Masked autoencoders have been extensively utilized in 3D point cloud self-supervised learning, where the fundamental approach involves masking a portion of the point cloud and subsequently reconstructing it. This process is hypothesized to enhance model learning by leveraging the inherent structure of the point cloud data. However, the information density within point clouds is inherently uneven, contrasting with the more uniform distributions found in language and 2D image data. This uneven distribution suggests that the application of random masking strategies, commonly adopted from NLP and 2D vision, may not be optimal for point cloud data, potentially leading to suboptimal learning outcomes. Based on this observation, we propose a simple yet effective Density-directed Masked Autoencoders for Point Cloud Self-supervised Learning (Point-DMAE), which learns latent semantic point cloud features using a density-directed masking strategy. Specifically, our method employs a dual-branch Transformer architecture to extract both high-level and fine-grained point features through global and local block density-directed masking, respectively. Point-DMAE demonstrates high pre-training efficiency and significantly outperforms our baseline (Point-MAE) on 3D object classification tasks within the ScanObjectNN dataset by 4.13% on OBJ-BG, 5.17% on OBJ-ONLY, and 4.17% on PB-T50-RS. Codes are available at https://github.com/jinxianglong10/Point-DMAE.

Principal Component Analysis (PCA) is one of the most famous unsupervised dimensionality reduction algorithms and has been widely used in many fields. However, it is very sensitive to outliers, which reduces the robustness of the algorithm. In recent years, many studies have tried to employ \(\ell_{1}\) -norm to improve the robustness of PCA, but they all lack rotation invariance or the solution is expensive. In this article, we propose a novel robust PCA, namely, Fuzzy Weighted Principal Component Analysis (FWPCA), which still uses squared \(\ell_{2}\) -norm to minimize reconstruction error and maintains rotation invariance of PCA. The biggest bright spot is that the contribution of data is restricted by fuzzy weights, so that the contribution of normal samples is much greater than noise or abnormal data, and realizes anomaly detection. Besides, a more reasonable data center can be obtained by solving the optimal mean to make projection matrix more accurate. Subsequently, an effective iterative optimization algorithm is developed to solve this problem, and its convergence is strictly proved. Extensive experimental results on face datasets and RGB anomaly detection datasets show the superiority of our proposed method.

Structured proximity matrix learning, one of the mainstream directions in clustering research, refers to learning a proximity matrix with an explicit clustering structure from the original first-order proximity matrix. Due to the complexity of the data structure, the original first-order proximity matrix always lacks some must-links compared to the groundtruth proximity matrix. It is worth noting that high-order proximity matrices can provide missed must-link information. However, the computation of high-order proximity matrices and clustering based on them are expensive. To solve the above problem, inspired by the anchor bipartite graph, we present a novel high-order bipartite graph proximity matrix and a fast method to compute it. This proposed high-order bipartite graph proximity matrix contains high-order proximity information and can significantly reduce the computational complexity of the whole clustering process. Furthermore, we introduce an efficient and simple high-order bipartite graph fusion framework that can adaptively assign weights to each order of the high-order bipartite graph matrices. Finally, under the Laplace rank constraint, a consensus structured bipartite graph proximity matrix is obtained. At the same time, an efficient solution algorithm is proposed for this model. The model's efficacy is underscored through rigorous experiments, highlighting its superior clustering performance and time efficiency. Code available:https://anonymous.4open.science/r/HBGC-F6C4.

Feature selection is one of the important topics of machine learning, and it has a wide range of applications in data preprocessing. At present, feature selection based on$\ell _{2,1}$-norm regularization is a relatively mature method, but it is not enough to maximize the sparsity and parameter-tuning leads to increased costs. Later scholars found that the$\ell _{2,0}$-norm constraint is more conductive to feature selection, but it is difficult to solve and lacks convergence guarantees. To address these problems, we creatively propose a novel Outliers Robust Unsupervised Feature Selection for structured sparse subspace (ORUFS), which utilizes$\ell _{2,0}$-norm constraint to learn a structured sparse subspace and avoid tuning the regularization parameter. Moreover, by adding binary weights, outliers are directly eliminated and the robustness of model is improved. More importantly, a Re-Weighted (RW) algorithm is exploited to solve our$\ell _{p}$-norm problem. For the NP-hard problem of$\ell _{2,0}$-norm constraint, we develop an effective iterative optimization algorithm with strict convergence guarantees and closed-form solution. Subsequently, we provide theoretical analysis about convergence and computational complexity. Experimental results on real-world datasets illustrate that our method is superior to the state-of-the-art methods in clustering and anomaly detection tasks.

In recent years, graph-based multiview clustering methods have become a research hotspot in the clustering field. However, most existing methods lack consideration of cluster balance in their results. In fact, cluster balance is crucial in many real-world scenarios. Additionally, graph-based multiview clustering methods often suffer from high time consumption and cannot handle large-scale datasets. To address these issues, this paper proposes a novel graph-based multiview clustering method. The method is built upon the bipartite graph. Specifically, it employs a label propagation mechanism to update the smaller anchor label matrix rather than the sample label matrix, significantly reducing the computational cost. The introduced balance constraint in the proposed model contributes to achieving balanced clustering results. The entire clustering model combines information from multiple views through graph fusion. The joint graph and view weight parameters in the model are obtained through task-driven self-supervised learning. Moreover, the model can directly obtain clustering results without the need for the two-stage processing typically used in general spectral clustering. Finally, extensive experiments on toy datasets and real-world datasets are conducted to validate the superiority of the proposed method in terms of clustering performance, clustering balance, and time expenditure.

In recent decades, unsupervised feature selection methods have become increasingly popular. Nevertheless, most of the existing unsupervised feature selection methods suffer from two major problems that lead to suboptimal solutions. Many methods impose a hard linear projection constraint on original data, which is overly strict in nature and not suitable for dealing with data sampled from nonlinear manifolds. Second, most existing methods usel2,p-norm (02S and SF2SOG, which can simultaneously learn optimal flexible projections and obtain an orthogonal sparse projection to directly select discriminative features by applyingl2,0-norm constraint. Moreover, we propose to explore the local structure of flexible embedding through preserving the manifold structure of original data and adaptively constructing an optimal graph in subspace. Thirdly, the novel iterative optimization algorithms are presented to solve objective functions guaranteeing convergence theoretically. Various evaluation experiments on synthetic and real-world datasets demonstrate the effectiveness and superiority of our proposed methods.

Semi-supervised learning as one of most attractive problems in machine learning research field has aroused broad attentions in recent years. In this paper, we propose a novel locality preserved dimensionality reduction framework, named Semi-supervised Adaptive Local Embedding learning (SALE), which learns a local discriminative embedding by constructing a$k_1$Nearest Neighbors ($k_1$NN) graph on labeled data, so as to explore the intrinsic structure, i.e., sub-manifolds from non-Gaussian labeled data. Then, mapping all samples into learned embedding and constructing another$k_2$NN graph on all embedded data to explore the global structure of all samples. Therefore, the unlabeled data and their corresponding labeled neighbors can be clustered into same sub-manifold, so as to improve the discriminative power of embedded data. Furthermore, we propose two semi-supervised dimensionality reduction methods with orthogonal and whitening constraints based on proposed SALE framework. An efficient alternatively iterative optimization algorithm is developed to solve the NP-hard problem in our models. Extensive experiments conducted on several synthetic and real-world data sets demonstrate the superiorities of our methods on local structure exploration and classification task.

Dimensionality reduction plays a significant role in high-dimensional data processing, and Linear Discriminant Analysis (LDA) is a widely used supervised dimensionality reduction approach. However, a major drawback of LDA is that it is incapable of extracting the local structure information, which is crucial for handling multimodal data. In this article, we propose a novel supervised dimensionality reduction method named Adaptive Local Linear Discriminant Analysis (ALLDA), which adaptively learns a k -nearest neighbors graph from data themselves to extract the local connectivity of data. Furthermore, the original high-dimensional data usually contains noisy and redundant features, which has a negative impact on the evaluation of neighborships and degrades the subsequent classification performance. To address this issue, our method learns the similarity matrix and updates the subspace simultaneously so that the neighborships can be evaluated in the optimal subspaces where the noises have been removed. Through the optimal graph embedding, the underlying sub-manifolds of data in intra-class can be extracted precisely. Meanwhile, an efficient iterative optimization algorithm is proposed to solve the minimization problem. Promising experimental results on synthetic and real-world datasets are provided to evaluate the effectiveness of proposed method.

Dimensionality reduction is a critical technology in the domain of pattern recognition, and linear discriminant analysis (LDA) is one of the most popular supervised dimensionality reduction methods. However, whenever its distance criterion of objective function uses$L_2$-norm, it is sensitive to outliers. In this paper, we propose a new formulation of linear discriminant analysis via joint$L_{2,1}$-norm minimization on objective function to induce robustness, so as to efficiently alleviate the influence of outliers and improve the robustness of proposed method. An efficient iterative algorithm is proposed to solve the optimization problem and proved to be convergent. Extensive experiments are performed on an artificial data set, on UCI data sets, and on four face data sets, which sufficiently demonstrates the efficiency of comparing to other methods and robustness to outliers of our approach.