Bounded Variation Spaces Research Articles

Sparse additive support vector machines in bounded variation space

Abstract We propose the total variation penalized sparse additive support vector machine (TVSAM) for performing classification in the high-dimensional settings, using a mixed $l_{1}$-type functional regularization scheme to induce sparsity and smoothness simultaneously. We establish a representer theorem for TVSAM, which turns the infinite-dimensional problem into a finite-dimensional one, thereby providing computational feasibility. Even for the least squares loss, our result fills a gap in the literature when compared with the existing representer theorem. Theoretically, we derive some risk bounds for TVSAM under both exact sparsity and near sparsity, and with arbitrarily specified internal knots. In this process, we develop an important interpolation inequality for the space of functions of bounded variation, relying on analytic techniques such as mollification and partition of unity. An efficient implementation based on the alternating direction method of multipliers is employed.

A characterization of weighted Sobolev spaces via weighted Riesz bounded variation spaces

A characterization of weighted Sobolev spaces via weighted Riesz bounded variation spaces

Non-stationary [formula omitted]-fractal functions and their dimensions in various function spaces

In this article, we study the novel concept of non-stationary iterated function systems (IFSs) introduced by Massopust in 2019. At first, using a sequence of different contractive operators, we construct non-stationary α-fractal functions on the space of all continuous functions. Next, we provide some elementary properties of the fractal operator associated with the non-stationary α-fractal functions. Further, we show that the proposed interpolant generalizes the existing stationary interpolant in the sense of IFS. For a class of functions defined on an interval, we derive conditions on the IFS parameters so that the corresponding non-stationary α-fractal functions are elements of some standard spaces like bounded variation space, convex Lipschitz space, and other function spaces. Finally, we discuss the dimensional analysis of the corresponding non-stationary α-fractal functions on these spaces.

On the Gaussian Projection Inequalities

Abstract We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection-type inequality. This inequality concludes that Ehrhard symmetrization contracts the Minkowski symmetrization of the Gaussian projection body for set of finite Gaussian perimeter $E$. On the other hand, we investigate the functional “lifting” of Ehrhard symmetrization and establish the affine Gaussian Pólya–Szegö-type inequalities in terms of the functional Ehrhard symmetrization.

Convolution Integral Operators in Variable Bounded Variation Spaces

Working in the frame of variable bounded variation spaces in the sense of Wiener, introduced by Castillo, Merentes, and Rafeiro, we prove convergence in variable variation by means of the classical convolution integral operators. In the proposed approach, a crucial step is the convergence of the variable modulus of smoothness for absolutely continuous functions. Several preliminary properties of the variable p(cdot )-variation are also presented.

Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks

We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the second-order Radon-domain bounded variation space. This space of functions was recently proposed as the natural function space associated with shallow ReLU neural networks. We derive a minimax lower bound for the estimation problem for this function space and show that the neural network estimators are minimax optimal up to logarithmic factors. This minimax rate is immune to the curse of dimensionality. We quantify an explicit gap between neural networks and linear methods (which include kernel methods) by deriving a linear minimax lower bound for the estimation problem, showing that linear methods necessarily suffer the curse of dimensionality in this function space. As a result, this paper sheds light on the phenomenon that neural networks seem to break the curse of dimensionality.

Laguerre BV spaces, Laguerre perimeter and their applications

<abstract><p>In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.</p></abstract>

An effective LRTC model integrated with total α ‐order variation and boundary adjustment for multichannel visual data inpainting

Restoring damaged multichannel visual data with high loss ratio is quite a challenging task. To address this problem, an effective LRTC (low-rank tensor completion) model integrated with total α-order variation (TVα) in the fractional bounded variation space BVα is proposed to perform superior fractional-in-space regularization. Based on using LR constraint to restore global patterns, TVα regularization is integrated to exploit nonlocally-correlated information on each channel to infer the lost data and simultaneously effectively deal with complex details due to the powerful fractional calculus. Then, a nonlocal fractional regularization strategy for multi-dimensional data and an effective numerical optimization method are creatively designed to solve this problem. Two novel fractional derivative matrix approximations are derived and applied to the first two unfolding modes of the tensor respectively to conveniently solve the fractional regularization subproblem by using an element-wise shrinkage-thresholding operation. In addition, boundary extension and adjustment strategy are designed for the unfolded matrices to alleviate the influence of inaccurate boundary conditions in fractional derivative computations. Experiments are conducted to illustrate its performance and efficiency for YUV video, RGB and HSI restoration, especially its ability to effectively recover complex structures and the details of multi-component visual data with relatively high missing rate.

Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces

We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional bounded variation spaces of fractional order s, denoted by BVs(a,b). Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis.

A note on convolution operators on Riesz Bounded variation spaces

We show some estimates and approximation results of operators of convolution type defined on Riesz Bounded variation spaces in Rn. We also state some embedding results that involve the collection of generalized absolutely continuous functions.

On functions of (ϕ, 2, α)-bounded variation

We introduce the (ϕ, 2, α)-bounded variation spaces, which are a common generalization between Riesz’s spaces, p-variation and (ϕ, 2)-bounded variation spaces. We also study its structure as Banach spaces, as well as some embedding results.

Nemytskii operator on $$(\phi ,2,\alpha )$$-bounded variation space in the sense of Riesz

In this paper we show that if the Nemytskii operator maps the $$(\phi ,2,\alpha )$$ -bounded variation space into itself and satisfies some Lipschitz condition, then there are two functions g and h belonging to the $$(\phi ,2,\alpha )$$ -bounded variation space such that $$f(t,y)=g(t)y+h(t)$$ for all $$t\in [a,b]$$ , $$y\in {\mathbb {R}}$$ .

Robust Object Tracking via Reverse Low-Rank Sparse Learning and Fractional-Order Variation Regularization

Object tracking based on low-rank sparse learning usually makes the drift phenomenon occur when the target faces severe occlusion and fast motion. In this paper, we propose a novel tracking algorithm via reverse low-rank sparse learning and fractional-order variation regularization. Firstly, we utilize convex low-rank constraint to force the appearance similarity of the candidate particles, so as to prune the irrelevant particles. Secondly, fractional-order variation is introduced to constrain the sparse coefficient difference in the bounded variation space, which allows the difference between consecutive frames to exist, so as to adapt object fast motion. Meanwhile, fractional-order regularization can restrain severe occlusion by considering more adjacent frames information. Thirdly, we employ an inverse sparse representation method to model the relationship between target candidates and target template, which can reduce the computation complexity for online tracking. Finally, an online updating scheme based on alternating iteration is proposed for tracking computation. Experiments on benchmark sequences show that our algorithm outperforms several state-of-the-art methods, especially exhibiting better adaptability for fast motion and severe occlusion.

Linear functionals on variable exponent Bochner–Lebesgue spaces

We characterize the linear functionals on variable exponent Bochner–Lebesgue spaces in terms of the variable exponent Riesz bounded variation spaces for vector measures, which are introduced in this paper.

On multipliers between bounded variation spaces

Wiener-type variation spaces, also known as BVp-spaces (1≤p<∞), are complete normed linear spaces. A function g is called a multiplier from BVp to BVq if the pointwise multiplication fg belongs to BVq for each f∈BVp. In this article, we characterize the multipliers from BVp to BVq for the cases 1≤q<p and 1≤p≤q.

Multiplication operators on the space of functions of bounded variation

Abstract In this paper,we study the properties of the multiplication operator acting on the bounded variation space BV[0, 1]. In particular,we show the existence of non-null compact multiplication operators on BV[0, 1] and non-invertible Fredholm multiplication operators on BV[0, 1].

Characterization of inclusion among Riesz−Medvedev variation spaces

We present a characterization of inclusion among Riesz−Medvedev bounded variation spaces, i.e., we shall present necessary and sufficient conditions for the Young functions \(\varphi_1\) and \(\varphi_2\) so that \(RV_{\varphi_{1}}[a,b]\subset RV_{\varphi{_{2}}}[a,b]\) or \(RV^*_{\varphi_{1}}[a,b]\subset RV^*_{\varphi_{2}}[a,b]\).

Variable exponent bounded variation spaces in the Riesz sense

In this paper we introduce variable exponent bounded variation spaces in the Riesz sense, prove some embedding results and finally we show that a type of Riesz representation lemma is valid in the newly introduced spaces.

A combined total variation and bilateral filter approach for image robust super resolution

In this paper, we consider the image super-resolution (SR) reconstitution problem. The main goal consists of obtaining a high-resolution (HR) image from a set of low-resolution (LR) ones. For that, we propose a novel approach based on a regularized criterion. The criterion is composed of the classical generalized total variation (TV) but adding a bilateral filter (BTV) regularizer. The main goal of our approach consists of the derivation and the use of an efficient combined deblurring and denoising stage that is applied on the high-resolution image. We demonstrate the existence of minimizers of the combined variational problem in the bounded variation space, and we propose a minimization algorithm. The numerical results obtained by our approach are compared with the classical robust super-resolution (RSR) algorithm and the SR with TV regularization. They confirm that the proposed combined approach allows to overcome efficiently the blurring effect while removing the noise.

Bounded Variation Spaces with p-Variable

In this paper, we introduce bounded variation spaces in the Wiener sense with p-variable and study some of its basic properties.