Lq Norm Research Articles

Operator ℓp → ℓq norms of random matrices with iid entries

We prove that for every p,q∈[1,∞] and every random matrix X=(Xi,j)i≤m,j≤n with iid centered entries satisfying the α-regularity assumption ‖Xi,j‖2ρ≤α‖Xi,j‖ρ for every ρ≥1, the expectation of the operator norm of X from ℓpn to ℓqm is comparable, up to a constant depending only on α, tom1/qsupt∈Bpn⁡‖∑j=1ntjX1,j‖q∧Logm+n1/p⁎sups∈Bq⁎m⁡‖∑i=1msiXi,1‖p⁎∧Logn. We give more explicit formulas, expressed as exact functions of p, q, m, and n, for the two-sided bounds of the operator norms in the case when the entries Xi,j are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range 1≤q≤2≤p we provide two-sided bounds under the weaker regularity assumption (EX1,14)1/4≤α(EX1,12)1/2.

Nonconvex submodule clustering via joint sliced sparse gradient and cluster-aware approach

Most existing subspace clustering methods preprocess image data by converting them into vectors, which lacks exploration of the spatial structure of high-dimensional data. Therefore, we proposes a nonconvex submodule clustering model (NSSGCA) via joint sliced sparse gradient and cluster-aware approach. NSSGCA arranges each 2D image as lateral slices of a 3rd-order tensor, and utilizes the t-product under the model of the union of free submodules to represent 3rd-order tensor samples, thereby exploring the latent spatial structure of samples. To more accurately approximate tensor rank, a nonconvex Schatten p-norm constraint is imposed on the rotated representation tensor. Under the submodule framework, a consistent gradient matrix is derived based on the δ-nearest neighbor adjacency graph to construct sliced sparse gradient (SSG) regularization, which is more conducive to clustering tasks. NSSGCA learns representation tensor with clearer block-like structure based on ℓq norm and cluster-aware attention mechanism. The convergence of the constructed sequence to the stationary Karush–Kuhn–Tucker (KKT) point is proven. Experimental results on real-world image datasets confirm the effectiveness of NSSGCA.

3D inversion of gravity data using mixed Lp–Lq norm regularization, a case study for potash targeting at the Ghare-Aghaj deposit in Iran

3D inversion of gravity data using mixed Lp–Lq norm regularization, a case study for potash targeting at the Ghare-Aghaj deposit in Iran

Sparse Linear Discriminant Analysis Based on lq Regularization

Linear discriminant analysis plays an important role in feature extraction, data dimensionality reduction, and classification. With the progress of science and technology, the data that need to be processed are becoming increasingly large. However, in high-dimensional situations, linear discriminant analysis faces two problems: the lack of interpretability of the projected data since they all involve all p features, which are linear combinations of all features, as well as the singularity problem of the within-class covariance matrix. There are three different arguments for linear discriminant analysis: multivariate Gaussian model, Fisher discrimination problem, and optimal scoring problem. To solve these two problems, this article establishes a model for solving the kth discriminant component, which first transforms the original model of Fisher discriminant problem in linear discriminant analysis by using a diagonal estimated matrix for the within-class variance in place of the original within-class covariance matrix, which overcomes the singularity problem of the matrix and projects it to an orthogonal projection space to remove its orthogonal constraints, and subsequently an lq norm regularization term is added to enhance its interpretability for the purpose of dimensionality reduction and classification. Finally, an iterative algorithm for solving the model and a convergence analysis are given, and it is proved that the sequence generated by the algorithm is descended and converges to a local minimum of the problem for any initial value.

On generalizations of the nonwindowed scattering transform

In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as Lq(Rn) norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of C2 diffeomorphisms in specific cases. Lastly, we extend our results to formulate an operator invariant to the action of rotations R∈SO(n) and an operator that is equivariant to the action of rotations of R∈SO(n).

An Image-Denoising Framework Using ℓq Norm-Based Higher Order Variation and Fractional Variation with Overlapping Group Sparsity

As one of the most significant issues in imaging science, image denoising plays a major role in plenty of image processing applications. Due to the ill-posed nature of image denoising, total variation regularization is widely used in image denoising problems for its capability to suppress noise and preserve image edges. Nevertheless, traditional total variation will inevitably yield undesirable staircase artifacts when applied to recorded images. Inspired by the success of ℓq norm minimization and overlapping group sparsity in image denoising, and the effective staircase artifacts removal by fractional total variation, the hybrid model which combines the fractional order total variation with overlapping group sparsity and higher order total variation with ℓq norm is developed in this paper to restore images corrupted by Gaussian noise. An efficient algorithm based on the parallel linear alternating direction method of multipliers is developed for solving the corresponding model and the numerical experiments demonstrate the effectiveness of the proposed approach against several state-of-the-art methods, in terms of peak signal-to-noise ratio and structure similarity index measure values.

Lq regularization for fair artificial intelligence robust to covariate shift

AbstractIt is well recognized that historical biases exist in training data against a certain sensitive group (e.g., non‐White, women) which are socially unacceptable, and these unfair biases are inherited in trained artificial intelligence (AI) models. Various learning algorithms have been proposed to remove or alleviate unfair biases in trained AI models. In this paper, we consider another type of bias in training data so‐called covariate shift in view of fair AI. Here, covariate shift means that training data do not represent the population of interest well. Covariate shift occurs when special sampling designs (e.g., stratified sampling) are used when collecting training data, or the population where training data are collected is different from the population of interest. When covariate shift exists, fair AI models on training data may not be fair in test data. To ensure fairness on test data, we develop computationally efficient learning algorithms robust to covariate shifts. In particular, we propose a robust fairness constraint based on the Lq norm which is a generic algorithm to be applied to various fairness AI problems without much hampering. By analyzing multiple benchmark datasets, we show that our proposed robust fairness AI algorithm improves existing fair AI algorithms much in terms of the fairness‐accuracy tradeoff to covariate shift and has significant computational advantages compared to other robust fair AI algorithms.

Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber's-Type Inequalities for the Second Rényi Entropy.

Let , , be the noise operator acting on functions on the boolean cube . Let f be a distribution on and let . We prove tight Mrs. Gerber-type results for the second Rényi entropy of which take into account the value of the Rényi entropy of f. For a general function f on we prove tight hypercontractive inequalities for the norm of which take into account the ratio between and norms of f.

Generalized Gini indices: Complementary sparsity measures to Box-Cox sparsity measures for machine condition monitoring

Sparsity measures that can quantify the sparsity of signals are often used as objective functions of signal processing and machine learning algorithms (e.g., sparse filtering, compressive sensing, blind deconvolution, and the fast Kurtogram, etc.). Classic sparsity measures include kurtosis, Gini index, negative entropy, the ratio of Lp norm to Lq norm, smoothness index, L1 norm, L0 norm, etc. To enrich the library of sparsity measures, this paper aims to generalize a classic Gini index to construct generalized Gini indices (GGIs). Firstly, inspired by the ratio of different quasi-arithmetic means, investigations on the GGIs showed that the GGIs satisfy all six properties of good sparsity measures. The GGIs only have one parameter a⩾0. When a=1, the GGIs are reduced to the classic Gini index. Secondly, investigations reveal that: (1) the GGIs can monotonically quantify the sparsity changes of Bernoulli coefficients. Sparsity quantification curves given by the GGIs are complementary to those generated by our recently proposed Box-Cox sparsity measures (BCSMs). Especially, when the unique parameters of the GGIs and the BCSMs are equal to zero, these sparsity quantification curves of the GGIs and BCSMs are reduced to that of negative entropy; (2) the GGIs are convergent with an increase of signal lengths; (3) when the squared envelope of white Gaussian noise is quantified by the GGIs, theoretical values of the GGIs are derived as a/(a+1), which can be used as baselines for machine abnormality detection. At last, two bearing run-to-failure datasets are used to validate the effectiveness of the GGIs for machine condition monitoring and their results showed that the GGIs are effective in detecting incipient faults and indicating informative frequency bands. In future, the proposed GGIs can be used in any algorithms that need sparsity measures.

An asymptotic thin shell condition and large deviations for random multidimensional projections

It is well known that fluctuations of marginals of high-dimensional random vectors that satisfy a certain concentration estimate called the thin shell condition are approximately Gaussian. In this article we identify a general condition on a sequence of high-dimensional random vectors under which one can identify the exponential decay rate of large deviation probabilities of the corresponding sequence of marginals. More precisely, consider the projection of an n-dimensional random vector onto a random kn-dimensional basis, kn≤n, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal kn-frames in Rn, in three different asymptotic regimes as n→∞: “constant” (kn=k), “sublinear” (kn→∞ but kn/n→0) and “linear” (kn/n→λ with 0<λ≤1). When the sequence of random vectors satisfies a certain “asymptotic thin shell condition”, we establish large deviation principles for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish large deviation principles for scaled ℓq norms of the random projections in all three regimes. Moreover, we show that the asymptotic thin shell condition holds for various sequences of random vectors of interest, including the uniform measure on suitably scaled ℓpn balls, for p∈[1,∞), and generalized Orlicz balls defined via a superquadratic function, as well as a class of Gibbs measures with superquadratic interaction potential. Along the way, we obtain logarithmic asymptotics of volumes of high-dimensional Orlicz balls, which may be of independent interest. We also show that the decay rate of large deviation probabilities of Euclidean norms of multi-dimensional projections of ℓpn balls, when p∈[1,2), exhibits an unexpected phase transition in the sublinear regime, thus disproving an earlier conjecture due to Alonso-Gutiérrez et al. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.

Lp → Lq norm estimates of Cauchy transforms on the Dirichlet problem and their applications

Denote by Cα(D) the space of the functions f on the unit disk D which are Hölder continuous with the exponent α, and denote by C1,α(D) the space which consists of differentiable functions f such that their derivatives are in the space Cα(D). Let C be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of ‖C‖Lp→Lq, where 3/2<p<2 and q=p/(p−1). Suppose g∈Lp(D) and f=G[g] is the Green potential of g. By using Sobolev embedding theorem, we show that if 1<p≤2, then f∈Cμ(D), where μ=2−2/p. We also show that if 2<p<∞, then f∈C1,ν(D), where ν=1−2/p. Finally, for the case p=∞, we show that f is not necessarily in C1,1(D), but its gradient, i.e., |∇f| is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [9].

Investigations on quasi-arithmetic means for machine condition monitoring

Machine condition monitoring aims to use on-line sensor data to evaluate machine health conditions. One of the most crucial steps is construction of a health index for incipient fault detection and monotonic degradation assessment. Moreover, observations of a health index can be used as inputs to prognostic models for machine remaining useful life prediction. Even though significant outcomes about sparsity measures, such as kurtosis, the ratio of Lp to Lq norm, pq-mean, smoothness index, negative entropy, and Gini index, for machine health monitoring have been achieved during recent years, construction of a health index for simultaneously realizing incipient fault detection and monotonic degradation assessment is not fully explored due to unexpected variances of repetitive transients caused by rotating machine faults. To solve this problem, in this paper, quasi-arithmetic means (QAMs) are thoroughly investigated. Moreover, the aforementioned sparsity measures can be respectively reformulated as the ratios of different QAMs. Further, a generalized framework based on the ratio of different QAMs for machine health monitoring is proposed. Experimental results demonstrate that some special cases of the generalized framework can simultaneously detect incipient rotating faults, exhibit a monotonic degradation tendency and be robust to impulsive noises, and they are better than existing sparsity measures for machine health monitoring.

Global Existence and Asymptotic Behavior of Solutions for Compressible Two-Fluid Euler–Maxwell Equation

We study the global existence and asymptotic behavior of the solutions for two-fluid compressible isentropic Euler–Maxwell equations by the Fourier transform and energy method. We discuss the case when the pressure for two fluids is not identical, and we also add friction between the two fluids. In addition, we discuss the rates of decay of Lp−Lq norms for a linear system. Moreover, we use the result for Lp−Lq estimates to prove the decay rates for the nonlinear systems.

Blind image recovery approach combing sparse and low-rank regularizations

The only useful prior knowledge in blind compressive sensing is that a signal is sparse in an unknown dictionary. Usually, general dictionaries cannot sparsify all images well. It simultaneously optimizes the dictionary and sparse coefficient in the reconstruction process and has been demonstrated to obtain same results as those compressive sensing techniques based on the known dictionary. In this paper, we propose a novel blind compressive sensing method combing sparse and low-rank regularizations to obtain competitive recovery results. We employ truncated Schatten-p norm and lq norm to approximate rank and norm. At last, we give an optimization strategy based on alternating direction method of multipliers to solve the recovery model. Experimental results prove that our approach could obtain the higher Peak Signal to Noise Ratio values than other competitive methods.

Adjoint-based anisotropic hp-adaptation for discontinuous Galerkin methods using a continuous mesh model

In this paper we propose an adjoint-based hp-adaptation method for conservation laws, and corresponding numerical schemes based on piecewise polynomial approximation spaces. The method uses a continuous mesh framework, similar to that proposed in [1], where a global optimization scheme was formulated with respect to the error in the numerical solution, measured in any Lq norm. The novelty of the present work is the extension to more general optimization targets. Here, any solution-dependent functional, which is compatible with an adjoint equation, may be the target of the continuous-mesh optimization. We present the rationale behind the formulation of the optimization problem, with particular emphasis on the continuous mesh model, and the relevant adjoint-based error estimate. Additionally we combine the adjoint-based error estimates with the polynomial optimization strategy from [2] to present a complete hp-adaptation method which shows exponential convergence in the target function. The h-only mesh adaptation strategy of this work has been presented as a conference proceeding earlier [3]. Numerical experiments are carried out to demonstrate the viability of the scheme.

General Fitting Methods Based on Lq Norms and their Optimization

The widely used fitting method of least squares is neither unique nor does it provide the most accurate results. Other fitting methods exist which differ on the metric norm can be used for expressing the total deviations between the given data and the fitted statistical model. The least square method is based on the Euclidean norm L2, while the alternative least absolute deviations method is based on the Taxicab norm, L1. In general, there is an infinite number of fitting methods based on metric spaces induced by Lq norms. The most accurate, and thus optimal method, is the one with the (i) highest sensitivity, given by the curvature at the minimum of total deviations, (ii) the smallest errors of the fitting parameters, (iii) best goodness of fitting. The first two cases concern fitting methods where the given curve functions or datasets do not have any errors, while the third case deals with fitting methods where the given data are assigned with errors.

Optimal condition for blow-up of the critical Lq norm for the semilinear heat equation

We shed light on a long-standing open question for the semilinear heat equation ut=Δu+|u|p−1u. Namely, without any restriction on the exponent p>1 nor on the smooth domain Ω, we prove that the critical Lq norm blows up whenever the solution undergoes type I blow-up. A similar property is also obtained for the local critical Lq norm near any blow-up point.In view of recent results of existence of type II blow-up solutions with bounded critical Lq norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical Lq norm blow-up appears to be a completely new observation.Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with backward uniqueness and unique continuation properties for parabolic equations.As a by-product, we obtain the nonexistence of self-similar profiles in the critical Lq space. Such properties were up to now only known for p≤pS and in radially symmetric case for p>pS, where pS is the Sobolev exponent.

Locality preserving difference component analysis based on the Lq norm

This paper develops locality preserving difference component analysis in which the intrinsic and global structure of data is exploited, and the model we propose also provides the flexibility to adapt some characteristics of data by applying the Lq norm. In order to solve the proposed model that is non-convex or non-smooth, we resort to the proximal alternating linearized optimization approach where each subproblem has good optimization properties. It is observed that the objective function in the proposed model is a semi-algebraic function. This allows us to give the convergence analysis of algorithms in terms of the Kurdyka–Lojasiewicz property. To be specific, the sequence of iterations generated by the proposed approach converges to a critical point of the objective function. The experiments on several data sets have been conducted to demonstrate the effectiveness of the proposed approach.

General Mixed Norm Based Adaptive Control Algorithm for Distribution Static Compensator

This paper presents an implementation of distribution static compensator (DSTATCOM) using general mixed norm (GMN) based adaptive algorithm for the three-phase distribution system. By using the convex mixture of lp norm and lq norm as the cost function, the proposed general mixed norm based algorithm is effective for obtaining the corresponding active and reactive weights of load currents. The proposed algorithm is fast in convergence and has quick response. The performance of proposed control algorithm is observed under nonlinear load in simulation. The DSTATCOM using GMN algorithm is effective in achieving the functions such as reactive power compensation, harmonics elimination and load balancing under linear and nonlinear loads and the test results of Simulink are found satisfactory with harmonic distortion of the supply currents well meeting the power quality standards.

Optimal Seismic Reflectivity Inversion: Data-driven ℓ p -loss-ℓ q -regularization Sparse Regression.

Seismic reflectivity inversion is widely applied to improve the seismic resolution to obtain detailed underground understandings. Based on the convolution model, seismic inversion removes the wavelet effect by solving an optimization problem. Taking advantage of the sparsity property, the ℓ 1 norm is commonly adopted in the regularization terms to overcome the noise/interference vulnerability observed in the ℓ p -losses minimization. However, no one has provided a deterministic conclusion that ℓ1 norm regularization is the best choice for seismic reflectivity inversion. Instead of using an unproved fixed regularization norm, we propose an optimal seismic reflectivity inversion approach. Our method adaptively adopts a ℓ p -loss-ℓ q -regularization (i.e. ℓ p,q regularization) for p = 2, 0 < q < 1 to estimate a more accurate and detailed reflectivity profile. In addition, we employ a K fold cross-validation based approach to obtain the optimal damping factor λ to further improve the seismic inversion results. The letter starts with the introduction of non-convex constraint for seismic inversion, and the necessity of the ℓ q norm regularization. Then the majorization-minimization and cross validation algorithms are briefly described. The performance of the proposed seismic inversion approach is evaluated through synthetic examples and a field example from the Bohai Bay Basin, China.