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We introduce the notion of regularized Bayesian best response (RBBR) learning dynamic in heterogeneous population games. We obtain such a dynamic via perturbation by an arbitrary lower semicontinuous, strongly convex regularizer in Bayesian population games with finitely many strategies. We provide a sufficient condition for the existence of rest points of the RBBR learning dynamic, and hence the existence of regularized Bayesian equilibrium in Bayesian population games. These equilibria are shown to approximate the Bayesian equilibria of the game for vanishingly small regularizations. We also explore the fundamental properties of the RBBR learning dynamic, which includes the existence of unique solutions from arbitrary initial conditions, as well as the continuity of the solution trajectories thus obtained with respect to the initial conditions. Finally, as applications to the above theory, we introduce the notions of Bayesian potential and Bayesian negative semidefinite games and provide convergence results for such games. [ABSTRACT FROM AUTHOR]

Image blind deblurring is an ill-posed inverse problem in image processing. While deep learning approaches have demonstrated effectiveness, they often lack interpretability and require extensive data. To address these limitations, we propose a novel variational neural network based on algorithm unfolding. The model is solved using the half quadratic splitting (HQS) method and proximal gradient descent. For blur kernel estimation, we introduce an L 0 regularizer to constrain the gradient information and use the fast fourier transform (FFT) to solve the iterative results, thereby improving accuracy. Image restoration is initiated with Gabor filters for the convolution kernel, and the activation function is approximated using a Gaussian radial basis function (RBF). Additionally, two attention mechanisms improve feature selection. The experimental results on various datasets demonstrate that our model outperforms state-of-the-art algorithm unfolding networks and other blind deblurring models. Our approach enhances interpretability and generalization while utilizing fewer data and parameters. [ABSTRACT FROM AUTHOR]

We introduce an unsupervised feature selection method based on regularized weighted Fuzzy C-Means (WRFCM) clustering. When the target task is clustering, our objective should be to select a subset of features that can generate the same/similar partition matrix to the partition matrix obtained from the original high dimensional data by a clustering algorithm. To achieve this we propose a novel objective function keeping in view the Fuzzy-C-Means (FCM) clustering algorithm. This approach realizes feature selection within the WRFCM framework, emphasizing features to maintain the FCM-based target partition. We evaluate our method using Normalized Mutual Information (NMI), Adjusted Rand Index (ARI) and Kuhn-Munkres index (KM-index). NMI, and ARI measure the agreement between clusters, i.e, the partition in the lower dimension and the partition of the original data. On the other hand, KM-index measures the disagreement between the two partitions. Experimental results on synthetic and real datasets showcase our method's efficacy in selecting informative features. This approach fills a crucial gap in unsupervised feature selection, making it valuable for real-world applications. The approach is very general in the sense that the target partition can be generated by any clustering algorithm or even by the actual class labels of the data, when they are available. [ABSTRACT FROM AUTHOR]

We implement deep learning for predicting bitcoin closing prices. Identifying two new determiners, we propose a novel LSTM Autoencoder using Mean Squared Error (MSE) loss which is regularized by False Nearest Neighbor (FNN) algorithm. The method results in reduced error rates when compared to traditional forecasting algorithms and is statistically validated. This research contributes by developing a robust algorithm that accurately determines the fluctuation directions in bitcoin prices and results in values closer to the actual prices. [ABSTRACT FROM AUTHOR]

Image blind deblurring is an ill-posed inverse problem in image processing. While deep learning approaches have demonstrated effectiveness, they often lack interpretability and require extensive data. To address these limitations, we propose a novel variational neural network based on algorithm unfolding. The model is solved using the half quadratic splitting (HQS) method and proximal gradient descent. For blur kernel estimation, we introduce an L0 regularizer to constrain the gradient information and use the fast fourier transform (FFT) to solve the iterative results, thereby improving accuracy. Image restoration is initiated with Gabor filters for the convolution kernel, and the activation function is approximated using a Gaussian radial basis function (RBF). Additionally, two attention mechanisms improve feature selection. The experimental results on various datasets demonstrate that our model outperforms state-of-the-art algorithm unfolding networks and other blind deblurring models. Our approach enhances interpretability and generalization while utilizing fewer data and parameters.

Важным этапом в становлении теории краевых задач стали предложенные А.М. Нахушевым в 1969 году нелокальные задачи нового типа, впоследствии названные у нас краевыми задачами со смещением, а за рубежом — задачами (проблемами) Нахушева. Они являются обобщением задачи Трикоми, а так же содержат широкий класс корректных самосопряженных задач. Эти задачи сразу вызвали большой интерес многих авторов. За последние годы исследования задач со смещением для уравнений смешанного типа ведутся особенно интенсивно. Но в этих работах краевые условия, как правило, содержат классические операторы, в то время как нелокальным краевым задачам, содержащим операторы более сложной структуры и операторы дробного интегро-дифференцирования. Настоящая статья посвящена исследованию вопроса однозначной разрешимости краевых задач со смещением для уравнения смешанного эллиптико-гиперболического типа. Сформулированы корректные краевые задачи со смещением для уравнения смешанного типа. В данной работе исследованы вопросы однозначной разрешимости задач со смешением для уравнения смешанного эллиптико-гиперболического типа в смешанной области Ω, ограниченной в полуплоскости y>0 гладкой кривой Жордана, а в полуплоскости y

Pi-sigma neural network (PSNN) is a class of high order feed-forward neural networks with product units in the output layer, which leads to its fast convergence speed and a high degree of nonlinear mapping capability. Inspired by the sparse response character of human neuron system, this paper investigates the sparse-response feed-forward algorithm, i.e. the gradient descent based high order algorithm with self-adaptive momentum term and group lasso regularizer for training PSNN inference models. In this paper, we mainly focus on two challenging tasks. First, since the general group lasso regularizer is no differentiable at the original point, which will lead to the oscillation of the error function and the norm gradient during the training. A key point of this paper is to modify the usual group lasso regularization term by smoothing it at the origin. The advantage of this processing is that sparse and efficient neural networks can be obtained, and the theoretical analysis of the algorithm can also be obtained. Second, the adaptive momentum term is introduced in the iteration process to further accelerate the network learning speed. In addition, the numerical experiments show that the proposed algorithm eliminates the oscillation and increases learning rate in computation. And the convergence of the algorithm is also verified. [ABSTRACT FROM AUTHOR]

Brain–computer interfaces (BCIs) from electroencephalography (EEG) provide a practical approach to support human–technology interaction. In particular, motor imagery (MI) is a widely used BCI paradigm that guides the mental trial of motor tasks without physical movement. Here, we present a deep learning methodology, named kernel-based regularized EEGNet (KREEGNet), leveled on centered kernel alignment and Gaussian functional connectivity, explicitly designed for EEG-based MI classification. The approach proactively tackles the challenge of intrasubject variability brought on by noisy EEG records and the lack of spatial interpretability within end-to-end frameworks applied for MI classification. KREEGNet is a refinement of the widely accepted EEGNet architecture, featuring an additional kernel-based layer for regularized Gaussian functional connectivity estimation based on CKA. The superiority of KREEGNet is evidenced by our experimental results from binary and multiclass MI classification databases, outperforming the baseline EEGNet and other state-of-the-art methods. Further exploration of our model's interpretability is conducted at individual and group levels, utilizing classification performance measures and pruned functional connectivities. Our approach is a suitable alternative for interpretable end-to-end EEG-BCI based on deep learning. [ABSTRACT FROM AUTHOR]

We study the Fredholm property of regular hypoelliptic operators with special variable coefficients. In this paper, necessary and sufficient conditions are obtained for a priori estimates for differential operators acting in multianisotropic Sobolev spaces. Fredholm criteria are obtained for a wide class of regular hypoelliptic operators in multianisotropic weighted spaces in . [ABSTRACT FROM AUTHOR]

Ocean waves are the richest small-scale texture on the sea surface, from which valuable information can be inversed. In general, synthetic aperture radar (SAR) images of surface waves will inevitably have an impact on some oceanography follow-up applications due to the intricate motion of the waves. However, existing suppression methods based on image geometric properties or deep learning-based have limitations. Owing to the lack of clean images and the uncertainty of the statistical characteristics of seawater, these methods suppress ocean surface waves while harming other target information. To address this issue, we propose an unsupervised wave suppression algorithm based on sub-aperture SAR images named SAR-Noise2Noise (SAR-N2N). First, SAR-N2N exploits the geometric and time-varying relationship of the sub-aperture images to suppress the wave target selectively through sub-aperture image cancellation. Second, through the method of unsupervised learning, only noisy images are used to suppress ocean surface waves, avoiding heavy dependence on the assumption of the wave distribution. Finally, the proposed regularizer is used as an additional loss to form an end-to-end training process. We explain our approach from a theoretical perspective and further validate it through extensive experiments, including synthetic experiments with various ocean wave distributions in simulated SAR images and real-world images. Training on collocations over 9000 SAR images, we demonstrate on test data from simulated images that SAR-N2N can improve peak signal-to-noise ratio (PSNR) up to 30 dB compared with the classical approach. Furthermore, it improves the equivalent number of looks (ENL) value of real-world SAR images by a factor of 8. All results and the methods are novel in terms of the accuracy achieved, combining the classical approach with deep learning techniques. We conclude that SAR-N2N has the potential to make useful contributions to several oceanographic applications as well as to the near real time (NRT) processing of multiparametric sea states. [ABSTRACT FROM AUTHOR]

Brain–computer interfaces (BCIs) from electroencephalography (EEG) provide a practical approach to support human–technology interaction. In particular, motor imagery (MI) is a widely used BCI paradigm that guides the mental trial of motor tasks without physical movement. Here, we present a deep learning methodology, named kernel-based regularized EEGNet (KREEGNet), leveled on centered kernel alignment and Gaussian functional connectivity, explicitly designed for EEG-based MI classification. The approach proactively tackles the challenge of intrasubject variability brought on by noisy EEG records and the lack of spatial interpretability within end-to-end frameworks applied for MI classification. KREEGNet is a refinement of the widely accepted EEGNet architecture, featuring an additional kernel-based layer for regularized Gaussian functional connectivity estimation based on CKA. The superiority of KREEGNet is evidenced by our experimental results from binary and multiclass MI classification databases, outperforming the baseline EEGNet and other state-of-the-art methods. Further exploration of our model’s interpretability is conducted at individual and group levels, utilizing classification performance measures and pruned functional connectivities. Our approach is a suitable alternative for interpretable end-to-end EEG-BCI based on deep learning.

An improved version of the Selective Backpropagation (SBP+) is described which significantly reduces the training time and enhances the generalization. The algorithm eliminates correctly predicted instances from the Backpropagation (BP) in an iteration. The minimum useful content level to define a correct prediction is determined using the hyperparameter c. A new performance measure d m is introduced, which denotes the average of the fraction of remaining instances in the BP during an epoch. With a proper selection of c , a model can be adjusted to learn dominant patterns (mostly found in instances located further away from a decision hyperplane) and skip minor details (the very details which are due to the random error and can be found in instances close to a decision hyperplane). Since the SBP+ assigns a lower correct softmax score to instances located closer to a decision hyperplane, it is shown that — under certain conditions — a Neural Networks (NN) standard crisp output can be converted to a fuzzy membership scores set. The regulating effect of the content level c is compared with other standard techniques and its advantages are outlined. Lastly, the statistical process leading to minimization of the useful total error is explicated and the SBP+ ability to pinpoint outliers is illustrated with an example. • Improved Selective Backpropagation is described which accelerates training. • Acceleration is due to the elimination of accurate predictions from backpropagation. • Improvement in accuracy is due to the inclusion of random error term. • The minimum content to be learned from instances is determined with hyperparameter c. • A performance measure is introduced, denoting the fraction of remaining instances. [ABSTRACT FROM AUTHOR]

Limited-angle reconstruction is a very important but challenging problem in the field of computed tomography (CT) which has been extensively studied for many years. However, some difficulties still remain. Based on the theory of visible and invisible boundary developed by Quinto et.al, we propose a reconstruction model for limited-angle CT, which encodes the visible edges as priors to recover the invisible ones. The new model utilizes generalized shrinkage operators as regularizers to perform edge-preserving smoothing such that the visible edges are employed as anchors to recover piecewise-constant or piecewise-smooth reconstructions, while noises and artifacts are suppressed or removed. This work extends our previous research on limited-angle reconstruction which employs gradient ℓ 0 ℓ 0 and ℓ 1 ℓ 1 norm regularizers. The effectiveness of the proposed model and its corresponding solving algorithm shall be verified by numerical experiments with simulated data as well as real data. [ABSTRACT FROM AUTHOR]

In present study, we proposed a general framework based on a convolutional kernel and a variational autoencoder (CVAE) for anomaly detection on both complex image and vector datasets. The main idea is to maximize mutual information (MMI) through regularizing key information as follows: (1) the features between original input and the representation of latent space, (2) that between the first convolutional layer output and the last convolutional layer input, (3) original input and output of the decoder to train the model. Therefore, the proposed CVAE is optimized by combining the representations learned across the three different objectives targeted at MMI on both local and global variables with the original training objective function of Kullback–Leibler divergence distributions. It allowed achieving the additional supervision power for the detection of image and vector data anomalies using convolutional and fully connected layers, respectively. Our proposal CVAE combined by regularizing multiple discriminator spaces to detect anomalies was introduced for the first time as far as we know. To evaluate the reliability of the proposed CVAE-MMI, it was compared with the convolutional autoencoder-based model using the original objective function. Furthermore, the performance of our network was compared over state-of-the-art approaches in distinguishing anomalies concerning both image and vector datasets. The proposed structure outperformed the state-of-the-arts with high and stable area under the curve values. [ABSTRACT FROM AUTHOR]

Mesh denoising is a fundamental yet not well-solved problem in computer graphics. Many existing methods formulate the mesh denoising as an optimization problem, whereby the optimized mesh could best fit both the input and a set of constraints defined as an Lp norm regularizer. Instead of setting p as a static value for the whole surface, we adopt a dynamic Lp regularizer which imposes two different forms of regularization onto different surface patches for a better understanding of the local surface features. To help determine the appropriate p value for each facet, the guidance is constructed dynamically in a patch-based manner. We compare the proposed method with state-of-the-arts in both synthetic and real-scanned benchmark datasets, and show that the proposed method could produce comparable results to neural network based mesh denoising method, without collecting large training datasets.

• Introduce regularizer based on the number of isolated objects for small vessel segmentation. • The number of isolated objects estimate from the topology of segmentation result by using Euler characteristics. • Euler characteristic-based regularizers proposed to pay attention to the small and disconnected vessel regions. Segmentation of retinal blood vessels is important for the analysis of diabetic retinopathy (DR). Existing methods do not prioritize the small and disconnected vessels for DR. With the aim of paying attention to the small and disconnected vessel regions, this study introduced Euler characteristics (EC) from topology to calculate the number of isolated objects on segmented vessel regions, which is the key contribution of this study. In addition, we utilized the number of isolated objects in a U-Net-like deep convolutional neural network (CNN) architecture as a regularizer to train the network for improving the connectivity between the pixels of the vessel regions. The proposed network performance of the regularizer based on EC in reconstructing vessel regions is compared over the network without our regularizer. Furthermore, the capacity of the proposed regularizer approach in enhancing the smoothness and pixel connectivity of the vessels is compared with graph-based smoothing (GS) and combined GS with isolated objects (GISO) regularizers for delineating blood vessel regions. The proposed approach achieved the area under the curve value of 0.982, which is much higher than the state-of-the-arts, and thus it is suggested that the proposed system could support accuracy and reliability in decision-making for DR detection. [ABSTRACT FROM AUTHOR]

In the past few years, the sparse representation (SR) graph-based semi-supervised learning (SSL) has drawn a lot of attention for its impressive performance in hyperspectral image classification with small numbers of training samples. Among these methods, the probabilistic class structure regularized sparse representation (PCSSR) approach, which introduces the probabilistic relationship between samples into the SR process, has shown its superiority over state-of-the-art approaches. However, this category of classification methods only apply another SR process to generate the probabilistic relationship, which focuses only on the spectral information but fails to utilize the spatial information. In this paper, we propose using the class adjusted spatial distance (CASD) to measure the distance between each two samples. We incorporate the proposed a CASD-based distance information into PCSSR mode to further increase the discriminability of original PCSSR approach. The proposed method considers not only the spectral information but also the spatial information of the hyperspectral data, consequently leading to significant performance improvement. Experimental results on different datasets demonstrate that compared with state-of-the-start classification models, the proposed method achieves the highest overall accuracies of 99.71%, 97.13%, and 97.07% on Botswana (BOT), Kennedy Space Center (KSC) and the truncated Indian Pines (PINE) datasets, respectively, with a small number of training samples selected from each class. [ABSTRACT FROM AUTHOR]

Deep neural networks have presented superlative performance in many machine learning based perception and recognition tasks, where they have even outperformed human precision in some applications. However, it has been found that human perception system is much more robust to adversarial perturbation, as compared to these artificial networks. It has been shown that a deep architecture with a lower Lipschitz constant can generalize better and tolerate higher level of adversarial perturbation. Smooth regularization has been proposed to control the Lipschitz constant of a deep architecture and in this work, we show how a deep convolutional neural network (CNN), based on non-smooth regularization of convolution and fully connected layers, can present enhanced generalization and robustness to adversarial perturbation, simultaneously. We propose two non-smooth regularizers that present specific features for adversarial samples with different levels of signal-to-noise ratios. The regularizers build direct interconnections for the weight matrices in each layer, through which they control the Lipschitz constant of architecture and improve the consistency of input-output mapping of the network. This leads to more reliable and interpretable network mapping and reduces abrupt changes in the networks output. We develop an efficient algorithm to solve the non-smooth learning problems, which presents a gradual complexity addition property. Our simulation results over three benchmark datasets signify the superiority of the proposed formulations over previously reported methods for improving the robustness of deep architecture, towards human robustness to adversarial samples. [ABSTRACT FROM AUTHOR]

Deep structures consisting of many layers of nonlinearities have a high potential of expressing complex relations if properly initialized. Autoencoders play a complementary role in training a deep structure by initializing each layer in a greedy unsupervised manner. Due to the high capacity presented by autoencoders, these structures need to be regularized. While mathematical regularizers (based on weight decay, sparsity, etc.) and structural ones (by way of, e.g., denoising and dropout) have been well studied in the literature, quite a few papers have addressed the problem of training autoencoder with non-smooth regularization. In this paper, we address the problem of training autoencoder with non-smooth regularization. We propose an efficient algorithm and mathematically prove that it is convergent, where the regularizer needs to be proximable. As one of major applications of the proposed method, we get focused on the problem of sparse autoencoders and show that the new training method leads to better disentangling of factors of variation. [ABSTRACT FROM AUTHOR]

In the past few years, the sparse representation (SR) graph-based semi-supervised learning (SSL) has drawn a lot of attention for its impressive performance in hyperspectral image classification with small numbers of training samples. Among these methods, the probabilistic class structure regularized sparse representation (PCSSR) approach, which introduces the probabilistic relationship between samples into the SR process, has shown its superiority over state-of-the-art approaches. However, this category of classification methods only apply another SR process to generate the probabilistic relationship, which focuses only on the spectral information but fails to utilize the spatial information. In this paper, we propose using the class adjusted spatial distance (CASD) to measure the distance between each two samples. We incorporate the proposed a CASD-based distance information into PCSSR mode to further increase the discriminability of original PCSSR approach. The proposed method considers not only the spectral information but also the spatial information of the hyperspectral data, consequently leading to significant performance improvement. Experimental results on different datasets demonstrate that compared with state-of-the-start classification models, the proposed method achieves the highest overall accuracies of 99.71%, 97.13%, and 97.07% on Botswana (BOT), Kennedy Space Center (KSC) and the truncated Indian Pines (PINE) datasets, respectively, with a small number of training samples selected from each class.

It is challenging for semantic segmentation of buildings based on high-resolution remote sensing images, given high variability of appearance and complicated backgrounds of the buildings and their images. In this communication, we proposed an ensemble multi-scale residual deep learning method with the regularizer of shape representation for semantic segmentation of buildings. Based on the U-Net architecture using residual connections and multi-scale ASPP (atrous spatial pyramid pooling) modules, our method introduced the regularizer of shape representation and ensemble learning of multi-scale models to enhance model training and reduce over-fitting. In our method, the shape representation was coded in an antoencoder that was used to encode and reconstruct the shape characteristics of the buildings. In prediction, we consider multi-scale trained models for different resolution inputs and side effects to obtain an optimal semantic segmentation. With the high-resolution image of the Changshan, an island county in China, we used two-thirds of the study region image to train the model and the remaining one-third for the independent test. We obtained the accuracy of 0.98–0.99, mean intersection over union (MIoU) of 0.91–0.93 and Jaccard coefficient of 0.89–0.92 in validation. In the independent test, our method achieved state-of-the-art performance (MIoU: 0.83; Jaccard index: 0.81). By comparing with the existing representative methods on four different data sets, the proposed method consistently improved the learning process and generalization. The study shows important contributions of ensemble learning of multi-scale residual models and regularizer of shape representation to semantic segmentation of buildings.

This paper investigates the split complex gradient descent based neuro-fuzzy algorithm with self-adaptive momentum and L 2 regularizer for training TSK (Takagi–Sugeno–Kang) fuzzy inference models. The major threat for disposing complex data with fuzzy system is contradiction of boundedness and analyticity in the complex domain, as expressed by Liouville’s theorem. The proposed algorithm operates a couple of real-valued functions and splits the complex variables into real part and imaginary part. Dynamical momentum is included in the learning mechanism to promote learning speed. L 2 regularizer is also added to control the magnitude of the weight parameters. Furthermore, a detailed convergence analysis of the proposed algorithm is fully studied. The monotonic decreasing property of the error function and convergence of the weight sequence are guaranteed. Plus a mild condition, strong convergence of the weight sequence is deduced. Finally, the simulation results are also demonstrated to verify the theoretical analysis results. [ABSTRACT FROM AUTHOR]

Abstract The aim of Image super-resolution (SR) is to recover high-resolution images from low-resolution ones. By virtue of the great success in numerous computer vision tasks achieved by the convolutional neural networks (CNNs), it is a nice direction to tackle the SR problem using CNNs. Despite progress in accuracy of SR using deeper CNNs, those models are almost trained base upon supervised way. In this paper, we propose a deep unsupervised learning approach for SR with a Generative Adversarial Network (GAN) framework, which is composed of a deep convolutional generator network with dense connections and a discriminator. A sub-pixel convolutional layer is operated on the top of the generator to upscale the inputs, and the standard convolutions are all implemented in the LR space, which leads to a fast restoration. The generator is trained to directly recover the high-resolution image from the low-resolution image. Strided convolution and ReLU activations are employed in the discriminator to distinguish the HR images from the produced HR images. The generator model is optimized with a combination of a data error, a regular term and an adversarial loss, which ensures local–global contents consistency and pixel faithfulness. Note that no labeled training data is employed during the training. Comparisons with several state-of-the-art supervised learn-based methods, experimental results demonstrate that the proposed model achieves a comparable result in terms of both quantitative and qualitative measurements, and it also implies the feasibility and effectiveness of the proposed unsupervised learning-based single-image super-resolution algorithm. Highlights • A new Super-Resolution model to solve ill-posed problem with unsupervised learning. • A regularizer for smoothing continual section of image and preserving sharp edge. • A loss function for training the proposed SR model with end-to-end learning manner. • Detailed study for regular items and downsampling strategies. • Comparable SR performance with faster inference speed. [ABSTRACT FROM AUTHOR]

This paper suggests a convex regularized optimization model to produce recommendations, which is adaptable, fast, and scalable—while remaining very competitive to state-of-the-art methods in terms of accuracy. We introduce a regularizer based on the covariance matrix such that the model minimizes two measures ensuring that the recommendations provided to a user are guided by both the preferences of the other users in the system and the known preferences of the user being processed. It is adaptable since (1) it can be viewed from both user and item perspectives (allowing to choose, depending on the task, the formulation with fewer decision variables) and (2) multiple constraints depending on the context (and not only based on the accuracy, but also on the utility of personalized recommendations) can easily be added, as shown in this paper through two examples. Since our regularizer is based on the covariance matrix, this paper also describes how to improve computational and space complexities by using matrix factorization techniques in the optimization model, leading to a fast and scalable model. To illustrate all these concepts, experiments were conducted on four real datasets of different sizes (i.e., FilmTrust, Ciao, MovieLens, and Netflix) and comparisons with state-of-the-art methods are provided, showing that our context-sensitive approach is very competitive in terms of accuracy. [ABSTRACT FROM AUTHOR]

For a degenerate hyperbolic equation in characteristic region (lune) a boundary-value problem with operators of fractional integro-differentiation is studied. The solution of this equation on the characteristics is related point-to-point to the solution and its derivative on the degeneration line. The uniqueness theorem is proved by the modified Tricomi method with inequality-type constraints on the known functions. Question of the problem solution's existence is reduced to the solvability of a singular integral equation with Cauchy kernel of the normal type.

The nonlocal problem for mixed-type equation with perpendicular lines of degeneracy is investigated for the case when the Dirichlet condition is given on the elliptic boundary, and the generalized derivatives of the solution values on the characteristics are pointwise related to the solution and its normal derivatives values on the lines of a parabolic degeneracy in its hyperbolic parts.

In order to effectively exploit the intra-class and inter-class structure information, we propose a new class-wise dictionary learning method for hyperspectral image classification. First, we construct two special manifold regularizers to encourage intra-class basis sharing and inter-class basis competition, and the regularizers are incorporated into the objective function to learn a discriminative class-wise dictionary. Then the sparse representations can be obtained via the learned class-wise dictionary under the collaborative representation framework. Finally, we put the sparse representations of the data into the support vector machine (SVM) for training and then apply the SVM classifiers to predict labels for the test set. The experimental results obtained on two hyperspectral datasets demonstrate that the proposed method can obtain higher classification accuracy with much lower computational cost compared with other traditional classifiers. [ABSTRACT FROM AUTHOR]

Paper presents a unique novel online learning algorithm for eight popular nonlinear (i.e., kernel), classifiers based on a classic stochastic gradient descent in primal domain. In particular, the online learning algorithm is derived for following classifiers: L1 and L2 support vector machines with both a quadratic regularizer w w and the l regularizer | w|; regularized huberized hinge loss; regularized kernel logistic regression; regularized exponential loss with l regularizer | w| and Least squares support vector machines. The online learning algorithm is aimed primarily for designing classifiers for large datasets. The novel learning model is accurate, fast and extremely simple (i.e., comprised of few coding lines only). Comparisons of performances of the proposed algorithm with the state of the art support vector machine algorithm on few real datasets are shown. [ABSTRACT FROM AUTHOR]

Recently, sparse coding has been widely adopted for data representation in real-world applications. In order to consider the geometric structure of data, we propose a novel method, local and global regularized sparse coding (LGSC) , for data representation. LGSC not only models the global geometric structure by a global regression regularizer, but also takes into account the manifold structure using a local regression regularizer. Compared with traditional sparse coding methods, the proposed method can preserve both global and local geometric structures of the original high-dimensional data in a new representation space. Experimental results on benchmark datasets show that the proposed method can improve the performance of clustering. [ABSTRACT FROM AUTHOR]

We study the two-dimensional quasistationary Stefan probem (the Hele-Shaw problem) in which the motion of the free boundary is described by a 'fractional' Darcy law. We prove the existence and uniqueness of a classical solution to the free boundary problem for a small time interval. [ABSTRACT FROM AUTHOR]

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra ...(ℝ+, V(ℝ)) of slowlyoscillating functions of limited smoothness introduced in [12]. We show that if a ∈ ...(ℝ+, V(ℝ)) does not degenerate on the "boundary" of ℝ+ x ℝ in a certain sense, then the Mellin PDO Op(a) is Fredholm on the space Lp for p ∈ (1, ∞) and each its regularizer is of the form Op(b) + K where K is a compact operator on Lp and b is a certain explicitly constructed function in the same algebra ...(ℝ+, V(ℝ)) such that b = 1/a on the "boundary" of ℝ+ x ℝ. This result complements theknown Fredholm criterion from [12] for Mellin PDO's with symbols in the closure of ...(ℝ+, V(ℝ)) and extends the corresponding result by V.S. Rabinovich (see [16]) on Mellin PDO's with slowly oscillating symbols in C∞(ℝ+ xℝ). [ABSTRACT FROM AUTHOR]

Abstract: A popular and feasible approach to determine the appropriate size of a neural network is to remove unnecessary connections from an oversized network. The advantage of L 1/2 regularization has been recognized for sparse modeling. However, the nonsmoothness of L 1/2 regularization may lead to oscillation phenomenon. An approach with smoothing L 1/2 regularization is proposed in this paper for Takagi–Sugeno (T–S) fuzzy models, in order to improve the learning efficiency and to promote sparsity of the models. The new smoothing L 1/2 regularizer removes the oscillation. Besides, it also enables us to prove the weak and strong convergence results for T–S fuzzy neural networks with zero-order. Furthermore, a relationship between the learning rate parameter and the penalty parameter is given to guarantee the convergence. Simulation results are provided to support the theoretical findings, and they show the superiority of the smoothing L 1/2 regularization over the original L 1/2 regularization. [Copyright &y& Elsevier]

Abstract: A learning scheme based on Extreme Learning Machine (ELM) and regularization is proposed for a double parallel feedforward neural network. ELM has been widely used as a fast learning method for feedforward networks with a single hidden layer. A key problem for ELM is the choice of the (minimum) number of the hidden nodes. To resolve this problem, we propose to combine the regularization method, that becomes popular in recent years in informatics, with ELM. It is shown in our experiments that the involvement of the regularizer in DPFNN with ELM results in less hidden nodes but equally good performance. [Copyright &y& Elsevier]