Extreme events frequently occur in real-world time series and often carry significant practical implications. In domains such as climate and healthcare, these events, such as floods, heatwaves, or acute medical episodes, can lead to serious consequences. Accurate forecasting of such events is therefore of substantial importance. Most existing time series forecasting models are optimized for overall performance within the prediction window, but often struggle to accurately predict extreme events, such as high temperatures or heart rate spikes. The main challenges are data imbalance and the neglect of valuable information contained in intermediate events that precede extreme events. In this paper, we propose xTime, a novel framework for extreme event forecasting in time series. xTime leverages knowledge distillation to transfer information from models trained on lower-rarity events, thereby improving prediction performance on rarer ones. In addition, we introduce a MoE mechanism that dynamically selects and fuses outputs from expert models across different rarity levels, which further improves the forecasting performance for extreme events. Experiments on multiple datasets show that xTime achieves consistent improvements, with forecasting accuracy on extreme events improving from 3% to 78%.

Graph Neural Networks (GNNs) resurge as a trending research subject owing to their impressive ability to capture representations from graph-structured data. However, the black-box nature of GNNs presents a significant challenge in terms of comprehending and trusting these models, thereby limiting their practical applications in mission-critical scenarios. Although there has been substantial progress in the field of explaining GNNs in recent years, the majority of these studies are centered on static graphs, leaving the explanation of dynamic GNNs less explored. Dynamic GNNs, with their ever-evolving graph structures, pose a unique challenge and require additional efforts to effectively capture temporal dependencies and structural relationships. To address this challenge, we present DyExplainer, a novel approach to explaining dynamic GNNs on the fly. DyExplainer trains a dynamic GNN backbone to extract representations of the graph at each snapshot, while simultaneously exploring structural relationships and temporal dependencies through a sparse attention technique. To preserve the desired properties of the explanation, such as structural consistency and temporal continuity, we augment our approach with contrastive learning techniques to provide a priori -guided regularization. To model longer-term temporal dependencies, we develop a buffer-based live-updating scheme for training. The results of our extensive experiments on various datasets demonstrate the superiority of DyExplainer, not only providing faithful explainability of the model predictions but also significantly improving the model prediction accuracy, as evidenced in the link prediction task.

Time series domain adaptation stands as a pivotal and intricate challenge with diverse applications, including but not limited to human activity recognition, sleep stage classification, and machine fault diagnosis. Despite the numerous domain adaptation techniques proposed to tackle this complex problem, they primarily focus on domain adaptation from a single source domain. Yet, it is more crucial to investigate domain adaptation from multiple domains due to the potential for greater improvements. To address this, three important challenges need to be overcome: 1). The lack of exploration to utilize domain-specific information for domain adaptation, 2). The difficulty to learn domain-specific information that changes over time, and 3). The difficulty to evaluate learned domain-specific information. In order to tackle these challenges simultaneously, in this paper, we introduce PrOmpt-based domaiN Discrimination (POND), the first framework to utilize prompts for time series domain adaptation. Specifically, to address Challenge 1, we extend the idea of prompt tuning to time series analysis and learn prompts to capture common and domain-specific information from all source domains. To handle Challenge 2, we introduce a conditional module for each source domain to generate prompts from time series input data. For Challenge 3, we propose two criteria to select good prompts, which are used to choose the most suitable source domain for domain adaptation. The efficacy and robustness of our proposed POND model are extensively validated through experiments across 50 scenarios encompassing four datasets. Experimental results demonstrate that our proposed POND model outperforms all state-of-the-art comparison methods by up to 66% on the F1-score.

In the context of long-tail classification on graphs, the vast majority of existing work primarily revolves around the development of model debiasing strategies, intending to mitigate class imbalances and enhance the overall performance. Despite the notable success, there is very limited literature that provides a theoretical tool for characterizing the behaviors of long-tail classes in graphs and gaining insight into generalization performance in real-world scenarios. To bridge this gap, we propose a generalization bound for long-tail classification on graphs by formulating the problem in the fashion of multi-task learning, i.e., each task corresponds to the prediction of one particular class. Our theoretical results show that the generalization performance of long-tail classification is dominated by the overall loss range and the task complexity. Building upon the theoretical findings, we propose a novel generic framework HierTail for long-tail classification on graphs. In particular, we start with a hierarchical task grouping module that allows us to assign related tasks into hypertasks and thus control the complexity of the task space; then, we further design a balanced contrastive learning module to adaptively balance the gradients of both head and tail classes to control the loss range across all tasks in a unified fashion. Extensive experiments demonstrate the effectiveness of HierTail in characterizing long-tail classes on real graphs, which achieves up to 12.9% improvement over the leading baseline method in balanced accuracy.

Imitation learning, which learns agent policy by mimicking expert demonstration, has shown promising results in many applications such as medical treatment regimes and self-driving vehicles. However, it remains a difficult task to interpret control policies learned by the agent. Difficulties mainly come from two aspects: 1) agents in imitation learning are usually implemented as deep neural networks, which are black-box models and lack interpretability; 2) the latent causal mechanism behind agents' decisions may vary along the trajectory, rather than staying static throughout time steps. To increase transparency and offer better interpretability of the neural agent, we propose to expose its captured knowledge in the form of a directed acyclic causal graph, with nodes being action and state variables and edges denoting the causal relations behind predictions. Furthermore, we design this causal discovery process to be state-dependent, enabling it to model the dynamics in latent causal graphs. Concretely, we conduct causal discovery from the perspective of Granger causality and propose a self-explainable imitation learning framework, CAIL. The proposed framework is composed of three parts: a dynamic causal discovery module, a causality encoding module, and a prediction module, and is trained in an end-to-end manner. After the model is learned, we can obtain causal relations among states and action variables behind its decisions, exposing policies learned by it. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of the proposed CAIL in learning the dynamic causal graphs for understanding the decision-making of imitation learning meanwhile maintaining high prediction accuracy.

Unsupervised anomaly detection approaches have been widely accepted in applications for industrial systems. Industrial systems often operate with multiple modes since they work for multiple purposes or under different conditions. In order to deal with the difficulty of anomaly detection due to multiple operating modes, multiple modeling strategies are employed. However, estimating the optimal set of models is a challenging problem due to the lack of supervision and computational burden. In this paper, we propose DeconAnomaly, a deep learning framework to estimate the optimal set of models using transfer learning for unsupervised anomaly detection under a multiple modeling strategy. It reduces computational burden with transfer learning and optimizes the number of models based on a surrogate metric of detection performance. The experimental results show clear advantages of DeconAnomaly.

Imitation learning that replicates experts' skills via their demonstrations has shown significant success in various decision-making tasks. However, two critical challenges still hinder the deployment of imitation learning techniques in real-world application scenarios. First, existing methods lack the intrinsic interpretability to explicitly explain the underlying rationale of the learned skill and thus making learned policy untrustworthy. Second, due to the scarcity of expert demonstrations from each end user (client), learning a policy based on different data silos is necessary but challenging in privacy-sensitive applications such as finance and healthcare. To this end, we present a privacy-preserved interpretable skill learning framework (FedSkill) that enables global policy learning to incorporate data from different sources and provides explainable interpretations to each local user without violating privacy and data sovereignty. Specifically, our proposed interpretable skill learning model can capture the varying patterns in the trajectories of expert demonstrations, and extract prototypical information as skills that provide implicit guidance for policy learning and explicit explanations in the reasoning process. Moreover, we design a novel aggregation mechanism coupled with the based skill learning model to preserve global information utilization and maintain local interpretability under the federated framework. Thoroughly experiments on three datasets and empirical studies demonstrate that our proposed FedSkill framework not only outperforms state-of-the-art imitation learning methods but also exhibits good interpretability under a federated setting. Our proposed FedSkill framework is the first attempt to bridge the gaps among federated learning, interpretable machine learning, and imitation learning.

Imitation learning has achieved great success in many sequential decision-making tasks, in which a neural agent is learned by imitating collected human demonstrations. However, existing algorithms typically require a large number of high-quality demonstrations that are difficult and expensive to collect. Usually, a trade-off needs to be made between demonstration quality and quantity in practice. Targeting this problem, in this work we consider the imitation of sub-optimal demonstrations, with both a small clean demonstration set and a large noisy set. Some pioneering works have been proposed, but they suffer from many limitations, e.g., assuming a demonstration to be of the same optimality throughout time steps and failing to provide any interpretation w.r.t knowledge learned from the noisy set. Addressing these problems, we propose \method by evaluating and imitating at the sub-demonstration level, encoding action primitives of varying quality into different skills. Concretely, SDIL consists of a high-level controller to discover skills and a skill-conditioned module to capture action-taking policies, and is trained following a two-phase pipeline by first discovering skills with all demonstrations and then adapting the controller to only the clean set. A mutual-information-based regularization and a dynamic sub-demonstration optimality estimator are designed to promote disentanglement in the skill space. Extensive experiments are conducted over two gym environments and a real-world healthcare dataset to demonstrate the superiority of SDIL in learning from sub-optimal demonstrations and its improved interpretability by examining learned skills.

Although many anomaly detection approaches have been developed for multivariate time series data, limited effort has been made in federated settings in which multivariate time series data are heterogeneously distributed among different edge devices while data sharing is prohibited. In this paper, we investigate the problem of federated unsupervised anomaly detection and present a Federated Exemplar-based Deep Neural Network (Fed-ExDNN) to conduct anomaly detection for multivariate time series data on different edge devices. Specifically, we first design an Exemplar-based Deep Neural network (ExDNN) for learning local time series representations based on their compatibility with an exemplar module which consists of hidden parameters learned to capture varieties of normal patterns on each edge device. Next, a constrained clustering mechanism (FedCC) is employed on the centralized server to align and aggregate the parameters of different local exemplar modules to obtain a unified global exemplar module. Finally, the global exemplar module is deployed together with a shared feature encoder to each edge device, and anomaly detection is conducted by examining the compatibility of testing data to the exemplar module. Fed-ExDNN captures local normal time series patterns with ExDNN and aggregates these patterns by FedCC, and thus can handle the heterogeneous data distributed over different edge devices simultaneously. Thoroughly empirical studies on six public datasets show that ExDNN and Fed-ExDNN can outperform state-of-the-art anomaly detection algorithms and federated learning techniques, respectively.

Personalized Federated Learning (PFL) which collaboratively trains a federated model while considering local clients under privacy constraints has attracted much attention. Despite its popularity, it has been observed that existing PFL approaches result in sub-optimal solutions when the joint distribution among local clients diverges. To address this issue, we present Federated Modular Network (FedMN), a novel PFL approach that adaptively selects sub-modules from a module pool to assemble heterogeneous neural architectures for different clients. FedMN adopts a light-weighted routing hypernetwork to model the joint distribution on each client and produce the personalized selection of the module blocks for each client. To reduce the communication burden in existing FL, we develop an efficient way to interact between the clients and the server. We conduct extensive experiments on the real-world test beds and the results show both effectiveness and efficiency of the proposed FedMN over the baselines.

It is critical and important to detect anomalies in event sequences, which becomes widely available in many application domains. Indeed, various efforts have been made to capture abnormal patterns from event sequences through sequential pattern analysis or event representation learning. However, existing approaches usually ignore the semantic information of event content. To this end, in this paper, we propose a self-attentive encoder-decoder transformer framework, Content-Aware Transformer CAT, for anomaly detection in event sequences. In CAT, the encoder learns preamble event sequence representations with content awareness, and the decoder embeds sequences under detection into a latent space, where anomalies are distinguishable. Specifically, the event content is first fed to a content-awareness layer, generating representations of each event. The encoder accepts preamble event representation sequence, generating feature maps. In the decoder, an additional token is added at the beginning of the sequence under detection, denoting the sequence status. A one-class objective together with sequence reconstruction loss is collectively applied to train our framework under the label efficiency scheme. Furthermore, CAT is optimized under a scalable and efficient setting. Finally, extensive experiments on three real-world datasets demonstrate the superiority of CAT.

Point-of-interest (POI) recommendation is an emerging area of research on location-based social networks to analyze user behaviors and contextual check-in information. For this problem, existing approaches, with shallow or deep architectures, have two major drawbacks. First, for these approaches, the attributes of individuals have been largely ignored. Therefore, it would be hard, if not impossible, to gather sufficient user attribute features to have complete coverage of possible motivation factors. Second, most existing models preserve the information of users or POIs by latent representations without explicitly highlighting salient factors or signals. Consequently, the trained models with unjustifiable parameters provide few persuasive rationales to explain why users favor or dislike certain POIs and what really causes a visit. To overcome these drawbacks, we propose GEAPR, a POI recommender that is able to interpret the POI prediction in an end-to-end fashion. Specifically, GEAPR learns user representations by aggregating different factors, such as structural context, neighbor impact, user attributes, and geolocation influence. GEAPR takes advantage of a triple attention mechanism to quantify the influences of different factors for each resulting recommendation and performs a thorough analysis of the model interpretability. Extensive experiments on real-world datasets demonstrate the effectiveness of the proposed model. GEAPR is deployed and under test on an internal web server. An example interface is presented to showcase its application on explainable POI recommendation.

In many high-stakes applications of machine learning models, outputting only predictions or providing statistical confidence is usually insufficient to gain trust from end users, who often prefer a transparent reasoning paradigm. Despite the recent encouraging developments on deep networks for sequential data modeling, due to the highly recursive functions, the underlying rationales of their predictions are difficult to explain. Thus, in this paper, we aim to develop a sequence modeling approach that explains its own predictions by breaking input sequences down into evidencing segments (i.e., sub-sequences) in its reasoning. To this end, we build our model upon convolutional neural networks, which, in their vanilla forms, associates local receptive fields with outputs in an obscure manner. To unveil it, we resort to case-based reasoning, and design prototype modules whose units (i.e., prototypes) resemble exemplar segments in the problem domain. Each prediction is obtained by combining the comparisons between the prototypes and the segments of an input. To enhance interpretability, we propose a training objective that delicately adapts the distribution of prototypes to the data distribution in latent spaces, and design an algorithm to map prototypes to human-understandable segments. Through extensive experiments in a variety of domains, we demonstrate that our model can achieve high interpretability generally, together with a competitive accuracy to the state-of-the-art approaches.

Aspect-based sentiment classification aims to predict sentimental polarities of one or multiple aspects in texts. As texts always contain a large proportion of task-irrelevant words, accurate alignment between aspects and their sentimental descriptions is the most crucial and challenging step. State-of-the-art approaches are mainly based on word-level attention learned from recurrent neural network variants (e.g., LSTM) or graph neural networks. From another view, these methods essentially weight and aggregate all possible alignments. However, this mechanism heavily relies on large-scale supervision training: without enough labels, it could easily overfit with difficulty in generalization. To address this challenge, we propose SentRL, a reinforcement learning-based framework for aspect-based sentiment classification. In this framework, input texts are transformed into their dependency graphs. Then, an agent is deployed to walk on the graphs, explores paths from target aspect nodes to their potential sentimental regions, and differentiates the effectiveness of different paths. By limiting the agent’s exploration budget, our method encourages the agent to skip task-irrelevant information and focus on the most effective paths for alignment purpose. Our method considerably reduces the impact of task-irrelevant words and improves generalization performance. Compared with competitive baseline methods, our approach achieves the highest performance on public benchmark datasets with up to 3.7% improvement.

Graph Neural Networks (GNNs) have shown to be powerful tools for graph analytics. The key idea is to recursively propagate and aggregate information along the edges of the given graph. Despite their success, however, the existing GNNs are usually sensitive to the quality of the input graph. Real-world graphs are often noisy and contain task-irrelevant edges, which may lead to suboptimal generalization performance in the learned GNN models. In this paper, we propose PTDNet, a parameterized topological denoising network, to improve the robustness and generalization performance of GNNs by learning to drop task-irrelevant edges. PTDNet prunes task-irrelevant edges by penalizing the number of edges in the sparsified graph with parameterized networks. To take into consideration the topology of the entire graph, the nuclear norm regularization is applied to impose the low-rank constraint on the resulting sparsified graph for better generalization. PTDNet can be used as a key component in GNN models to improve their performances on various tasks, such as node classification and link prediction. Experimental studies on both synthetic and benchmark datasets show that PTDNet can improve the performance of GNNs significantly and the performance gain becomes larger for more noisy datasets.

Accurate air turbulence forecasting can help airlines avoid hazardous turbulence, guide the routes that keep passengers safe, maximize efficiency, and reduce costs. Traditional turbulence forecasting approaches heavily rely on painstakingly customized turbulence indexes, which are less effective in dynamic and complex weather conditions. The recent availability of high-resolution weather data and turbulence records allows more accurate forecasting of the turbulence in a data-driven way. However, it is a non-trivial task for developing a machine learning based turbulence forecasting system due to two challenges: (1) Complex spatio-temporal correlations, turbulence is caused by air movement with complex spatio-temporal patterns, (2) Label scarcity, very limited turbulence labels can be obtained. To this end, in this paper, we develop a unified semi-supervised framework, T2-Net, to address the above challenges. Specifically, we first build an encoder-decoder paradigm based on the convolutional LSTM to model the spatio-temporal correlations. Then, to tackle the label scarcity problem, we propose a novel Dual Label Guessing method to take advantage of massive unlabeled turbulence data. It integrates complementary signals from the main Turbulence Forecasting task and the auxiliary Turbulence Detection task to generate pseudo-labels, which are dynamically utilized as additional training data. Finally, extensive experimental results on a real-world turbulence dataset validate the superiority of our method on turbulence forecasting.

Deep neural networks have achieved state-of-the-art performance in many fields including image classification. However, recent studies show these models are vulnerable to adversarial examples formed by adding small but intentional perturbations to clean examples. In this paper, we introduce a significant defense method against adversarial examples. The key idea is to leverage a super-resolution coding (SR-coding) network to eliminate noise from adversarial examples. Furthermore, to boost the effect of defending noise, we propose a novel hybrid approach that incorporates SR-coding and adversarial training to train robust neural networks. Experiments on benchmark datasets demonstrate the effectiveness of our method against both the state-of-the-art white-box attacks and black-box attacks. The proposed approach significantly improves defense performance and achieves up to 41.26% improvement based on the accuracy by ResNet18 on PGD white-box attack.

Click-through rate (CTR) prediction is a critical task in online advertising and marketing. For this problem, existing approaches, with shallow or deep architectures, have three major drawbacks. First, they typically lack persuasive rationales to explain the outcomes of the models. Unexplainable predictions and recommendations may be difficult to validate and thus unreliable and untrustworthy. In many applications, inappropriate suggestions may even bring severe consequences. Second, existing approaches have poor efficiency in analyzing high-order feature interactions. Third, the polysemy of feature interactions in different semantic subspaces is largely ignored. In this paper, we propose InterHAt that employs a Transformer with multi-head self-attention for feature learning. On top of that, hierarchical attention layers are utilized for predicting CTR while simultaneously providing interpretable insights of the prediction results. InterHAt captures high-order feature interactions by an efficient attentional aggregation strategy with low computational complexity. Extensive experiments on four public real datasets and one synthetic dataset demonstrate the effectiveness and efficiency of InterHAt.

Question routing (QR) aims at recommending newly posted questions to the potential answerers who are most likely to answer the questions. The existing approaches that learn users' expertise from their past question-answering activities usually suffer from challenges in two aspects: 1) multi-faceted expertise and 2) temporal dynamics in the answering behavior. This paper proposes a novel temporal context-aware model in multiple granularities of temporal dynamics that concurrently address the above challenges. Specifically, the temporal context-aware attention characterizes the answerer's multi-faceted expertise in terms of the questions' semantic and temporal information simultaneously. Moreover, the design of the multi-shift and multi-resolution module enables our model to handle temporal impact on different time granularities. Extensive experiments on six datasets from different domains demonstrate that the proposed model significantly outperforms competitive baseline models.

Recent developments in discovering dynamic treatment regimes (DTRs) have heightened the importance of deep reinforcement learning (DRL) which are used to recover the doctor’s treatment policies. However, existing DRL-based methods expose the following limitations: 1) supervised methods based on behavior cloning suffer from compounding errors; 2) the self-defined reward signals in reinforcement learning models are either too sparse or need clinical guidance; 3) only positive trajectories (e.g. survived patients) are considered in current imitation learning models, with negative trajectories (e.g. deceased patients) been largely ignored, which are examples of what not to do and could help the learned policy avoid repeating mistakes. To address these limitations, in this paper, we propose the adversarial cooperative imitation learning model, ACIL, to deduce the optimal dynamic treatment regimes that mimics the positive trajectories while differs from the negative trajectories. Specifically, two discriminators are used to help achieve this goal: an adversarial discriminator is designed to minimize the discrepancies between the trajectories generated from the policy and the positive trajectories, and a cooperative discriminator is used to distinguish the negative trajectories from the positive and generated trajectories. The reward signals from the discriminators are utilized to refine the policy for dynamic treatment regimes. Experiments on the publicly real-world medical data demonstrate that ACIL improves the likelihood of patient survival and provides better dynamic treatment regimes with the exploitation of information from both positive and negative trajectories.

Existing representation learning methods based on graph neural networks and their variants rely on the aggregation of neighborhood information, which makes it sensitive to noises in the graph, e.g. erroneous links between nodes, incorrect/missing node features. In this paper, we propose Graph Denoising Policy Network (short for GDPNet) to learn robust representations from noisy graph data through reinforcement learning. GDPNet first selects signal neighborhoods for each node, and then aggregates the information from the selected neighborhoods to learn node representations for the down-stream tasks. Specifically, in the signal neighborhood selection phase, GDPNet optimizes the neighborhood for each target node by formulating the process of removing noisy neighborhoods as a Markov decision process and learning a policy with task-specific rewards received from the representation learning phase. In the representation learning phase, GDPNet aggregates features from signal neighbors to generate node representations for down-stream tasks, and provides task-specific rewards to the signal neighbor selection phase. These two phases are jointly trained to select optimal sets of neighbors for target nodes with maximum cumulative task-specific rewards, and to learn robust representations for nodes. Experimental results on node classification task demonstrate the effectiveness of GDNet, outperforming the state-of-the-art graph representation learning methods on several well-studied datasets.

Given a network with the labels for a subset of nodes, transductive node classification targets to predict the labels for the remaining nodes in the network. This technique has been used in a variety of applications such as voxel functionality detection in brain network and group label prediction in social network. Most existing node classification approaches are performed in static networks. However, many real-world networks are dynamic and evolve over time. The dynamics of both node attributes and network topology jointly determine the node labels. In this paper, we study the problem of classifying the nodes in dynamic networks. The task is challenging for three reasons. First, it is hard to effectively learn the spatial and temporal information simultaneously. Second, the network evolution is complex. The evolving patterns lie in both node attributes and network topology. Third, for different networks or even different nodes in the same network, the node attributes, the neighborhood node representations and the network topology usually affect the node labels differently, it is desirable to assess the relative importance of different factors over evolutionary time scales. To address the challenges, we propose AdaNN, an adaptive neural network for transductive node classification. AdaNN learns node attribute information by aggregating the node and its neighbors, and extracts network topology information with a random walk strategy. The attribute information and topology information are further fed into two connected gated recurrent units to learn the spatio-temporal contextual information. Additionally, a triple attention module is designed to automatically model the different factors that influence the node representations. AdaNN is the first node classification model that is adaptive to different kinds of dynamic networks. Extensive experiments on real datasets demonstrate the effectiveness of AdaNN.

Network embedding aims to learn the low-dimensional representations/embeddings of vertices which preserve the structure and inherent properties of the networks. The resultant embeddings are beneficial to downstream tasks such as vertex classification and link prediction. A vast majority of real-world networks are coupled with a rich set of vertex attributes, which could be potentially complementary in learning better embeddings. Existing attributed network embedding models, with shallow or deep architectures, typically seek to match the representations in topology space and attribute space for each individual vertex by assuming that the samples from the two spaces are drawn uniformly. The assumption, however, can hardly be guaranteed in practice. Due to the intrinsic sparsity of sampled vertex sequences and incompleteness in vertex attributes, the discrepancy between the attribute space and the network topology space inevitably exists. Furthermore, the interactions among vertex attributes, a.k.a cross features, have been largely ignored by existing approaches. To address the above issues, in this paper, we propose Nettention, a self-attentive network embedding approach that can efficiently learn vertex embeddings on attributed network. Instead of sample-wise optimization, Nettention aggregates the two types of information through minimizing the difference between the representation distributions in the low-dimensional topology and attribute spaces. The joint inference is encapsulated in a generative adversarial training process, yielding better generalization performance and robustness. The learned distributions consider both locality-preserving and global reconstruction constraints which can be inferred from the learning of the adversarially regularized autoencoders. Additionally, a multi-head self-attention module is developed to explicitly model the attribute interactions. Extensive experiments on benchmark datasets have verified the effectiveness of the proposed Nettention model on a variety of tasks, including vertex classification and link prediction.

Co-clustering partitions instances and features simultaneously by leveraging the duality between them and it often yields impressive performance improvement over traditional clustering algorithms. The recent development in learning deep representations has demonstrated the advantage in extracting effective features. However, the research on leveraging deep learning frameworks for co-clustering is limited for two reasons: 1) current deep clustering approaches usually decouple feature learning and cluster assignment as two separate steps, which cannot yield the task-specific feature representation; 2) existing deep clustering approaches cannot learn representations for instances and features simultaneously. In this paper, we propose a deep learning model for co-clustering called DeepCC. DeepCC utilizes the deep autoencoder for dimension reduction, and employs a variant of Gaussian Mixture Model (GMM) to infer the cluster assignments. A mutual information loss is proposed to bridge the training of instances and features. DeepCC jointly optimizes the parameters of the deep autoencoder and the mixture model in an end-to-end fashion on both the instance and the feature spaces, which can help the deep autoencoder escape from local optima and the mixture model circumvent the Expectation-Maximization (EM) algorithm. To the best of our knowledge, DeepCC is the first deep learning model for co-clustering. Experimental results on various dataseis demonstrate the effectiveness of DeepCC.

Given a large number of low-quality heterogeneous categorical alerts collected from an anomaly detection system, how to characterize the complex relationships between different alerts and deliver trustworthy rankings to end users? While existing techniques focus on either mining alert patterns or filtering out false positive alerts, it can be more advantageous to consider the two perspectives simultaneously in order to improve detection accuracy and better understand abnormal system behaviors. In this paper, we propose CAR, a collaborative alert ranking framework that exploits both temporal and content correlations from heterogeneous categorical alerts. CAR first builds a hierarchical Bayesian model to capture both short-term and long-term dependencies in each alert sequence. Then, an entity embedding-based model is proposed to learn the content correlations between alerts via their heterogeneous categorical attributes. Finally, by incorporating both temporal and content dependencies into a unified optimization framework, CAR ranks both alerts and their corresponding alert patterns. Our experiments-using both synthetic and real-world enterprise security alert data-show that CAR can accurately identify true positive alerts and successfully reconstruct the attack scenarios at the same time.

Local community detection, which aims to find a target community containing a set of query nodes, has recently drawn intense research interest. The existing local community detection methods usually assume all query nodes are from the same community and only find a single target community. This is a strict requirement and does not allow much flexibility. In many real-world applications, however, we may not have any prior knowledge about the community memberships of the query nodes, and different query nodes may be from different communities. To address this limitation of the existing methods, we propose a novel memory-based random walk method, MRW, that can simultaneously identify multiple target local communities to which the query nodes belong. In MRW, each query node is associated with a random walker. Different from commonly used memoryless random walk models, MRW records the entire visiting history of each walker. The visiting histories of walkers can help unravel whether they are from the same community or not. Intuitively, walkers with similar visiting histories are more likely to be in the same community. Moreover, MRW allows walkers with similar visiting histories to reinforce each other so that they can better capture the community structure instead of being biased to the query nodes. We provide rigorous theoretical foundation for the proposed method and develop efficient algorithms to identify multiple target local communities simultaneously. Comprehensive experimental evaluations on a variety of real-world datasets demonstrate the effectiveness and efficiency of the proposed method.

Multivariate time series data are becoming increasingly common in numerous real world applications, e.g., power plant monitoring, health care, wearable devices, automobile, etc. As a result, multivariate time series retrieval, i.e., given the current multivariate time series segment, how to obtain its relevant time series segments in the historical data (or in the database), attracts significant amount of interest in many fields. Building such a system, however, is challenging since it requires a compact representation of the raw time series which can explicitly encode the temporal dynamics as well as the correlations (interactions) between different pairs of time series (sensors). Furthermore, it requires query efficiency and expects a returned ranking list with high precision on the top. Despite the fact that various approaches have been developed, few of them can jointly resolve these two challenges. To cope with this issue, in this paper we propose a Deep r-th root of Rank Supervised Joint Binary Embedding (Deep r-RSJBE) to perform multivariate time series retrieval. Given a raw multivariate time series segment, we employ Long Short-Term Memory (LSTM) units to encode the temporal dynamics and utilize Convolutional Neural Networks (CNNs) to encode the correlations (interactions) between different pairs of time series (sensors). Subsequently, a joint binary embedding is pursued to incorporate both the temporal dynamics and the correlations. Finally, we develop a novel r-th root ranking loss to optimize the precision at the top of a Hamming distance ranking list. Thoroughly empirical studies based upon three publicly available time series datasets demonstrate the effectiveness and the efficiency of Deep r-RSJBE.

Massive and dynamic networks arise in many practical applications such as social media, security and public health. Given an evolutionary network, it is crucial to detect structural anomalies, such as vertices and edges whose "behaviors'' deviate from underlying majority of the network, in a real-time fashion. Recently, network embedding has proven a powerful tool in learning the low-dimensional representations of vertices in networks that can capture and preserve the network structure. However, most existing network embedding approaches are designed for static networks, and thus may not be perfectly suited for a dynamic environment in which the network representation has to be constantly updated. In this paper, we propose a novel approach, NetWalk, for anomaly detection in dynamic networks by learning network representations which can be updated dynamically as the network evolves. We first encode the vertices of the dynamic network to vector representations by clique embedding, which jointly minimizes the pairwise distance of vertex representations of each walk derived from the dynamic networks, and the deep autoencoder reconstruction error serving as a global regularization. The vector representations can be computed with constant space requirements using reservoir sampling. On the basis of the learned low-dimensional vertex representations, a clustering-based technique is employed to incrementally and dynamically detect network anomalies. Compared with existing approaches, NetWalk has several advantages: 1) the network embedding can be updated dynamically, 2) streaming network nodes and edges can be encoded efficiently with constant memory space usage, 3). flexible to be applied on different types of networks, and 4) network anomalies can be detected in real-time. Extensive experiments on four real datasets demonstrate the effectiveness of NetWalk.

The problem of network representation learning, also known as network embedding, arises in many machine learning tasks assuming that there exist a small number of variabilities in the vertex representations which can capture the "semantics" of the original network structure. Most existing network embedding models, with shallow or deep architectures, learn vertex representations from the sampled vertex sequences such that the low-dimensional embeddings preserve the locality property and/or global reconstruction capability. The resultant representations, however, are difficult for model generalization due to the intrinsic sparsity of sampled sequences from the input network. As such, an ideal approach to address the problem is to generate vertex representations by learning a probability density function over the sampled sequences. However, in many cases, such a distribution in a low-dimensional manifold may not always have an analytic form. In this study, we propose to learn the network representations with adversarially regularized autoencoders (NetRA). NetRA learns smoothly regularized vertex representations that well capture the network structure through jointly considering both locality-preserving and global reconstruction constraints. The joint inference is encapsulated in a generative adversarial training process to circumvent the requirement of an explicit prior distribution, and thus obtains better generalization performance. We demonstrate empirically how well key properties of the network structure are captured and the effectiveness of NetRA on a variety of tasks, including network reconstruction, link prediction, and multi-label classification.

Network embedding aims to learn a low-dimensional vector representation for each node in the social and information networks, with the constraint to preserve network structures. Most existing methods focus on single network embedding, ignoring the relationship between multiple networks. In many real-world applications, however, multiple networks may contain complementary information, which can lead to further refined node embeddings. Thus, in this paper, we propose a novel multi-network embedding method, DMNE. DMNE is flexible. It allows different networks to have different sizes, to be (un)weighted and (un)directed. It leverages multiple networks via cross-network relationships between nodes in different networks, which may form many-to-many node mappings, and be associated with weights. To model the non-linearity of the network data, we develop DMNE to have a new deep learning architecture, which coordinates multiple neural networks (one for each input network data) with a co-regularized loss function. With multiple layers of non-linear mappings, DMNE progressively transforms each input network to a highly non-linear latent space, and in the meantime, adapts different spaces to each other through a co-regularized learning schema. Extensive experimental results on real-life datasets demonstrate the effectiveness of our method.

Joint clustering of multiple networks has been shown to be more accurate than performing clustering on individual networks separately. This is because multi-network clustering algorithms typically assume there is a common clustering structure shared by all networks, and different networks can provide compatible and complementary information for uncovering this underlying clustering structure. However, this assumption is too strict to hold in many emerging applications, where multiple networks usually have diverse data distributions. More popularly, the networks in consideration belong to different underlying groups. Only networks in the same underlying group share similar clustering structures. Better clustering performance can be achieved by considering such groups differently. As a result, an ideal method should be able to automatically detect network groups so that networks in the same group share a common clustering structure. To address this problem, we propose a new method, ComClus, to simultaneously group and cluster multiple networks. ComClus is novel in combining the clustering approach of non-negative matrix factorization (NMF) and the feature subspace learning approach of metric learning. Specifically, it treats node clusters as features of networks and learns proper subspaces from such features to differentiate different network groups. During the learning process, the two procedures of network grouping and clustering are coupled and mutually enhanced. Moreover, ComClus can effectively leverage prior knowledge on how to group networks such that network grouping can be conducted in a semi-supervised manner. This will enable users to guide the grouping process using domain knowledge so that network clustering accuracy can be further boosted. Extensive experimental evaluations on a variety of synthetic and real datasets demonstrate the effectiveness and scalability of the proposed method.

Accurately identifying time-invariant operational relationships among different components is critical to autonomic management of complex manufactural systems. In this paper, we collect time series of sensor readings from manufacturing systems, and propose a solution leveraging Sparse Group LASSO to discover structured pairwise nonlinear relationships and quantify them by mathematical formulas. We consider both real-life operational patterns and underlying physical reactions inside the manufactural systems, which leads to a learning formulation for combined periodic and aperiodic system behaviors. An accelerated gradient descent algorithm is developed to efficiently solve the related optimization problem. We estimate sample correlations between proximal time points to improve the accuracy of the discovered relationships and the nonlinear quantitative formulas. The method is evaluated using both synthetic and real-world datasets, which shows superior performance over the state of the art in discovering nonlinear relationships in manufactural systems.

Local community detection (or local clustering) is of fundamental importance in large network analysis. Random walk based methods have been routinely used in this task. Most existing random walk methods are based on the single-walker model. However, without any guidance, a single-walker may not be adequate to effectively capture the local cluster. In this paper, we study a multi-walker chain (MWC) model, which allows multiple walkers to explore the network. Each walker is influenced (or pulled back) by all other walkers when deciding the next steps. This helps the walkers to stay as a group and within the cluster. We introduce two measures based on the mean and standard deviation of the visiting probabilities of the walkers. These measures not only can accurately identify the local cluster, but also help detect the cluster center and boundary, which cannot be achieved by the existing single-walker methods. We provide rigorous theoretical foundation for MWC, and devise efficient algorithms to compute it. Extensive experimental results on a variety of real-world networks demonstrate that MWC outperforms the state-of-the-art local community detection methods by a large margin.

Sparse Discriminant Analysis (SDA) has been widely used to improve the performance of classical Fisher's Linear Discriminant Analysis in supervised metric learning, feature selection and classification. With the increasing needs of distributed data collection, storage and processing, enabling the Sparse Discriminant Learning to embrace the Multi-Party distributed computing environments becomes an emerging research topic. This paper proposes a novel Multi-Party SDA algorithm, which can learn SDA models effectively without sharing any raw dataand basic statistics among machines. The proposed algorithm 1) leverages the direct estimation of SDA [1] to derive a distributed loss function for the discriminant learning, 2) parameterizes the distributed loss function with local/global estimates through bootstrapping, and 3) approximates a global estimation of linear discriminant projection vector by optimizing the "distributed bootstrapping loss function" with gossip-based stochastic gradient descent. Experimental results on both synthetic and real-world benchmark datasets show that our algorithm can compete with the centralized SDA with similar performance, and significantly outperforms the most recent distributed SDA [2] in terms of accuracy and F1-score.

Complex systems are prevalent in many fields such as finance, security and industry. A fundamental problem in system management is to perform diagnosis in case of system failure such that the causal anomalies, i.e., root causes, can be identified for system debugging and repair. Recently, invariant network has proven a powerful tool in characterizing complex system behaviors. In an invariant network, a node represents a system component, and an edge indicates a stable interaction between two components. Recent approaches have shown that by modeling fault propagation in the invariant network, causal anomalies can be effectively discovered. Despite their success, the existing methods have a major limitation: they typically assume there is only a single and global fault propagation in the entire network. However, in real-world large-scale complex systems, it's more common for multiple fault propagations to grow simultaneously and locally within different node clusters and jointly define the system failure status. Inspired by this key observation, we propose a two-phase framework to identify and rank causal anomalies. In the first phase, a probabilistic clustering is performed to uncover impaired node clusters in the invariant network. Then, in the second phase, a low-rank network diffusion model is designed to backtrack causal anomalies in different impaired clusters. Extensive experimental results on real-life datasets demonstrate the effectiveness of our method.

Linear Discriminant Analysis (LDA) is widely-used for supervised dimension reduction and linear classification. Classical LDA, however, suffers from the ill-posed estimation problem on data with high dimension and low sample size (HDLSS). To cope with this problem, in this paper, we propose an Adaptive Wishart Discriminant Analysis (AWDA) for classification, that makes predictions in an ensemble way. Comparing to existing approaches, AWDA has two advantages: 1) leveraging theWishart distribution, AWDA ensembles multiple LDA classifiers parameterized by the sampled covariance matrices via a Bayesian Voting Scheme, which theoretically improves the robustness of classification, compared to LDA classifiers using a single (probably ill-posed) covariance matrix estimator; 2) AWDA updates the weights for voting optimally to adapt the local information of each new input data, so as to enable the nonlinear classification. Theoretical analysis indicates that AWDA guarantees a close approximation to the optimal Bayesian inference and thus achieves robust performance on high dimensional data. Extensive experiments on real-world datasets show that our approach outperforms state-of-the-art algorithms by a large margin.

Detecting system anomalies is an important problem in many fields such as security, fault management, and industrial optimization. Recently, invariant network has shown to be powerful in characterizing complex system behaviours. In the invariant network, a node represents a system component and an edge indicates a stable, significant interaction between two components. Structures and evolutions of the invariance network, in particular the vanishing correlations, can shed important light on locating causal anomalies and performing diagnosis. However, existing approaches to detect causal anomalies with the invariant network often use the percentage of vanishing correlations to rank possible casual components, which have several limitations: (1) fault propagation in the network is ignored, (2) the root casual anomalies may not always be the nodes with a high percentage of vanishing correlations, (3) temporal patterns of vanishing correlations are not exploited for robust detection, and (4) prior knowledge on anomalous nodes are not exploited for (semi-)supervised detection. To address these limitations, in this article we propose a network diffusion based framework to identify significant causal anomalies and rank them. Our approach can effectively model fault propagation over the entire invariant network and can perform joint inference on both the structural and the time-evolving broken invariance patterns. As a result, it can locate high-confidence anomalies that are truly responsible for the vanishing correlations and can compensate for unstructured measurement noise in the system. Moreover, when the prior knowledge on the anomalous status of some nodes are available at certain time points, our approach is able to leverage them to further enhance the anomaly inference accuracy. When the prior knowledge is noisy, our approach also automatically learns reliable information and reduces impacts from noises. By performing extensive experiments on synthetic datasets, bank information system datasets, and coal plant cyber-physical system datasets, we demonstrate the effectiveness of our approach.

Joint clustering of multiple networks has been shown to be more accurate than performing clustering on individual networks separately. Many multi-view and multi-domain network clustering methods have been developed for joint multi-network clustering. These methods typically assume there is a common clustering structure shared by all networks, and different networks can provide complementary information on this underlying clustering structure. However, this assumption is too strict to hold in many emerging real-life applications, where multiple networks have diverse data distributions. More popularly, the networks in consideration belong to different underlying groups. Only networks in the same underlying group share similar clustering structures. Better clustering performance can be achieved by considering such groups differently. As a result, an ideal method should be able to automatically detect network groups so that networks in the same group share a common clustering structure. To address this problem, we propose a novel method, ComClus, to simultaneously group and cluster multiple networks. ComClus treats node clusters as features of networks and uses them to differentiate different network groups. Network grouping and clustering are coupled and mutually enhanced during the learning process. Extensive experimental evaluation on a variety of synthetic and real datasets demonstrates the effectiveness of our method.

Modern world has witnessed a dramatic increase in our ability to collect, transmit and distribute real-time monitoring and surveillance data from large-scale information systems and cyber-physical systems. Detecting system anomalies thus attracts significant amount of interest in many fields such as security, fault management, and industrial optimization. Recently, invariant network has shown to be a powerful way in characterizing complex system behaviours. In the invariant network, a node represents a system component and an edge indicates a stable, significant interaction between two components. Structures and evolutions of the invariance network, in particular the vanishing correlations, can shed important light on locating causal anomalies and performing diagnosis. However, existing approaches to detect causal anomalies with the invariant network often use the percentage of vanishing correlations to rank possible casual components, which have several limitations: 1) fault propagation in the network is ignored; 2) the root casual anomalies may not always be the nodes with a high-percentage of vanishing correlations; 3) temporal patterns of vanishing correlations are not exploited for robust detection. To address these limitations, in this paper we propose a network diffusion based framework to identify significant causal anomalies and rank them. Our approach can effectively model fault propagation over the entire invariant network, and can perform joint inference on both the structural, and the time-evolving broken invariance patterns. As a result, it can locate high-confidence anomalies that are truly responsible for the vanishing correlations, and can compensate for unstructured measurement noise in the system. Extensive experiments on synthetic datasets, bank information system datasets, and coal plant cyber-physical system datasets demonstrate the effectiveness of our approach.

Multi-view graph clustering aims to enhance clustering performance by integrating heterogeneous information collected in different domains. Each domain provides a different view of the data instances. Leveraging cross-domain information has been demonstrated an effective way to achieve better clustering results. Despite the previous success, existing multi-view graph clustering methods usually assume that different views are available for the same set of instances. Thus, instances in different domains can be treated as having strict one-to-one relationship. In many real-life applications, however, data instances in one domain may correspond to multiple instances in another domain. Moreover, relationships between instances in different domains may be associated with weights based on prior (partial) knowledge. In this article, we propose a flexible and robust framework, Co-regularized Graph Clustering (CGC), based on non-negative matrix factorization (NMF), to tackle these challenges. CGC has several advantages over the existing methods. First, it supports many-to-many cross-domain instance relationship. Second, it incorporates weight on cross-domain relationship. Third, it allows partial cross-domain mapping so that graphs in different domains may have different sizes. Finally, it provides users with the extent to which the cross-domain instance relationship violates the in-domain clustering structure, and thus enables users to re-evaluate the consistency of the relationship. We develop an efficient optimization method that guarantees to find the global optimal solution with a given confidence requirement. The proposed method can automatically identify noisy domains and assign smaller weights to them. This helps to obtain optimal graph partition for the focused domain. Extensive experimental results on UCI benchmark datasets, newsgroup datasets, and biological interaction networks demonstrate the effectiveness of our approach.

Network clustering is an important problem thathas recently drawn a lot of attentions. Most existing workfocuses on clustering nodes within a single network. In manyapplications, however, there exist multiple related networks, inwhich each network may be constructed from a different domainand instances in one domain may be related to instances in otherdomains. In this paper, we propose a robust algorithm, MCA, formulti-network clustering that takes into account cross-domain relationshipsbetween instances. MCA has several advantages overthe existing single network clustering methods. First, it is ableto detect associations between clusters from different domains, which, however, is not addressed by any existing methods. Second, it achieves more consistent clustering results on multiple networksby leveraging the duality between clustering individual networksand inferring cross-network cluster alignment. Finally, it providesa multi-network clustering solution that is more robust to noiseand errors. We perform extensive experiments on a variety ofreal and synthetic networks to demonstrate the effectiveness andefficiency of MCA.

Similarity query is a fundamental problem in database, data mining and information retrieval research. Recently, querying incomplete data has attracted extensive attention as it poses new challenges to traditional querying techniques. The existing work on querying incomplete data addresses the problem where the data values on certain dimensions are unknown. However, in many real-life applications, such as data collected by a sensor network in a noisy environment, not only the data values but also the dimension information may be missing. In this work, we propose to investigate the problem of similarity search on dimension incomplete data. A probabilistic framework is developed to model this problem so that the users can find objects in the database that are similar to the query with probability guarantee. Missing dimension information poses great computational challenge, since all possible combinations of missing dimensions need to be examined when evaluating the similarity between the query and the data objects. We develop the lower and upper bounds of the probability that a data object is similar to the query. These bounds enable efficient filtering of irrelevant data objects without explicitly examining all missing dimension combinations. A probability triangle inequality is also employed to further prune the search space and speed up the query process. The proposed probabilistic framework and techniques can be applied to both whole and subsequence queries. Extensive experimental results on real-life data sets demonstrate the effectiveness and efficiency of our approach.

Multi-view graph clustering aims to enhance clustering performance by integrating heterogeneous information collected in different domains. Each domain provides a different view of the data instances. Leveraging cross-domain information has been demonstrated an effective way to achieve better clustering results. Despite the previous success, existing multi-view graph clustering methods usually assume that different views are available for the same set of instances. Thus instances in different domains can be treated as having strict one-to-one relationship. In many real-life applications, however, data instances in one domain may correspond to multiple instances in another domain. Moreover, relationships between instances in different domains may be associated with weights based on prior (partial) knowledge. In this paper, we propose a flexible and robust framework, CGC (Co-regularized Graph Clustering), based on non-negative matrix factorization (NMF), to tackle these challenges. CGC has several advantages over the existing methods. First, it supports many-to-many cross-domain instance relationship. Second, it incorporates weight on cross-domain relationship. Third, it allows partial cross-domain mapping so that graphs in different domains may have different sizes. Finally, it provides users with the extent to which the cross-domain instance relationship violates the in-domain clustering structure, and thus enables users to re-evaluate the consistency of the relationship. Extensive experimental results on UCI benchmark data sets, newsgroup data sets and biological interaction networks demonstrate the effectiveness of our approach.

Two dimensional contingency tables or co-occurrence matrices arise frequently in various important applications such as text analysis and web-log mining. As a fundamental research topic, co-clustering aims to generate a meaningful partition of the contingency table to reveal hidden relationships between rows and columns. Traditional co-clustering algorithms usually produce a predefined number of flat partition of both rows and columns, which do not reveal relationship among clusters. To address this limitation, hierarchical co-clustering algorithms have attracted a lot of research interests recently. Although successful in various applications, the existing hierarchial co-clustering algorithms are usually based on certain heuristics and do not have solid theoretical background.

Transfer learning aims to leverage the knowledge in the source domain to facilitate the learning tasks in the target domain. It has attracted extensive research interests recently due to its effectiveness in a wide range of applications. The general idea of the existing methods is to utilize the common latent structure shared across domains as the bridge for knowledge transfer. These methods usually model the common latent structure by using either the marginal distribution or the conditional distribution. However, without exploring the duality between these two distributions, these single bridge methods may not achieve optimal capability of knowledge transfer. In this paper, we propose a novel approach, Dual Transfer Learning (DTL), which simultaneously learns the marginal and conditional distributions, and exploits the duality between them in a principled way. The key idea behind DTL is that learning one distribution can help to learn the other. This duality property leads to mutual reinforcement when adapting both distributions across domains to transfer knowledge. The proposed method is formulated as an optimization problem based on joint nonnegative matrix trifactorizations (NMTF). The two distributions are learned from the decomposed latent factors that exhibit the duality property. An efficient alternating minimization algorithm is developed to solve the optimization problem with convergence guarantee. Extensive experimental results demonstrate that DTL is more effective than alternative transfer learning methods.

Transfer learning targets to leverage knowledge from one domain for tasks in a new domain. It finds abundant applications, such as text/sentiment classification. Many previous works are based on cluster analysis, which assume some common clusters shared by both domains. They mainly focus on the one-to-one cluster correspondence to bridge different domains. However, such a correspondence scheme might be too strong for real applications where each cluster in one domain corresponds to many clusters in the other domain. In this paper, we propose a Cluster Correspondence Inference (CCI) method to iteratively infer many-to-many correspondence among clusters from different domains. Specifically, word clusters and document clusters are exploited for each domain using nonnegative matrix factorization, then the word clusters from different domains are corresponded in a many-to-many scheme, with the help of shared word space as a bridge. These two steps are run iteratively and label information is transferred from source domain to target domain through the inferred cluster correspondence. Experiments on various real data sets demonstrate that our method outperforms several state-of-the-art approaches for cross-domain text classification.

Retrieving similar data has drawn many research efforts in the literature due to its importance in data mining, database and information retrieval. This problem is challenging when the data is incomplete. In previous research, data incompleteness refers to the fact that data values for some dimensions are unknown. However, in many practical applications (e.g., data collection by sensor network under bad environment), not only data values but even data dimension information may also be missing, which will make most similarity query algorithms infeasible. In this work, we propose the novel similarity query problem on dimension incomplete data and adopt a probabilistic framework to model this problem. For this problem, users can give a distance threshold and a probability threshold to specify their retrieval requirements. The distance threshold is used to specify the allowed distance between query and data objects and the probability threshold is used to require that the retrieval results satisfy the distance condition at least with the given probability. Instead of enumerating all possible cases to recover the missed dimensions, we propose an efficient approach to speed up the retrieval process by leveraging the inherent relations between query and dimension incomplete data objects. During the query process, we estimate the lower/upper bounds of the probability that the query is satisfied by a given data object, and utilize these bounds to filter irrelevant data objects efficiently. Furthermore, a probability triangle inequality is proposed to further speed up query processing. According to our experiments on real data sets, the proposed similarity query method is verified to be effective and efficient on dimension incomplete data.