Truncated Cauchy Non-Negative Matrix Factorization

Optimizing wastewater treatment through artificial intelligence: recent advances and future prospects

Artificial intelligence (AI) is increasingly being applied to wastewater treatment to enhance efficiency, improve processes, and optimize resource utilization. This review focuses on objectives, advantages, outputs, and major findings of various AI models in the three key aspects: the prediction of removal efficiency for both organic and inorganic pollutants, real-time monitoring of essential water quality parameters (such as pH, COD, BOD, turbidity, TDS, and conductivity), and fault detection in the processes and equipment integral to wastewater treatment. The prediction accuracy (R2 value) of AI technologies for pollutant removal has been reported to vary between 0.64 and 1.00. A critical aspect explored in this review is the cost-effectiveness of implementing AI systems in wastewater treatment. Numerous countries and municipalities are actively engaging in pilot projects and demonstrations to assess the feasibility and effectiveness of AI applications in wastewater treatment. Notably, the review highlights successful outcomes from these initiatives across diverse geographical contexts, showcasing the adaptability and positive impact of AI in revolutionizing wastewater treatment on a global scale. Further, insights on the ethical considerations and potential future directions for the use of AI in wastewater treatment plants have also been provided.

Regularly Truncated M-Estimators for Learning With Noisy Labels

The sample selection approach is very popular in learning with noisy labels. As deep networks "learn pattern first", prior methods built on sample selection share a similar training procedure: the small-loss examples can be regarded as clean examples and used for helping generalization, while the large-loss examples are treated as mislabeled ones and excluded from network parameter updates. However, such a procedure is arguably debatable from two folds: (a) it does not consider the bad influence of noisy labels in selected small-loss examples; (b) it does not make good use of the discarded large-loss examples, which may be clean or have meaningful information for generalization. In this paper, we propose regularly truncated M-estimators (RTME) to address the above two issues simultaneously. Specifically, RTME can alternately switch modes between truncated M-estimators and original M-estimators. The former can adaptively select small-losses examples without knowing the noise rate and reduce the side-effects of noisy labels in them. The latter makes the possibly clean examples but with large losses involved to help generalization. Theoretically, we demonstrate that our strategies are label-noise-tolerant. Empirically, comprehensive experimental results show that our method can outperform multiple baselines and is robust to broad noise types and levels.

Entropy Minimizing Matrix Factorization

Nonnegative matrix factorization (NMF) is a widely used data analysis technique and has yielded impressive results in many real-world tasks. Generally, existing NMF methods represent each sample with several centroids and find the optimal centroids by minimizing the sum of the residual errors. However, outliers deviating from the normal data distribution may have large residues and then dominate the objective value. In this study, an entropy minimizing matrix factorization (EMMF) framework is developed to tackle the above problem. Considering that outliers are usually much less than the normal samples, a new entropy loss function is established for matrix factorization, which minimizes the entropy of the residue distribution and allows a few samples to have large errors. In this way, the outliers do not affect the approximation of normal samples. Multiplicative updating rules for EMMF are derived, and the convergence is proven theoretically. In addition, a Graph regularized version of EMMF (G-EMMF) is also presented, which uses a data graph to capture the data relationship. Clustering results on various synthetic and real-world datasets demonstrate the advantages of the proposed models, and the effectiveness is also verified through the comparison with state-of-the-art methods.

An Entropy Weighted Nonnegative Matrix Factorization Algorithm for Feature Representation

Nonnegative matrix factorization (NMF) has been widely used to learn low-dimensional representations of data. However, NMF pays the same attention to all attributes of a data point, which inevitably leads to inaccurate representations. For example, in a human-face dataset, if an image contains a hat on a head, the hat should be removed or the importance of its corresponding attributes should be decreased during matrix factorization. This article proposes a new type of NMF called entropy weighted NMF (EWNMF), which uses an optimizable weight for each attribute of each data point to emphasize their importance. This process is achieved by adding an entropy regularizer to the cost function and then using the Lagrange multiplier method to solve the problem. Experimental results with several datasets demonstrate the feasibility and effectiveness of the proposed method. The code developed in this study is available at https://github.com/Poisson-EM/Entropy-weighted-NMF.

Robust capped norm dual hyper-graph regularized non-negative matrix tri-factorization

Non-negative matrix factorization (NMF) has been widely used in machine learning and data mining fields. As an extension of NMF, non-negative matrix tri-factorization (NMTF) provides more degrees of freedom than NMF. However, standard NMTF algorithm utilizes Frobenius norm to calculate residual error, which can be dramatically affected by noise and outliers. Moreover, the hidden geometric information in feature manifold and sample manifold is rarely learned. Hence, a novel robust capped norm dual hyper-graph regularized non-negative matrix tri-factorization (RCHNMTF) is proposed. First, a robust capped norm is adopted to handle extreme outliers. Second, dual hyper-graph regularization is considered to exploit intrinsic geometric information in feature manifold and sample manifold. Third, orthogonality constraints are added to learn unique data presentation and improve clustering performance. The experiments on seven datasets testify the robustness and superiority of RCHNMTF.

Robust semi-supervised non-negative matrix factorization for binary subspace learning

Non-negative matrix factorization and its extensions were applied to various areas (i.e., dimensionality reduction, clustering, etc.). When the original data are corrupted by outliers and noise, most of non-negative matrix factorization methods cannot achieve robust factorization and learn a subspace with binary codes. This paper puts forward a robust semi-supervised non-negative matrix factorization method for binary subspace learning, called RSNMF, for image clustering. For better clustering performance on the dataset contaminated by outliers and noise, we propose a weighted constraint on the noise matrix and impose manifold learning into non-negative matrix factorization. Moreover, we utilize the discrete hashing learning method to constrain the learned subspace, which can achieve a binary subspace from the original data. Experimental results validate the robustness and effectiveness of RSNMF in binary subspace learning and image clustering on the face dataset corrupted by Salt and Pepper noise and Contiguous Occlusion.

Symmetric Nonnegative Matrix Factorization-Based Community Detection Models and Their Convergence Analysis

Community detection is a popular yet thorny issue in social network analysis. A symmetric and nonnegative matrix factorization (SNMF) model based on a nonnegative multiplicative update (NMU) scheme is frequently adopted to address it. Current research mainly focuses on integrating additional information into it without considering the effects of a learning scheme. This study aims to implement highly accurate community detectors via the connections between an SNMF-based community detector's detection accuracy and an NMU scheme's scaling factor. The main idea is to adjust such scaling factor via a linear or nonlinear strategy, thereby innovatively implementing several scaling-factor-adjusted NMU schemes. They are applied to SNMF and graph-regularized SNMF models to achieve four novel SNMF-based community detectors. Theoretical studies indicate that with the proposed schemes and proper hyperparameter settings, each model can: 1) keep its loss function nonincreasing during its training process and 2) converge to a stationary point. Empirical studies on eight social networks show that they achieve significant accuracy gain in community detection over the state-of-the-art community detectors.

Kernel Correntropy Conjugate Gradient Algorithms Based on Half-Quadratic Optimization

As a nonlinear similarity measure defined in the kernel space, the correntropic loss (C-Loss) can address the stability issues of second-order similarity measures thanks to its ability to extract high-order statistics of data. However, the kernel adaptive filter (KAF) based on the C-Loss uses the stochastic gradient descent (SGD) method to update its weights and, thus, suffers from poor performance and a slow convergence rate. To address these issues, the conjugate gradient (CG)-based correntropy algorithm is developed by solving the combination of half-quadratic (HQ) optimization and weighted least-squares (LS) problems, generating a novel robust kernel correntropy CG (KCCG) algorithm. The proposed KCCG with less computational complexity achieves comparable performance to the kernel recursive maximum correntropy (KRMC) algorithm. To further curb the growth of the network in KCCG, the random Fourier features KCCG (RFFKCCG) algorithm is proposed by transforming the original input data into a fixed-dimensional random Fourier features space (RFFS). Since only one current error information is used in the loss function of RFFKCCG, it can provide a more efficient filter structure than the other KAFs with sparsification. The Monte Carlo simulations conducted in the prediction of synthetic and real-world chaotic time series and the regression for large-scale datasets validate the superiorities of the proposed algorithms in terms of robustness, filtering accuracy, and complexity.

Correntropy-Based Hypergraph Regularized NMF for Clustering and Feature Selection on Multi-Cancer Integrated Data

Non-negative matrix factorization (NMF) has become one of the most powerful methods for clustering and feature selection. However, the performance of the traditional NMF method severely degrades when the data contain noises and outliers or the manifold structure of the data is not taken into account. In this article, a novel method called correntropy-based hypergraph regularized NMF (CHNMF) is proposed to solve the above problem. Specifically, we use the correntropy instead of the Euclidean norm in the loss term of CHNMF, which will improve the robustness of the algorithm. And the hypergraph regularization term is also applied to the objective function, which can explore the high-order geometric information in more sample points. Then, the half-quadratic (HQ) optimization technique is adopted to solve the complex optimization problem of CHNMF. Finally, extensive experimental results on multi-cancer integrated data indicate that the proposed CHNMF method is superior to other state-of-the-art methods for clustering and feature selection.

Simultaneous Dimensionality Reduction and Classification via Dual Embedding Regularized Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a well-known paradigm for data representation. Traditional NMF-based classification methods first perform NMF or one of its variants on input data samples to obtain their low-dimensional representations, which are successively classified by means of a typical classifier [e.g., k -nearest neighbors (KNN) and support vector machine (SVM)]. Such a stepwise manner may overlook the dependency between the two processes, resulting in the compromise of the classification accuracy. In this paper, we elegantly unify the two processes by formulating a novel constrained optimization model, namely dual embedding regularized NMF (DENMF), which is semi-supervised. Our DENMF solution simultaneously finds the low-dimensional representations and assignment matrix via joint optimization for better classification. Specifically, input data samples are projected onto a couple of low-dimensional spaces (i.e., feature and label spaces), and locally linear embedding is employed to preserve the identical local geometric structure in different spaces. Moreover, we propose an alternating iteration algorithm to solve the resulting DENMF, whose convergence is theoretically proven. Experimental results over five benchmark datasets demonstrate that DENMF can achieve higher classification accuracy than state-of-the-art algorithms.

Matrix Completion Based on Non-convex Low Rank Approximation

Without any prior structure information, Nuclear Norm Minimization (NNM), a convex relaxation for Rank Minimization (RM), is a widespread tool for matrix completion and relevant low rank approximation problems. Nevertheless, the result derivated by NNM generally deviates the solution we desired, because NNM ignores the difference between different singular values. In this paper, we present a non-convex regularizer and utilize it to construct two matrix completion models. In order to solve the constructed models efficiently, we develop an efficient optimization method with convergence guarantee, which can achieve faster convergence speed compared to conventional approaches. Particularly, we show that the proposed regularizer as well as optimization method are suitable for other RM problems, such as subspace clustering based on low rank representation. Extensive experimental results on real images demonstrate that the constructed models provide significant advantages over several state-of-the-art matrix completion algorithms. In addition, we implement numerous experiments to investigate the convergence speed of developed optimization method.