This paper uses Gaussian mixture model instead of linear Gaussian model to fit the distribution of every node in Bayesian network. We will explain why and how we use Gaussian mixture models in Bayesian network. Meanwhile we propose a new method, called double iteration algorithm, to optimize the mixture model, the double iteration algorithm combines the expectation maximization algorithm and gradient descent algorithm, and it performs perfectly on the Bayesian network with mixture models. In experiments we test the Gaussian mixture model and the optimization algorithm on different graphs which is generated by different structure learning algorithm on real data sets, and give the details of every experiment.
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Network inference has been extensively studied in several fields, such as systems biology and social sciences. Learning network topology and internal dynamics is essential to understand mechanisms of complex systems. In particular, sparse topologies and stable dynamics are fundamental features of many real-world continuous-time (CT) networks. Given that usually only a partial set of nodes are able to observe, in this paper, we consider linear CT systems to depict networks since they can model unmeasured nodes via transfer functions. Additionally, measurements tend to be noisy and with low and varying sampling frequencies. For this reason, we consider CT models since discrete-time approximations often require fine-grained measurements and uniform sampling steps. The developed method applies dynamical structure functions (DSFs) derived from linear stochastic differential equations (SDEs) to describe networks of measured nodes. A numerical sampling method, preconditioned Crank-Nicolson (pCN), is used to refine coarse-grained trajectories to improve inference accuracy. The convergence property of the developed method is robust to the dimension of data sources. Monte Carlo simulations indicate that the developed method outperforms state-of-the-art methods including group sparse Bayesian learning (GSBL), BINGO, kernel-based methods, dynGENIE3, GENIE3, and ARNI. The simulations include random and ring networks, and a synthetic biological network. These are challenging networks, suggesting that the developed method can be applied under a wide range of contexts, such as gene regulatory networks, social networks, and communication systems.
Neural network models have become the leading solution for a large variety of tasks, such as classification, language processing, protein folding, and others. However, their reliability is heavily plagued by adversarial inputs: small input perturbations that cause the model to produce erroneous outputs. Adversarial inputs can occur naturally when the system's environment behaves randomly, even in the absence of a malicious adversary, and are a severe cause for concern when attempting to deploy neural networks within critical systems. In this paper, we present a new statistical method, called Robustness Measurement and Assessment (RoMA), which can measure the expected robustness of a neural network model. Specifically, RoMA determines the probability that a random input perturbation might cause misclassification. The method allows us to provide formal guarantees regarding the expected frequency of errors that a trained model will encounter after deployment. Our approach can be applied to large-scale, black-box neural networks, which is a significant advantage compared to recently proposed verification methods. We apply our approach in two ways: comparing the robustness of different models, and measuring how a model's robustness is affected by the magnitude of input perturbation. One interesting insight obtained through this work is that, in a classification network, different output labels can exhibit very different robustness levels. We term this phenomenon categorial robustness. Our ability to perform risk and robustness assessments on a categorial basis opens the door to risk mitigation, which may prove to be a significant step towards neural network certification in safety-critical applications.
Major depressive disorder (MDD) requires study of brain functional connectivity alterations for patients, which can be uncovered by resting-state functional magnetic resonance imaging (rs-fMRI) data. We consider the problem of identifying alterations of brain functional connectivity for a single MDD patient. This is particularly difficult since the amount of data collected during an fMRI scan is too limited to provide sufficient information for individual analysis. Additionally, rs-fMRI data usually has the characteristics of incompleteness, sparsity, variability, high dimensionality and high noise. To address these problems, we proposed a multitask Gaussian Bayesian network (MTGBN) framework capable for identifying individual disease-induced alterations for MDD patients. We assume that such disease-induced alterations show some degrees of similarity with the tool to learn such network structures from observations to understanding of how system are structured jointly from related tasks. First, we treat each patient in a class of observation as a task and then learn the Gaussian Bayesian networks (GBNs) of this data class by learning from all tasks that share a default covariance matrix that encodes prior knowledge. This setting can help us to learn more information from limited data. Next, we derive a closed-form formula of the complete likelihood function and use the Monte-Carlo Expectation-Maximization(MCEM) algorithm to search for the approximately best Bayesian network structures efficiently. Finally, we assess the performance of our methods with simulated and real-world rs-fMRI data.
Domain generalization (DG) aims at learning generalizable models under distribution shifts to avoid redundantly overfitting massive training data. Previous works with complex loss design and gradient constraint have not yet led to empirical success on large-scale benchmarks. In this work, we reveal the mixture-of-experts (MoE) model's generalizability on DG by leveraging to distributively handle multiple aspects of the predictive features across domains. To this end, we propose Sparse Fusion Mixture-of-Experts (SF-MoE), which incorporates sparsity and fusion mechanisms into the MoE framework to keep the model both sparse and predictive. SF-MoE has two dedicated modules: 1) sparse block and 2) fusion block, which disentangle and aggregate the diverse learned signals of an object, respectively. Extensive experiments demonstrate that SF-MoE is a domain-generalizable learner on large-scale benchmarks. It outperforms state-of-the-art counterparts by more than 2% across 5 large-scale DG datasets (e.g., DomainNet), with the same or even lower computational costs. We further reveal the internal mechanism of SF-MoE from distributed representation perspective (e.g., visual attributes). We hope this framework could facilitate future research to push generalizable object recognition to the real world. Code and models are released at //github.com/Luodian/SF-MoE-DG.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork linearized Laplace: We first obtain a MAP estimate of all weights and then infer a full-covariance Gaussian posterior over a subnetwork. We propose a subnetwork selection strategy that aims to maximally preserve the model's predictive uncertainty. Empirically, our approach is effective compared to ensembles and less expressive posterior approximations over full networks.
Graph Convolutional Networks (GCNs) have received increasing attention in recent machine learning. How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly optimizing the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the GEneralized Multi-relational Graph Convolutional Networks (GEM-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge-base embedding methods, and goes beyond. Our theoretical analysis shows that GEM-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of GEM-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.
This paper aims at revisiting Graph Convolutional Neural Networks by bridging the gap between spectral and spatial design of graph convolutions. We theoretically demonstrate some equivalence of the graph convolution process regardless it is designed in the spatial or the spectral domain. The obtained general framework allows to lead a spectral analysis of the most popular ConvGNNs, explaining their performance and showing their limits. Moreover, the proposed framework is used to design new convolutions in spectral domain with a custom frequency profile while applying them in the spatial domain. We also propose a generalization of the depthwise separable convolution framework for graph convolutional networks, what allows to decrease the total number of trainable parameters by keeping the capacity of the model. To the best of our knowledge, such a framework has never been used in the GNNs literature. Our proposals are evaluated on both transductive and inductive graph learning problems. Obtained results show the relevance of the proposed method and provide one of the first experimental evidence of transferability of spectral filter coefficients from one graph to another. Our source codes are publicly available at: //github.com/balcilar/Spectral-Designed-Graph-Convolutions
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.
Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to increase the receptive field, we propose a novel deep Hierarchical Graph Convolutional Network (H-GCN) for semi-supervised node classification. H-GCN first repeatedly aggregates structurally similar nodes to hyper-nodes and then refines the coarsened graph to the original to restore the representation for each node. Instead of merely aggregating one- or two-hop neighborhood information, the proposed coarsening procedure enlarges the receptive field for each node, hence more global information can be learned. Comprehensive experiments conducted on public datasets demonstrate the effectiveness of the proposed method over the state-of-art methods. Notably, our model gains substantial improvements when only a few labeled samples are provided.
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