differential privacy. Experiments on multiple datasets demonstrate that, compared to the original algorithms, the accuracy drops of our federated algorithms are tiny.

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AAAI Conference 2025 Conference Paper

Multi-Subspace Matrix Recovery from Permuted Data

Liangqi Xie
Jicong Fan

This paper aims to recover a multi-subspace matrix from permuted data: given a matrix, in which the columns are drawn from a union of low-dimensional subspaces and some columns are corrupted by permutations on their entries, recover the original matrix. The task has numerous practical applications such as data cleaning, integration, and de-anonymization, but it remains challenging and cannot be well addressed by existing techniques such as robust principal component analysis because of the presence of multiple subspaces and the permutations on the elements of vectors. To solve the challenge, we develop a novel four-stage algorithm pipeline including outlier identification, subspace reconstruction, outlier classification, and unsupervised sensing for permuted vector recovery. Particularly, we provide theoretical guarantees for the outlier classification step, ensuring reliable multi-subspace matrix recovery. Our pipeline is compared with state-of-the-art competitors on multiple benchmarks and shows superior performance.

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AAAI Conference 2025 Conference Paper

Non-Convex Tensor Recovery from Local Measurements

Tongle Wu
Ying Sun
Jicong Fan

Motivated by the settings where sensing the entire tensor is infeasible, this paper proposes a novel tensor compressed sensing model, where measurements are only obtained from sensing each lateral slice via mutually independent matrices. Leveraging the low tubal rank structure, we reparameterize the unknown tensor?* using two compact tensor factors and formulate the recovery problem as a nonconvex minimization problem. To solve the problem, we first propose an alternating minimization algorithm, termed Alt-PGD-Min, that iteratively optimizes the two factors using a projected gradient descent and an exact minimization step, respectively. Despite nonconvexity, we prove that Alt-PGD-Min achieves ϵ-accuracy recovery with?(?²log1/?) iteration complexity and?(?⁶rn₃logn₃(?²r(n₁+n₂)+n₁log1/ε)) sample complexity, where? denotes tensor condition number of?*. To further accelerate the convergence, especially when the tensor is ill-conditioned with large?, we prove Alt-ScalePGD-Min that preconditions the gradient update using an approximate Hessian that can be computed efficiently. We show that Alt-ScalePGD-Min achieves? independent iteration complexity?(log1/ε) and improves the sample complexity to?(?⁴rn₃log n₃(?⁴ r(n₁ + n₂)+n₁log 1/ε)). Experiments validate the effectiveness of the proposed methods.

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AAAI Conference 2025 Conference Paper

Unsupervised Anomaly Detection for Tabular Data Using Deep Noise Evaluation

Wei Dai
Kai Hwang
Jicong Fan

Unsupervised anomaly detection (UAD) plays an important role in modern data analytics and it is crucial to provide simple yet effective and guaranteed UAD algorithms for real applications. In this paper, we present a novel UAD method for tabular data by evaluating how much noise is in the data. Specifically, we propose to learn a deep neural network from the clean (normal) training dataset and a noisy dataset, where the latter is generated by adding highly diverse noises to the clean data. The neural network can learn a reliable decision boundary between normal data and anomalous data when the diversity of the generated noisy data is sufficiently high so that the hard abnormal samples lie in the noisy region. Importantly, we provide theoretical guarantees, proving that the proposed method can detect anomalous data successfully, although the method does not utilize any real anomalous data in the training stage. Extensive experiments through more than 60 benchmark datasets demonstrate the effectiveness of the proposed method in comparison to 12 baselines of UAD. Our method obtains a 92.27% AUC score and a 1.68 ranking score on average. Moreover, compared to the state-of-the-art UAD methods, our method is easier to implement.

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AAAI Conference 2024 Conference Paper

Dense Projection for Anomaly Detection

Dazhi Fu
Zhao Zhang
Jicong Fan

This work presents a novel method called dense projection for unsupervised anomaly detection (DPAD). The main idea is maximizing the local density of (normal) training data and then determining whether a test data is anomalous or not by evaluating its density. Specifically, DPAD uses a deep neural network to learn locally dense representations of normal data. Since density estimation is computationally expensive, we minimize the local distances of the representations in an iteratively reweighting manner, where the weights are updated adaptively and the parameters are regularized to avoid model collapse (all representations collapse to a single point). Compared with many state-of-the-art methods of anomaly detection, our DPAD does not rely on any assumption about the distribution or spatial structure of the normal data and representations. Moreover, we provide theoretical guarantees for the effectiveness of DPAD. The experiments show that our method DPAD is effective not only in traditional one-class classification problems but also in scenarios with complex normal data composed of multiple classes.

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IJCAI Conference 2024 Conference Paper

Dual Contrastive Graph-Level Clustering with Multiple Cluster Perspectives Alignment

Jinyu Cai
Yunhe Zhang
Jicong Fan
Yali Du
Wenzhong Guo

Graph-level clustering, which is essential in medical, biomedical, and social network data analysis, aims to group a set of graphs into various clusters. However, existing methods generally rely on a single clustering criterion, e. g. , $k$-means, which limits their abilities to fully exploit the complex Euclidean and structural information inherent in graphs. To bridge this gap, we propose a dual contrastive graph-level clustering (DCGLC) method in this paper. DCGLC leverages graph contrastive learning and introduces the Euclidian-based and subspace-based cluster heads to capture the cluster information from different cluster perspectives. To overcome the inconsistency estimations and fuse the cluster information of multiple cluster heads, we propose a contrastive mechanism to align the cluster information derived from them. The cluster-perspective contrast facilitates the capture of more comprehensive cluster information. Importantly, DCGLC is an end-to-end framework in which graph contrastive learning and cluster-perspective contrast are mutually improved. We demonstrate the superiority of DCGLC over the state-of-the-art baselines on numerous graph benchmarks.

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NeurIPS Conference 2024 Conference Paper

Graph Classification via Reference Distribution Learning: Theory and Practice

Zixiao Wang
Jicong Fan

Graph classification is a challenging problem owing to the difficulty in quantifying the similarity between graphs or representing graphs as vectors, though there have been a few methods using graph kernels or graph neural networks (GNNs). Graph kernels often suffer from computational costs and manual feature engineering, while GNNs commonly utilize global pooling operations, risking the loss of structural or semantic information. This work introduces Graph Reference Distribution Learning (GRDL), an efficient and accurate graph classification method. GRDL treats each graph's latent node embeddings given by GNN layers as a discrete distribution, enabling direct classification without global pooling, based on maximum mean discrepancy to adaptively learned reference distributions. To fully understand this new model (the existing theories do not apply) and guide its configuration (e. g. , network architecture, references' sizes, number, and regularization) for practical use, we derive generalization error bounds for GRDL and verify them numerically. More importantly, our theoretical and numerical results both show that GRDL has a stronger generalization ability than GNNs with global pooling operations. Experiments on moderate-scale and large-scale graph datasets show the superiority of GRDL over the state-of-the-art, emphasizing its remarkable efficiency, being at least 10 times faster than leading competitors in both training and inference stages.

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IJCAI Conference 2024 Conference Paper

Kernel Readout for Graph Neural Networks

Jiajun Yu
Zhihao Wu
Jinyu Cai
Adele Lu Jia
Jicong Fan

Graph neural networks (GNNs) for graph classification or representation learning require a pooling operation to convert the nodes' embeddings of each graph to a vector as the graph-level representation and the operation has a significant impact on model accuracy. The paper presents a novel graph pooling method called Kernel Readout (KerRead). KerRead maps the node embeddings from the sample space with limited nodes to an augmented sample space with infinite nodes, and then calculates the inner product between some learnable adaptive centers and the augmented node embeddings, which forms a final graph-level feature vector. We apply the proposed strategy to six supervised and two unsupervised graph neural networks such as GCN, GIN, GUNet, InfoGraph, and GraphCL, and the experiments on eight benchmark datasets show that the proposed readout outperforms classical pooling methods such as Sum and seven state-of-the-art pooling methods such as SRead and Janossy GRU. Code and Appendix are both available at https: //github. com/jiajunCAU/KerRead.

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IJCAI Conference 2024 Conference Paper

LG-FGAD: An Effective Federated Graph Anomaly Detection Framework

Jinyu Cai
Yunhe Zhang
Jicong Fan
See-Kiong Ng

Graph anomaly detection (GAD), which aims to identify those graphs that are significantly different from other ones, has gained growing attention in many real-world scenarios. However, existing GAD methods are generally designed for centralized training, while in real-world collaboration, graph data is generally distributed across various clients and exhibits significant non-IID characteristics. To tackle this challenge, we propose a federated graph anomaly detection framework with local-global anomaly awareness (LG-FGAD). We first introduce a self-adversarial generation module and train a discriminator to identify the generated anomalous graphs from the normal graph. To enhance the anomaly awareness of the model, we propose to maximize/minimize the mutual information from local and global perspectives. Importantly, to alleviate the impact of non-IID problems in collaborative learning, we propose a dual knowledge distillation module. The knowledge distillation is conducted over both logits and embedding distributions, and only the student model engages in collaboration to preserve the personalization of each client. Empirical results on various types of real-world datasets prove the superiority of our method.

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NeurIPS Conference 2024 Conference Paper

Unsupervised Anomaly Detection in The Presence of Missing Values

Feng Xiao
Jicong Fan

Anomaly detection methods typically require fully observed data for model training and inference and cannot handle incomplete data, while the missing data problem is pervasive in science and engineering, leading to challenges in many important applications such as abnormal user detection in recommendation systems and novel or anomalous cell detection in bioinformatics, where the missing rates can be higher than 30\% or even 80\%. In this work, first, we construct and evaluate a straightforward strategy, ''impute-then-detect'', via combining state-of-the-art imputation methods with unsupervised anomaly detection methods, where the training data are composed of normal samples only. We observe that such two-stage methods frequently yield imputation bias from normal data, namely, the imputation methods are inclined to make incomplete samples ''normal", where the fundamental reason is that the imputation models learned only on normal data and cannot generalize well to abnormal data in the inference stage. To address this challenge, we propose an end-to-end method that integrates data imputation with anomaly detection into a unified optimization problem. The proposed model learns to generate well-designed pseudo-abnormal samples to mitigate the imputation bias and ensure the discrimination ability of both the imputation and detection processes. Furthermore, we provide theoretical guarantees for the effectiveness of the proposed method, proving that the proposed method can correctly detect anomalies with high probability. Experimental results on datasets with manually constructed missing values and inherent missing values demonstrate that our proposed method effectively mitigates the imputation bias and surpasses the baseline methods significantly. The source code of our method is available at https: //github. com/jicongfan/ImAD-Anomaly-Detection-With-Missing-Data.

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NeurIPS Conference 2023 Conference Paper

Boosting Spectral Clustering on Incomplete Data via Kernel Correction and Affinity Learning

Fangchen Yu
Runze Zhao
Zhan Shi
Yiwen Lu
Jicong Fan
Yicheng Zeng
Jianfeng Mao
Wenye Li

Spectral clustering has gained popularity for clustering non-convex data due to its simplicity and effectiveness. It is essential to construct a similarity graph using a high-quality affinity measure that models the local neighborhood relations among the data samples. However, incomplete data can lead to inaccurate affinity measures, resulting in degraded clustering performance. To address these issues, we propose an imputation-free framework with two novel approaches to improve spectral clustering on incomplete data. Firstly, we introduce a new kernel correction method that enhances the quality of the kernel matrix estimated on incomplete data with a theoretical guarantee, benefiting classical spectral clustering on pre-defined kernels. Secondly, we develop a series of affinity learning methods that equip the self-expressive framework with $\ell_p$-norm to construct an intrinsic affinity matrix with an adaptive extension. Our methods outperform existing data imputation and distance calibration techniques on benchmark datasets, offering a promising solution to spectral clustering on incomplete data in various real-world applications.

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TMLR Journal 2023 Journal Article

Euclidean-Norm-Induced Schatten-p Quasi-Norm Regularization for Low-Rank Tensor Completion and Tensor Robust Principal Component Analysis

Jicong Fan
Lijun Ding
Chengrun Yang
Zhao Zhang
Madeleine Udell

The nuclear norm and Schatten-$p$ quasi-norm are popular rank proxies in low-rank matrix recovery. However, computing the nuclear norm or Schatten-$p$ quasi-norm of a tensor is hard in both theory and practice, hindering their application to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). In this paper, we propose a new class of tensor rank regularizers based on the Euclidean norms of the CP component vectors of a tensor and show that these regularizers are monotonic transformations of tensor Schatten-$p$ quasi-norm. This connection enables us to minimize the Schatten-$p$ quasi-norm in LRTC and TRPCA implicitly via the component vectors. The method scales to big tensors and provides an arbitrarily sharper rank proxy for low-rank tensor recovery compared to the nuclear norm. On the other hand, we study the generalization abilities of LRTC with the Schatten-$p$ quasi-norm regularizer and LRTC with the proposed regularizers. The theorems show that a relatively sharper regularizer leads to a tighter error bound, which is consistent with our numerical results. Particularly, we prove that for LRTC with Schatten-$p$ quasi-norm regularizer on $d$-order tensors, $p=1/d$ is always better than any $p>1/d$ in terms of the generalization ability. We also provide a recovery error bound to verify the usefulness of small $p$ in the Schatten-$p$ quasi-norm for TRPCA. Numerical results on synthetic data and real data demonstrate the effectiveness of the regularization methods and theorems.

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NeurIPS Conference 2023 Conference Paper

Federated Spectral Clustering via Secure Similarity Reconstruction

Dong Qiao
Chris Ding
Jicong Fan

Federated learning has a significant advantage in protecting information privacy. Many scholars proposed various secure learning methods within the framework of federated learning but the study on secure federated unsupervised learning especially clustering is limited. We in this work propose a secure kernelized factorization method for federated spectral clustering on distributed dataset. The method is non-trivial because the kernel or similarity matrix for spectral clustering is computed by data pairs, which violates the principle of privacy protection. Our method implicitly constructs an approximation for the kernel matrix on distributed data such that we can perform spectral clustering under the constraint of privacy protection. We provide a convergence guarantee of the optimization algorithm, reconstruction error bounds of the Gaussian kernel matrix, and the sufficient condition of correct clustering of our method. We also present some results of differential privacy. Numerical results on synthetic and real datasets demonstrate that the proposed method is efficient and accurate in comparison to the baselines.

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NeurIPS Conference 2023 Conference Paper

Graph Convolutional Kernel Machine versus Graph Convolutional Networks

Zhihao Wu
Zhao Zhang
Jicong Fan

Graph convolutional networks (GCN) with one or two hidden layers have been widely used in handling graph data that are prevalent in various disciplines. Many studies showed that the gain of making GCNs deeper is tiny or even negative. This implies that the complexity of graph data is often limited and shallow models are often sufficient to extract expressive features for various tasks such as node classification. Therefore, in this work, we present a framework called graph convolutional kernel machine (GCKM) for graph-based machine learning. GCKMs are built upon kernel functions integrated with graph convolution. An example is the graph convolutional kernel support vector machine (GCKSVM) for node classification, for which we analyze the generalization error bound and discuss the impact of the graph structure. Compared to GCNs, GCKMs require much less effort in architecture design, hyperparameter tuning, and optimization. More importantly, GCKMs are guaranteed to obtain globally optimal solutions and have strong generalization ability and high interpretability. GCKMs are composable, can be extended to large-scale data, and are applicable to various tasks (e. g. , node or graph classification, clustering, feature extraction, dimensionality reduction). The numerical results on benchmark datasets show that, besides the aforementioned advantages, GCKMs have at least competitive accuracy compared to GCNs.

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NeurIPS Conference 2023 Conference Paper

Lovász Principle for Unsupervised Graph Representation Learning

Ziheng Sun
Chris Ding
Jicong Fan

This paper focuses on graph-level representation learning that aims to represent graphs as vectors that can be directly utilized in downstream tasks such as graph classification. We propose a novel graph-level representation learning principle called Lovász principle, which is motivated by the Lovász number in graph theory. The Lovász number of a graph is a real number that is an upper bound for graph Shannon capacity and is strongly connected with various global characteristics of the graph. Specifically, we show that the handle vector for computing the Lovász number is potentially a suitable choice for graph representation, as it captures a graph's global properties, though a direct application of the handle vector is difficult and problematic. We propose to use neural networks to address the problems and hence provide the Lovász principle. Moreover, we propose an enhanced Lovász principle that is able to exploit the subgraph Lovász numbers directly and efficiently. The experiments demonstrate that our Lovász principles achieve competitive performance compared to the baselines in unsupervised and semi-supervised graph-level representation learning tasks. The code of our Lovász principles is publicly available on GitHub.

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NeurIPS Conference 2022 Conference Paper

A Simple Approach to Automated Spectral Clustering

Jicong Fan
Yiheng Tu
Zhao Zhang
Mingbo Zhao
Haijun Zhang

The performance of spectral clustering heavily relies on the quality of affinity matrix. A variety of affinity-matrix-construction (AMC) methods have been proposed but they have hyperparameters to determine beforehand, which requires strong experience and leads to difficulty in real applications, especially when the inter-cluster similarity is high and/or the dataset is large. In addition, we often need to choose different AMC methods for different datasets, which still depends on experience. To solve these two challenging problems, in this paper, we present a simple yet effective method for automated spectral clustering. First, we propose to find the most reliable affinity matrix via grid search or Bayesian optimization among a set of candidates given by different AMC methods with different hyperparameters, where the reliability is quantified by the \textit{relative-eigen-gap} of graph Laplacian introduced in this paper. Second, we propose a fast and accurate AMC method based on least squares representation and thresholding and prove its effectiveness theoretically. Finally, we provide a large-scale extension for the automated spectral clustering method, of which the time complexity is linear with the number of data points. Extensive experiments of natural image clustering show that our method is more versatile, accurate, and efficient than baseline methods.

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AAAI Conference 2022 Conference Paper

Dynamic Nonlinear Matrix Completion for Time-Varying Data Imputation

Jicong Fan

Classical matrix completion methods focus on data with stationary latent structure and hence are not effective in missing value imputation when the latent structure changes with time. This paper proposes a dynamic nonlinear matrix completion (D-NLMC) method, which is able to recover the missing values of streaming data when the low-dimensional nonlinear latent structure of the data changes with time. The paper provides an efficient approach to updating the nonlinear model dynamically. D-NLMC incorporates the information of new data and remove the information of earlier data recursively. The paper shows that the missing data can be estimated if the change of latent structure is slow enough. Different from existing online or adaptive low-rank matrix completion methods, D-NLMC does not require the local low-rank assumption and is able to adaptively recover high-rank matrices with lowdimensional latent structures. Note that existing high-rank matrix completion methods have high-computational costs and are not applicable to streaming data with varying latent structures, which fortunately can be handled by D-NLMC efficiently and accurately. Numerical results show that D- NLMC outperforms the baselines in real applications.

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NeurIPS Conference 2022 Conference Paper

Perturbation Learning Based Anomaly Detection

Jinyu Cai
Jicong Fan

This paper presents a simple yet effective method for anomaly detection. The main idea is to learn small perturbations to perturb normal data and learn a classifier to classify the normal data and the perturbed data into two different classes. The perturbator and classifier are jointly learned using deep neural networks. Importantly, the perturbations should be as small as possible but the classifier is still able to recognize the perturbed data from unperturbed data. Therefore, the perturbed data are regarded as abnormal data and the classifier provides a decision boundary between the normal data and abnormal data, although the training data do not include any abnormal data. Compared with the state-of-the-art of anomaly detection, our method does not require any assumption about the shape (e. g. hypersphere) of the decision boundary and has fewer hyper-parameters to determine. Empirical studies on benchmark datasets verify the effectiveness and superiority of our method.

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AAAI Conference 2020 Conference Paper

Polynomial Matrix Completion for Missing Data Imputation and Transductive Learning

Jicong Fan
Yuqian Zhang
Madeleine Udell

This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Speciﬁcally, we assume that the columns of a matrix are generated by polynomials acting on a low-dimensional intrinsic variable, and wish to recover the missing entries under this assumption. We show that we can identify the complete matrix of minimum intrinsic dimension by minimizing the rank of the matrix in a high dimensional feature space. We develop a new formulation of the resulting problem using the kernel trick together with a new relaxation of the rank objective, and propose an efﬁcient optimization method. We also show how to use our methods to complete data drawn from multiple nonlinear manifolds. Comparative studies on synthetic data, subspace clustering with missing data, motion capture data recovery, and transductive learning verify the superiority of our methods over the state-of-the-art.

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NeurIPS Conference 2019 Conference Paper

Factor Group-Sparse Regularization for Efficient Low-Rank Matrix Recovery

Jicong Fan
Lijun Ding
Yudong Chen
Madeleine Udell

This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the \emph{matrix rank} function. Our new factor group-sparse regularizers are motivated as a relaxation of the \emph{number of nonzero columns} in a factorization of the matrix. These nonconvex regularizers are sharper than the nuclear norm; indeed, we show they are related to Schatten-$p$ norms with arbitrarily small $0 < p \leq 1$. Moreover, these factor group-sparse regularizers can be written in a factored form that enables efficient and effective nonconvex optimization; notably, the method does not use singular value decomposition. We provide generalization error bounds for low-rank matrix completion which show improved upper bounds for Schatten-$p$ norm reglarization as $p$ decreases. Compared to the max norm and the factored formulation of the nuclear norm, factor group-sparse regularizers are more efficient, accurate, and robust to the initial guess of rank. Experiments show promising performance of factor group-sparse regularization for low-rank matrix completion and robust principal component analysis.

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