L1-norm Research Articles

Multi-view hypergraph regularized Lp norm least squares twin support vector machines for semi-supervised learning

In recent years, multi-view semi-supervised learning has gradually become a popular research direction. The classic binary classification methods in this field are multi-view Laplacian support vector machines (MvLapSVM) and multi-view Laplacian twin support vector machines (MvLapTSVM), which extend semi-supervised support vector machine to multi-view learning. Nevertheless, similar to the majority of SVM-based multi-view methods, the above methods are two-view methods that cannot fully leverage the information from all views and are constructed based on the L2 norm. Additionally, in semi-supervised graph learning, the quality of the graph often has a significant impact on the results. Therefore, we propose a novel multi-view hypergraph regularized Lp norm least squares twin support vector machines (MvHGLpLSTSVM) that can handle general multi-view data for semi-supervised learning. It extends hypergraph learning to multi-view learning and combines Lp norm to further explore the manifold structure and embedded geometric information of multi-view data. By using equality constraints, we design a simple and effective iterative algorithm. In the classification of six multi-view datasets, we compare the proposed method with some other state-of-the-art methods, and the results show that the proposed method is effective.

Some characterizations of ergodicity in Riesz spaces

In the recent surge of papers on ergodic theory within Riesz spaces, this article contributes by introducing enhanced characterizations of ergodicity. Our work extends and strengthens prior results of both the authors and Homann, Kuo, and Watson. Specifically, we show that in a conditional expectation preserving system (E, T, S, e), S can be extended to L 1(T) and operates as an isometry on Lp (T) spaces.

WELL-TO-WELL (W2W) ELECTROMAGNETIC TOMOGRAPHY MODELING ADVANCEMENT: IMPROVING PRECISION AND EFFECTIVENESS WITH REGULARIZATION

This research describes the direct and inverse problems of cross-well electromagnetic tomography. The model geometry has azimuthal symmetry, which simplifies the forward modeling and the inversion procedure. In the direct problem, the finite element method is used in the numerical solution of the Helmholtz equation. In the inverse problem, the study discusses the use of three stabilizer functionals: Global Smoothness (GS), Total Variation (TV), and Absolute Equality (AE). The first uses a smoothing function in the L2 norm, while the latter uses smoothing in the L1 norm, for it accepts abrupt changes between adjacent parameters. The results show that the TV method generated good estimates of both geometry and conductivity of the bodies, both for small and large conductivity contrast between the targets and the surrounding environment. Through the results, one can also observe that the regularization of the Total Variation presented a better estimate of the parameters than the Global Smoothness. In most of the synthetic models used in this work, the best estimates of the proposed model occurred when Absolute Equality constraints were used on the cells at the edges of the inversion grid, in addition to the stabilizer functional.

Total Third-Degree Variation for Noise Reduction in Atomic-Resolution STEM Images.

Scanning Transmission Electron Microscopy (STEM) enables direct determination of atomic arrangements in materials and devices. However, materials such as battery components are weak for electron beam irradiation and low electron doses are required to prevent beam-induced damages. Noise removal is thus essential for precise structural analysis of electron beam sensitive materials at atomic resolution. Total square variation (TSV) regularization is an algorithm that exhibits high noise removal performance. However, the use of the TSV regularization term leads to significant image blurring and intensity reduction. To address these problems, we here propose a new approach adopting L2 norm regularization based on higher-order total variation. An atomic-resolution STEM image can be approximated as a set of smooth curves represented by quadratic functions. Since the third-degree derivative of any quadratic function is 0, total third-degree variation (TTDV) is suitable for a regularization term. The application of TTDV for denoising the atomic-resolution STEM image of CaF2 observed along the [001] zone axis is shown, where we can clearly see the Ca and F atomic columns without compromising image quality.

Sparse portfolio optimization via [formula omitted] over [formula omitted] regularization

Sparse portfolio optimization, which significantly boosts the out-of-sample performance of traditional mean–variance methods, is widely studied in the fields of optimization and financial economics. In this paper, we explore the ℓ1/ℓ2 fractional regularization constructed by the ratio of the ℓ1 and ℓ2 norms on the mean–variance model to promote sparse portfolio selection. We present an ℓ1/ℓ2 regularized sparse portfolio optimization model and provide financial insights regarding short positions and estimation errors. Then, we develop an efficient alternating direction method of multipliers (ADMM) method to solve it numerically. Due to the nonconvexity and noncoercivity of the ℓ1/ℓ2 term, we give the convergence analysis for the proposed ADMM based on the nonconvex optimization framework. Furthermore, we discuss an extension of the model to incorporate a more general ℓ1/ℓq regularization, where q>1. Moreover, we conduct numerical experiments on four stock datasets to demonstrate the effectiveness and superiority of the proposed model in promoting sparse portfolios while achieving the desired level of expected return.

Fire detection and risk assessment via Support Vector Regression with Flattening-Samples Based Augmented Regularization

We propose a Hybrid Support Vector Regression (SVR) with Flattening-Samples Based Augmented Regularization (Hybrid FSR-SVR) architecture for multi-sensor fire detection and forest fire risk assessment. The Hybrid FSR-SVR is a lightweight architecture built upon the novel Flattening-Samples Based Augmented Regularization (FSR) approach and temporal trends of environmental variables. The FSR approach augments l2 norm based smoothing term into an l1-l2 combination, facilitating the integration of l1 regularization into the SVR method, thereby enhancing generalization with minimal computational load. We evaluate the performance of Hybrid FSR-SVR using two distinct datasets covering indoor and forest fires, benchmarking against 15 machine learning models, including state-of-the-art techniques, such as Recurrent Trend Predictive Neural Network (rTPNN), Long-Short Term Memory (LSTM), Multi-Layer Perceptron (MLP), Gated Recurrent Unit (GRU), and Gradient Boosting. Our findings demonstrate that Hybrid FSR-SVR effectively assesses the risk of forest fire, enabling early preventive measures. Notably, it achieves a remarkable accuracy of 0.95 for forest fire detection and ranks third with 0.88 accuracy for indoor fire detection. Importantly, it exhibits computation times significantly lower – by 1 to 2 orders of magnitude – than the majority of compared techniques. The superior generalization ability of Hybrid FSR-SVR, facilitated by flattening-samples based augmented regularization, allows for high detection performance even with smaller training sets.

Approximation of time separating stochastic processes by neural networks revisited

Here we study the univariate quantitative approximation of time separating stochastic process over the whole real line by the normalized bell and squashing type neural network operators. Activation functions here are of compact support. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged stochastic function or its high order derivative. The approximations are pointwise and with respect to the Lp norm. The feed-forward neural networks are with one hidden layer. We finish with a great variety of special applications.

From Schrödinger equation to tempered distribution space on Minkowski chronotope and Schwartz Linear Algebra

In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

Optimal convex approximations of qubit states under Pauli distance

Utilizing the l1 norm as the quantum state distance metric, we consider the Pauli distance of arbitrary target quantum states, that is, the optimal convex approximation of any qubit mixed states relative to the available state of pairs of eigenstates in three Pauli matrices. We get the complete analytical expressions for B1 -distance, B2 -distance and B3 -distance. We determine the parameter range for the optimal approximation of each B2 -distance in eight distinct scenarios. Furthermore, some trade-off relations about B2 -distances are also studied.

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁p(·)

Let C denote a convex subset within the vector space 𝓁p(·), and let T represent a mapping from C onto itself. Assume α=(α1,⋯,αn) is a multi-index in [0,1]n such that ∑i=1nαi=1, where α1>0 and αn>0. We define Tα:C→C as Tα=∑i=1nαiTi, known as the mean average of the mapping T. While every fixed point of T remains fixed for Tα, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for Tα and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in R. Our focus then shifts to variable exponent modular vector spaces 𝓁p(·), where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework.

Combinatorial algorithms for restricted inverse optimal value problems on minimum spanning tree under weighted [formula omitted] norm

We study the restricted inverse optimal value problem on minimum spanning tree under weighted l1 norm. In a connected edge-weighted network G(V,E,w), we are given a spanning tree T0, a cost vector c and a value K. We aim to obtain a new weight vector w̄ satisfying the bounded constraint such that T0 is a minimum spanning tree under w̄ whose weight is just K. We focus on minimizing the modification cost under weighted l1 norm. We first convert its mathematical model into a linear programming problem (P). Then we solve its dual problem (D) by a sub-problem (Dz∗) corresponding to the critical value z∗ which can be calculated by a binary search method. Impressively the sub-problem Dz for z∈R can be transformed into a minimum cost flow problem on an auxiliary network. Finally, we propose an O(|E|2|V|2log|V|log(|V|cmax)) algorithm, where cmax is the maximum cost of the vector c.

Seismic wavelet shape-oriented reflectivity inversion method

Abstract Reflectivity inversion plays a pivotal role in reservoir prediction. Conventional sparse-spike deconvolution assumes that the reflectivity (reflection coefficient) is sparse, which is solved based on the l1 norm. However, the restricted isometry property (RIP) of wavelet matrix and seismic effective bandwidth limits the accuracy of the sparse-spike reflectivity inversion. Consequently, we investigate the connection between seismic amplitude shape and reflectivity. When the reflectivity contains more non-zero values, the wavelet bandwidth within the effective seismic data bandwidth approaches a limit corresponding to the Sinc wavelet, where the main-lobe amplitude closely approximates the reflectivity. Conversely, when the reflectivity has fewer non-zero values, a wavelet with a smaller sidelobe provides a more accurate approximation of the reflectivity. In this paper, we propose a high-resolution inversion optimization method based on joint l2 norm and l1 norm constraints. By parameter tuning, we construct the Sinc wavelet or the wavelet with a weak-sidelobe corresponding to the seismic spectrum. Subsequently, we determine the extremum to approximate the reflectivity. To mitigate the RIP condition's constraints, we employ the l2 norm to balance the l1 norm (joint constraint) by introducing l2 norm with low-pass filtering characteristics. This approach yields more accurate reflectivity estimates. By taking the extremum, this approach yields more accurate reflectivity estimates. The synthetic test demonstrates that our method achieves better reflectivity inversion accuracy compared to sparse-spike inversion with l1–l2 norm constraint. Furthermore, field tests indicate that the proposed reflectivity inversion method not only can better match the well curve, but also exhibits excellent resolution.

Properties of eigenfunctions of a boundary value problem for ordinary differential equations of fourth-order with boundary conditions depending on the spectral parameter

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space Lp,1<p<∞, after removing four functions.

An effective smoothing Newton projection algorithm for finding sparse solutions to NP-hard tensor complementarity problems

In this paper, we study the sparse solution of the tensor complementarity problem (TCP), which is an NP hard problem due to the nonconvexity and noncontinuity of the ℓ0 norm. We transform the complementarity constraints into a fixed point equation with projection type and use 0/1-loss function to relax ℓ0 norm. We develop a smoothing Newton projection (SNP) algorithm to solve the transformed problem. The subproblem with 0/1-loss function is solved by Newton method. Finally, numerical results show that the SNP algorithm can effectively get the sparse solution to a TCP.

Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H1 and Lp norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H1 and Lp norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

Machine Learning Algorithms for Predicting Wear Rates on the Basis of Friction Noise

Under varying operational conditions, the contact and relative movement of a polymer and metal result in surface wear, accompanied by the emission of noise. The relationship between friction noise and wear is inherently complex and nonlinear. In light of these tribological characteristics, this paper introduces the implementation of a random forest algorithm and generalized regression neural network algorithm to establish a mathematical model for predicting the wear rate based on friction noise. To enhance the accuracy of wear rate regression, this study incorporates L2 norm feature selection and the sparrow search algorithm, which are tailored toward the friction characteristics. These techniques optimize the machine learning-based friction model, ultimately improving the regression accuracy of the wear rate.

LP norm regularized deep CNN classifier based on biwolf optimization for mitosis detection in histopathology images

LP norm regularized deep CNN classifier based on biwolf optimization for mitosis detection in histopathology images

Semi-blind sparse channel estimation using regularized expectation maximization

In Massive multiple-input multiple-output (MIMO) systems, channel estimation is crucial. The large size of the antennas causes a significant pilot and feedback overhead, making it challenging to estimate channels in massive MIMO systems. Besides, the studies have shown that mmWave channels are sparse due to the limited number of dominant propagation paths. Therefore, the motivation of this paper is to exploit the inherent sparsity of the massive mmWave MIMO channels to develop a semi-blind channel estimation with reduced number of pilots. To this goal, an expectation maximization (EM) based technique has been developed which leverages the sparsity of the underlying channel for its better estimation. An iterative approach is proposed to solve the modeled problem which simultaneously updates the channel coefficients and data symbols using available data and system structure at each iteration. The proposed method imposes sparsity with the aid of Smoothed L0 norm (SL0) in the M-step. The simulation results demonstrate the proposed method have quick convergence and lower channel estimation error compared to the existing methods. As a quantitative evaluation, the proposed method attains the normalized mean square error of 9×10−2 and 7×10−3 at SNR=5dB and SNR=15dB, respectively.

Differentially private federated learning with local momentum updates and gradients filtering

Differential Privacy (DP) is applied in Federated Learning (FL) for defending against various privacy attacks. Existing methods based on Gaussian mechanism require the operations of clipping and adding noise, leading to significant accuracy degradation. In this paper, we propose a novel FL scheme named DPFL-LMG to provide user-level DP guarantee while maintaining a high model accuracy. Our main idea is to mitigate the negative effects of the clipping on the model convergence by decreasing the L2 norm of local updates and the cross-client update variance. Specifically, our method includes two techniques, Local Momentum Updates (LMU) and Gradients Filtering (GF). LMU combines local updates of different rounds in a momentum way. It can significantly decrease the cross-client update variance by weakening the gradient noise in local updates caused by stochastic gradient descent (SGD) algorithm. GF estimates the gradient noise in each element of local updates by observing the element-wise variance. Elements with large noise are considered unnecessary and are zeroed out for the reduction of local update norms. We theoretically analyze the privacy guarantee and the convergence of our method. Experiments demonstrate that DPFL-LMG can effectively mitigate the accuracy degradation caused by clipping and outperform previous DPFL methods in the accuracy.