Consider our approach for local Lipschitz bound computation. The third entry varies under perturbation. A.2 Why not consider linear ReLU outputs? We refer to these outputs as linear ReLU outputs. For simplicity, let us consider the standard ReLU activation.

Min cut is an important graph partitioning method. However, current solutions to the min cut problem suffer from slow speeds, difficulty in solving, and often converge to simple solutions. To address these issues, we relax the min cut problem into a dual-bounded constraint and, for the first time, treat the min cut problem as a dual-bounded nonlinear optimal transport problem. Additionally, we develop a method for solving dual-bounded nonlinear optimal transport based on the Frank-Wolfe method (abbreviated as DNF). Notably, DNF not only solves the size constrained min cut problem but is also applicable to all dual-bounded nonlinear optimal transport problems. We prove that for convex problems satisfying Lipschitz smoothness, the DNF method can achieve a convergence rate of \(\mathcal{O}(\frac{1}{t})\). We apply the DNF method to the min cut problem and find that it achieves state-of-the-art performance in terms of both the loss function and clustering accuracy at the fastest speed, with a convergence rate of \(\mathcal{O}(\frac{1}{\sqrt{t}})\). Moreover, the DNF method for the size constrained min cut problem requires no parameters and exhibits better stability.

Multi-view unsupervised feature selection has been proven to be efficient in reducing the dimensionality of multi-view unlabeled data with high dimensions. The previous methods assume all of the views are complete. However, in real applications, the multi-view data are often incomplete, i.e., some views of instances are missing, which will result in the failure of these methods. Besides, while the data arrive in form of streams, these existing methods will suffer the issues of high storage cost and expensive computation time. To address these issues, we propose an Incremental Incomplete Multi-view Unsupervised Feature Selection method (I$^2$MUFS) on incomplete multi-view streaming data. By jointly considering the consistent and complementary information across different views, I$^2$MUFS embeds the unsupervised feature selection into an extended weighted non-negative matrix factorization model, which can learn a consensus clustering indicator matrix and fuse different latent feature matrices with adaptive view weights. Furthermore, we introduce the incremental leaning mechanisms to develop an alternative iterative algorithm, where the feature selection matrix is incrementally updated, rather than recomputing on the entire updated data from scratch. A series of experiments are conducted to verify the effectiveness of the proposed method by comparing with several state-of-the-art methods. The experimental results demonstrate the effectiveness and efficiency of the proposed method in terms of the clustering metrics and the computational cost.

Multi-view unsupervised feature selection (MUFS) has been demonstrated as an effective technique to reduce the dimensionality of multi-view unlabeled data. The existing methods assume that all of views are complete. However, multi-view data are usually incomplete, i.e., a part of instances are presented on some views but not all views. Besides, learning the complete similarity graph, as an important promising technology in existing MUFS methods, cannot achieve due to the missing views. In this paper, we propose a complementary and consensus learning-based incomplete multi-view unsupervised feature selection method (C$^{2}$IMUFS) to address the aforementioned issues. Concretely, C$^{2}$IMUFS integrates feature selection into an extended weighted non-negative matrix factorization model equipped with adaptive learning of view-weights and a sparse $\ell_{2,p}$-norm, which can offer better adaptability and flexibility. By the sparse linear combinations of multiple similarity matrices derived from different views, a complementary learning-guided similarity matrix reconstruction model is presented to obtain the complete similarity graph in each view. Furthermore, C$^{2}$IMUFS learns a consensus clustering indicator matrix across different views and embeds it into a spectral graph term to preserve the local geometric structure. Comprehensive experimental results on real-world datasets demonstrate the effectiveness of C$^{2}$IMUFS compared with state-of-the-art methods.

Certified robustness is a desirable property for deep neural networks in safety-critical applications, and popular training algorithms can certify robustness of a neural network by computing a global bound on its Lipschitz constant. However, such a bound is often loose: it tends to over-regularize the neural network and degrade its natural accuracy. A tighter Lipschitz bound may provide a better tradeoff between natural and certified accuracy, but is generally hard to compute exactly due to non-convexity of the network. In this work, we propose an efficient and trainable \emph{local} Lipschitz upper bound by considering the interactions between activation functions (e.g. ReLU) and weight matrices. Specifically, when computing the induced norm of a weight matrix, we eliminate the corresponding rows and columns where the activation function is guaranteed to be a constant in the neighborhood of each given data point, which provides a provably tighter bound than the global Lipschitz constant of the neural network. Our method can be used as a plug-in module to tighten the Lipschitz bound in many certifiable training algorithms. Furthermore, we propose to clip activation functions (e.g., ReLU and MaxMin) with a learnable upper threshold and a sparsity loss to assist the network to achieve an even tighter local Lipschitz bound. Experimentally, we show that our method consistently outperforms state-of-the-art methods in both clean and certified accuracy on MNIST, CIFAR-10 and TinyImageNet datasets with various network architectures.

Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models that are widely used in modern educational, psychological, social and biological sciences. A key component of CDMs is a binary $Q$-matrix characterizing the dependence structure between the items and the latent attributes. Additionally, researchers also assume in many applications certain hierarchical structures among the latent attributes to characterize their dependence. In most CDM applications, the attribute-attribute hierarchical structures, the item-attribute $Q$-matrix, the item-level diagnostic model, as well as the number of latent attributes, need to be fully or partially pre-specified, which however may be subjective and misspecified as noted by many recent studies. This paper considers the problem of jointly learning these latent and hierarchical structures in CDMs from observed data with minimal model assumptions. Specifically, a penalized likelihood approach is proposed to select the number of attributes and estimate the latent and hierarchical structures simultaneously. An efficient expectation-maximization (EM) algorithm and a latent structure recovery algorithm are developed, and statistical consistency theory is also established under mild conditions. The good performance of the proposed method is illustrated by simulation studies and a real data application in educational assessment.

In this paper, we demonstrate how deep autoencoders can be generalized to the case of inpainting and denoising, even when no clean training data is available. In particular, we show how neural networks can be trained to perform all of these tasks simultaneously. While, deep autoencoders implemented by way of neural networks have demonstrated potential for denoising and anomaly detection, standard autoencoders have the drawback that they require access to clean data for training. However, recent work in Robust Deep Autoencoders (RDAEs) shows how autoencoders can be trained to eliminate outliers and noise in a dataset without access to any clean training data. Inspired by this work, we extend RDAEs to the case where data are not only noisy and have outliers, but also only partially observed. Moreover, the dataset we train the neural network on has the properties that all entries have noise, some entries are corrupted by large mistakes, and many entries are not even known. Given such an algorithm, many standard tasks, such as denoising, image inpainting, and unobserved entry imputation can all be accomplished simultaneously within the same framework. Herein we demonstrate these techniques on standard machine learning tasks, such as image inpainting and denoising for the MNIST and CIFAR10 datasets. However, these approaches are not only applicable to image processing problems, but also have wide ranging impacts on datasets arising from real-world problems, such as manufacturing and network processing, where noisy, partially observed data naturally arise.

Incomplete multi-view clustering is an important technique to deal with real-world incomplete multi-view data. Previous works assume that all views have the same incompleteness, i.e., balanced incompleteness. However, different views often have distinct incompleteness, i.e., unbalanced incompleteness, which results in strong views (low-incompleteness views) and weak views (high-incompleteness views). The unbalanced incompleteness prevents us from directly using the previous methods for clustering. In this paper, inspired by the effective biological evolution theory, we design the novel scheme of view evolution to cluster strong and weak views. Moreover, we propose an Unbalanced Incomplete Multi-view Clustering method (UIMC), which is the first effective method based on view evolution for unbalanced incomplete multi-view clustering. Compared with previous methods, UIMC has two unique advantages: 1) it proposes weighted multi-view subspace clustering to integrate these unbalanced incomplete views, which effectively solves the unbalanced incomplete multi-view problem; 2) it designs the low-rank and robust representation to recover the data, which diminishes the impact of the incompleteness and noises. Extensive experimental results demonstrate that UIMC improves the clustering performance by up to 40% on three evaluation metrics over other state-of-the-art methods.

Multi-view clustering is an important yet challenging task due to the difficulty of integrating the information from multiple representations. Most existing multi-view clustering methods explore the heterogeneous information in the space where the data points lie. Such common practice may cause significant information loss because of unavoidable noise or inconsistency among views. Since different views admit the same cluster structure, the natural space should be all partitions. Orthogonal to existing techniques, in this paper, we propose to leverage the multi-view information by fusing partitions. Specifically, we align each partition to form a consensus cluster indicator matrix through a distinct rotation matrix. Moreover, a weight is assigned for each view to account for the clustering capacity differences of views. Finally, the basic partitions, weights, and consensus clustering are jointly learned in a unified framework. We demonstrate the effectiveness of our approach on several real datasets, where significant improvement is found over other state-of-the-art multi-view clustering methods.

The K-means algorithm is arguably the most popular data clustering method, commonly applied to processed datasets in some "feature spaces", as is in spectral clustering. Highly sensitive to initializations, however, K-means encounters a scalability bottleneck with respect to the number of clusters K as this number grows in big data applications. In this work, we promote a closely related model called K-indicators model and construct an efficient, semi-convex-relaxation algorithm that requires no randomized initializations. We present extensive empirical results to show advantages of the new algorithm when K is large. In particular, using the new algorithm to start the K-means algorithm, without any replication, can significantly outperform the standard K-means with a large number of currently state-of-the-art random replications.