Background subtraction refers to extracting the foreground from an observed video, and is the fundamental problem of various applications. There are two kinds of popular methods to deal with background separation, namely, robust principal component analysis (RPCA) and low-rank matrix factorization (LRMF). Nevertheless, the drawback of RPCA requires tuning penalty parameter to attain an ideal result. Compared with RPCA, the ℓ1 -norm based LRMF does not involve extra parameters tuning, but it is challenging to optimize the ℓ1 -norm based minimization because of the nonsmooth ℓ1 -norm. In addition, it becomes time-consuming to find the optimal solution. In this work, we propose to employ smooth ℓ1,ϵ -norm, an approximation of ℓ1 -norm, to tackle background subtraction. Thus, the proposed model inherits the superiority of LRMF and even becomes tractable. Then the resultant optimization problem is solved by alternating minimization and gradient descent where the step-size of the gradient descent is adaptively updated via backtracking line searching approach. The proposed method is proved to be locally convergent. Experimental results on synthetic and real-world data demonstrate that our method outperforms the state-of-the-art algorithms in terms of reconstruction loss, computational speed and hardware performance.
With the wide deployment of digital image capturing equipment, the need of denoising to produce a crystal clear image from noisy capture environment has become indispensable. In this article, a novel type-2 fuzzy-based filter is proposed for denoising images corrupted by impulse noise, especially for the high density of salt-and-pepper noise. It operates two stages, namely, type-2 fuzzy identifier and matrix completion denoiser. In the proposed method, the type-2 fuzzy identifier is first employed to identify and trim the entries contaminated by impulse noise in the data matrix from fuzzy system. Then, the trimmed data matrix is utilized to retrieve the noiseless data matrix with the matrix completion technology. Herein, a novel matrix completion technique is developed without a priori rank information compared to its counterparts. Simulation results are presented, which vividly show the denoised images obtained by the proposed method can achieve crystal clear image with strong structural integrity, and are showing good performance in terms of peak signal-to-noise ratio.
Recently, nonnegative matrix factorization (NMF) has been widely adopted for community detection, because of its better interpretability. However, the existing NMF-based methods have the following three problems: 1) they directly transform the original network into community membership space, so it is difficult for them to capture the hierarchical information; 2) they often only pay attention to the topology of the network and ignore its node attributes; 3) it is hard for them to learn the global structure information necessary for community detection. Therefore, we propose a new community detection algorithm, named Contrastive Deep Nonnegative Matrix Factorization (CDNMF). Firstly, we deepen NMF to strengthen its capacity for information extraction. Subsequently, inspired by contrastive learning, our algorithm creatively constructs network topology and node attributes as two contrasting views. Furthermore, we utilize a debiased negative sampling layer and learn node similarity at the community level, thereby enhancing the suitability of our model for community detection. We conduct experiments on three public real graph datasets and the proposed model has achieved better results than state-of-the-art methods. Code available at https://github.com/6lyc/CDNMF.git.
One underlying assumption of recent federated learning (FL) paradigms is that all local models usually share the same network architecture and size, which becomes impractical for devices with different hardware resources. A scalable federated learning framework should address the heterogeneity that clients have different computing capacities and communication capabilities. To this end, this paper proposes FedHM, a novel heterogeneous federated model compression framework, distributing the heterogeneous low-rank models to clients and then aggregating them into a full-rank model. Our solution enables the training of heterogeneous models with varying computational complexities and aggregates them into a single global model. Furthermore, FedHM significantly reduces the communication cost by using low-rank models. Extensive experimental results demonstrate that FedHM is superior in the performance and robustness of models of different sizes, compared with state-of-the-art heterogeneous FL methods under various FL settings. Additionally, the convergence guarantee of FL for heterogeneous devices is first theoretically analyzed.
We propose a fast non-gradient-based method of rank-1 non-negative matrix factorization (NMF) for missing data, called A1GM, that minimizes the KL divergence from an input matrix to the reconstructed rank-1 matrix. Our method is based on our new finding of an analytical closed-formula of the best rank-1 non-negative multiple matrix factorization (NMMF), a variety of NMF. NMMF is known to exactly solve NMF for missing data if positions of missing values satisfy a certain condition, and A1GM transforms a given matrix so that the analytical solution to NMMF can be applied. We empirically show that A1GM is more efficient than a gradient method with competitive reconstruction errors.
Deep convolutional neural networks (CNNs) with a large number of parameters require intensive computational resources, and thus are hard to be deployed in resource-constrained platforms. Decomposition-based methods, therefore, have been utilized to compress CNNs in recent years. However, since the compression factor and performance are negatively correlated, the state-of-the-art works either suffer from severe performance degradation or have relatively low compression factors. To overcome this problem, we propose to compress CNNs and alleviate performance degradation via joint matrix decomposition, which is different from existing works that compressed layers separately. The idea is inspired by the fact that there are lots of repeated modules in CNNs. By projecting weights with the same structures into the same subspace, networks can be jointly compressed with larger ranks. In particular, three joint matrix decomposition schemes are developed, and the corresponding optimization approaches based on Singular Value Decomposition are proposed. Extensive experiments are conducted across three challenging compact CNNs for different benchmark data sets to demonstrate the superior performance of our proposed algorithms. As a result, our methods can compress the size of ResNet-34 by 22X with slighter accuracy degradation compared with several state-of-the-art methods.
In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the accuracy of the desired singular values and also all of their corresponding singular vectors are essential. As a real application, we evaluate the performance of our methods on the problem of Riemannian similarity learning between two various image datasets of MNIST and USPS.
Parameter servers (PSs) facilitate the implementation of distributed training for large machine learning tasks. In this paper, we argue that existing PSs are inefficient for tasks that exhibit non-uniform parameter access; their performance may even fall behind that of single node baselines. We identify two major sources of such non-uniform access: skew and sampling. Existing PSs are ill-suited for managing skew because they uniformly apply the same parameter management technique to all parameters. They are inefficient for sampling because the PS is oblivious to the associated randomized accesses and cannot exploit locality. To overcome these performance limitations, we introduce NuPS, a novel PS architecture that (i) integrates multiple management techniques and employs a suitable technique for each parameter and (ii) supports sampling directly via suitable sampling primitives and sampling schemes that allow for a controlled quality--efficiency trade-off. In our experimental study, NuPS outperformed existing PSs by up to one order of magnitude and provided up to linear scalability across multiple machine learning tasks.
Active and passive thermography are two efficient techniques extensively used to measure heterogeneous thermal patterns leading to subsurface defects for diagnostic evaluations. This study conducts a comparative analysis on low-rank matrix approximation methods in thermography with applications of semi-, convex-, and sparse- non-negative matrix factorization (NMF) methods for detecting subsurface thermal patterns. These methods inherit the advantages of principal component thermography (PCT) and sparse PCT, whereas tackle negative bases in sparse PCT with non-negative constraints, and exhibit clustering property in processing data. The practicality and efficiency of these methods are demonstrated by the experimental results for subsurface defect detection in three specimens (for different depth and size defects) and preserving thermal heterogeneity for distinguishing breast abnormality in breast cancer screening dataset (accuracy of 74.1%, 75.8%, and 77.8%).
We propose an efficient matrix rank reduction method for non-negative matrices, whose time complexity is quadratic in the number of rows or columns of a matrix. Our key insight is to formulate rank reduction as a mean-field approximation by modeling matrices via a log-linear model on structured sample space, which allows us to solve the rank reduction as convex optimization. The highlight of this formulation is that the optimal solution that minimizes the KL divergence from a given matrix can be analytically computed in a closed form. We empirically show that our rank reduction method is faster than NMF and its popular variant, lraNMF, while achieving competitive low rank approximation error on synthetic and real-world datasets.
Bayesian low-rank matrix factorization techniques have become an essential tool for relational data analysis and matrix completion. A standard approach is to assign zero-mean Gaussian priors on the columns or rows of factor matrices to create a conjugate system. This choice of prior leads to simple implementations; however it also causes symmetries in the posterior distribution that can severely reduce the efficiency of Markov-chain Monte-Carlo (MCMC) sampling approaches. In this paper, we propose a simple modification to the prior choice that provably breaks these symmetries and maintains/improves accuracy. Specifically, we provide conditions that the Gaussian prior mean and covariance must satisfy so the posterior does not exhibit invariances that yield sampling difficulties. For example, we show that using non-zero linearly independent prior means significantly lowers the autocorrelation of MCMC samples, and can also lead to lower reconstruction errors.
Text clustering is arguably one of the most important topics in modern data mining. Nevertheless, text data require tokenization which usually yields a very large and highly sparse term-document matrix, which is usually difficult to process using conventional machine learning algorithms. Methods such as Latent Semantic Analysis have helped mitigate this issue, but are nevertheless not completely stable in practice. As a result, we propose a new feature agglomeration method based on Nonnegative Matrix Factorization, which is employed to separate the terms into groups, and then each group's term vectors are agglomerated into a new feature vector. Together, these feature vectors create a new feature space much more suitable for clustering. In addition, we propose a new deterministic initialization for spherical K-Means, which proves very useful for this specific type of data. In order to evaluate the proposed method, we compare it to some of the latest research done in this field, as well as some of the most practiced methods. In our experiments, we conclude that the proposed method either significantly improves clustering performance, or maintains the performance of other methods, while improving stability in results.
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also provide examples from machine learning, including principal component analysis, kernel supervised principal component analysis, and Fisher discriminant analysis, which result in eigenvalue and generalized eigenvalue problems. Finally, we introduce the solutions to both eigenvalue and generalized eigenvalue problems.
The refugee crisis is perhaps the single most challenging problem for Europe today. Hundreds of thousands of people have already traveled across dangerous sea passages from Turkish shores to Greek islands, resulting in thousands of dead and missing, despite the best rescue efforts from both sides. One of the main reasons is the total lack of any early warning-alerting system, which could provide some preparation time for the prompt and effective deployment of resources at the hot zones. This work is such an attempt for a systemic analysis of the refugee influx in Greece, aiming at (a) the statistical and signal-level characterization of the smuggling networks and (b) the formulation and preliminary assessment of such models for predictive purposes, i.e., as the basis of such an early warning-alerting protocol. To our knowledge, this is the first-ever attempt to design such a system, since this refugee crisis itself and its geographical properties are unique (intense event handling, little or no warning). The analysis employs a wide range of statistical, signal-based and matrix factorization (decomposition) techniques, including linear & linear-cosine regression, spectral analysis, ARMA, SVD, Probabilistic PCA, ICA, K-SVD for Dictionary Learning, as well as fractal dimension analysis. It is established that the behavioral patterns of the smuggling networks closely match (as expected) the regular burst and pause periods of store-and-forward networks in digital communications. There are also major periodic trends in the range of 6.2-6.5 days and strong correlations in lags of four or more days, with distinct preference in the Sunday-Monday 48-hour time frame. These results show that such models can be used successfully for short-term forecasting of the influx intensity, producing an invaluable operational asset for planners, decision-makers and first-responders.
While several matrix factorization (MF) and tensor factorization (TF) models have been proposed for knowledge base (KB) inference, they have rarely been compared across various datasets. Is there a single model that performs well across datasets? If not, what characteristics of a dataset determine the performance of MF and TF models? Is there a joint TF+MF model that performs robustly on all datasets? We perform an extensive evaluation to compare popular KB inference models across popular datasets in the literature. In addition to answering the questions above, we remove a limitation in the standard evaluation protocol for MF models, propose an extension to MF models so that they can better handle out-of-vocabulary (OOV) entity pairs, and develop a novel combination of TF and MF models. We also analyze and explain the results based on models and dataset characteristics. Our best model is robust, and obtains strong results across all datasets.