Discretize Relaxed Solution of Spectral Clustering via a Non-Heuristic Algorithm

Spectral clustering and its extensions usually consist of two steps: (1) constructing a graph and computing the relaxed solution; (2) discretizing relaxed solutions. Although the former has been extensively investigated, the discretization techniques are mainly heuristic methods, e.g., k-means, spectral rotation. Unfortunately, the goal of the existing methods is not to find a discrete solution that minimizes the original objective. In other words, the primary drawback is the neglect of the original objective when computing the discrete solution. Inspired by the first-order optimization algorithms, we propose to develop a first-order term to bridge the original problem and discretization algorithm, which is the first non-heuristic to the best of our knowledge. Since the non-heuristic method is aware of the original graph cut problem, the final discrete solution is more reliable and achieves the preferable loss value. We also theoretically show that the continuous optimum is beneficial to discretization algorithms though simply finding its closest discrete solution is an existing heuristic algorithm which is also unreliable. Sufficient experiments significantly show the superiority of our method.

Variational Positive-incentive Noise: How Noise Benefits Models

A large number of works aim to alleviate the impact of noise due to an underlying conventional assumption of the negative role of noise. However, some existing works show that the assumption does not always hold. In this paper, we investigate how to benefit the classical models by random noise under the framework of Positive-incentive Noise (Pi-Noise). Since the ideal objective of Pi-Noise is intractable, we propose to optimize its variational bound instead, namely variational Pi-Noise (VPN). With the variational inference, a VPN generator implemented by neural networks is designed for enhancing base models and simplifying the inference of base models, without changing the architecture of base models. Benefiting from the independent design of base models and VPN generators, the VPN generator can work with most existing models. From the experiments, it is shown that the proposed VPN generator can improve the base models. It is appealing that the trained variational VPN generator prefers to blur the irrelevant ingredients in complicated images, which meets our expectations.

Decouple Graph Neural Networks: Train Multiple Simple GNNs Simultaneously Instead of One

Graph neural networks (GNN) suffer from severe inefficiency. It is mainly caused by the exponential growth of node dependency with the increase of layers. It extremely limits the application of stochastic optimization algorithms so that the training of GNN is usually time-consuming. To address this problem, we propose to decouple a multi-layer GNN as multiple simple modules for more efficient training, which is comprised of classical forward training (FT)and designed backward training (BT). Under the proposed framework, each module can be trained efficiently in FT by stochastic algorithms without distortion of graph information owing to its simplicity. To avoid the only unidirectional information delivery of FT and sufficiently train shallow modules with the deeper ones, we develop a backward training mechanism that makes the former modules perceive the latter modules. The backward training introduces the reversed information delivery into the decoupled modules as well as the forward information delivery. To investigate how the decoupling and greedy training affect the representational capacity, we theoretically prove that the error produced by linear modules will not accumulate on unsupervised tasks in most cases. The theoretical and experimental results show that the proposed framework is highly efficient with reasonable performance.

Deep Manifold Learning with Graph Mining

Admittedly, Graph Convolution Network (GCN) has achieved excellent results on graph datasets such as social networks, citation networks, etc. However, softmax used as the decision layer in these frameworks is generally optimized with thousands of iterations via gradient descent. Furthermore, due to ignoring the inner distribution of the graph nodes, the decision layer might lead to an unsatisfactory performance in semi-supervised learning with less label support. To address the referred issues, we propose a novel graph deep model with a non-gradient decision layer for graph mining. Firstly, manifold learning is unified with label local-structure preservation to capture the topological information of the nodes. Moreover, owing to the non-gradient property, closed-form solutions is achieved to be employed as the decision layer for GCN. Particularly, a joint optimization method is designed for this graph model, which extremely accelerates the convergence of the model. Finally, extensive experiments show that the proposed model has achieved state-of-the-art performance compared to the current models.

Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning

The existing matrix completion methods focus on optimizing the relaxation of rank function such as nuclear norm, Schatten-p norm, etc. They usually need many iterations to converge. Moreover, only the low-rank property of matrices is utilized in most existing models and several methods that incorporate other knowledge are quite time-consuming in practice. To address these issues, we propose a novel non-convex surrogate that can be optimized by closed-form solutions, such that it empirically converges within dozens of iterations. Besides, the optimization is parameter-free and the convergence is proved. Compared with the relaxation of rank, the surrogate is motivated by optimizing an upper-bound of rank. We theoretically validate that it is equivalent to the existing matrix completion models. Besides the low-rank assumption, we intend to exploit the column-wise correlation for matrix completion, and thus an adaptive correlation learning, which is scaling-invariant, is developed. More importantly, after incorporating the correlation learning, the model can be still solved by closed-form solutions such that it still converges fast. Experiments show the effectiveness of the non-convex surrogate and adaptive correlation learning.

AnchorGAE: General Data Clustering via $O(n)$ Bipartite Graph Convolution

Graph-based clustering plays an important role in clustering tasks. As graph convolution network (GCN), a variant of neural networks on graph-type data, has achieved impressive performance, it is attractive to find whether GCNs can be used to augment the graph-based clustering methods on non-graph data, i.e., general data. However, given $n$ samples, the graph-based clustering methods usually need at least $O(n^2)$ time to build graphs and the graph convolution requires nearly $O(n^2)$ for a dense graph and $O(|\mathcal{E}|)$ for a sparse one with $|\mathcal{E}|$ edges. In other words, both graph-based clustering and GCNs suffer from severe inefficiency problems. To tackle this problem and further employ GCN to promote the capacity of graph-based clustering, we propose a novel clustering method, AnchorGAE. As the graph structure is not provided in general clustering scenarios, we first show how to convert a non-graph dataset into a graph by introducing the generative graph model, which is used to build GCNs. Anchors are generated from the original data to construct a bipartite graph such that the computational complexity of graph convolution is reduced from $O(n^2)$ and $O(|\mathcal{E}|)$ to $O(n)$. The succeeding steps for clustering can be easily designed as $O(n)$ operations. Interestingly, the anchors naturally lead to a siamese GCN architecture. The bipartite graph constructed by anchors is updated dynamically to exploit the high-level information behind data. Eventually, we theoretically prove that the simple update will lead to degeneration and a specific strategy is accordingly designed.

Non-Gradient Manifold Neural Network

Deep neural network (DNN) generally takes thousands of iterations to optimize via gradient descent and thus has a slow convergence. In addition, softmax, as a decision layer, may ignore the distribution information of the data during classification. Aiming to tackle the referred problems, we propose a novel manifold neural network based on non-gradient optimization, i.e., the closed-form solutions. Considering that the activation function is generally invertible, we reconstruct the network via forward ridge regression and low rank backward approximation, which achieve the rapid convergence. Moreover, by unifying the flexible Stiefel manifold and adaptive support vector machine, we devise the novel decision layer which efficiently fits the manifold structure of the data and label information. Consequently, a jointly non-gradient optimization method is designed to generate the network with closed-form results. Eventually, extensive experiments validate the superior performance of the model.

Enhanced Principal Component Analysis under A Collaborative-Robust Framework

Principal component analysis (PCA) frequently suffers from the disturbance of outliers and thus a spectrum of robust extensions and variations of PCA have been developed. However, existing extensions of PCA treat all samples equally even those with large noise. In this paper, we first introduce a general collaborative-robust weight learning framework that combines weight learning and robust loss in a non-trivial way. More significantly, under the proposed framework, only a part of well-fitting samples are activated which indicates more importance during training, and others, whose errors are large, will not be ignored. In particular, the negative effects of inactivated samples are alleviated by the robust loss function. Then we furthermore develop an enhanced PCA which adopts a point-wise sigma-loss function that interpolates between L_2,1-norm and squared Frobenius-norm and meanwhile retains the rotational invariance property. Extensive experiments are conducted on occluded datasets from two aspects including reconstructed errors and clustering accuracy. The experimental results prove the superiority and effectiveness of our model.

Adaptive Graph Auto-Encoder for General Data Clustering

Graph based clustering plays an important role in clustering area. Recent studies about graph convolution neural networks have achieved impressive success on graph type data. However, in traditional clustering tasks, the graph structure of data does not exist such that the strategy to construct graph is crucial for performance. In addition, the existing graph auto-encoder based approaches perform poorly on weighted graph, which is widely used in graph based clustering. In this paper, we propose a graph auto-encoder with local structure preserving for general data clustering, which can update the constructed graph adaptively. The adaptive process is designed to utilize the non-Euclidean structure sufficiently. By combining generative model for graph embedding and graph based clustering, a graph auto-encoder with a novel decoder is developed and it performs well in weighted graph used scenarios. Extensive experiments prove the superiority of our model.