Smooth Norm Research Articles

Compressed sensing with smooth L0 constraints for moving force identification from bridge response measurements

Abstract Compressed sensing (CS), as an emerging information sampling technique, has been successfully applied in the field of moving force identification (MFI). However, existing MFI CS models often fail to obtain the optimal sparse solutions and frequently underestimate the amplitude of local impact forces. To effectively address this issue, a new CS method is proposed for MFI based on smooth L0 norm constraints and bridge response measurements. Firstly, a smooth function is used to approximate the L0 norm, establishing a noise CS reconstruction model for MFI. The introduction of the smoothing function can locally convexify the original MFI problem and enhance the smoothness and differentiability of the objective function, making the optimization problem easier to solve. Subsequently, the Polak–Ribiere–Polyak formula is adopted to point the descent direction of the new objective function, and the sparse solution is iteratively advanced through the conjugate gradient algorithm. Finally, the applicability and feasibility of the proposed method is confirmed by numerical simulations and vehicle–bridge interaction tests, respectively. The results show that the proposed method can accurately identify moving forces from limited measurements of bridge responses. Compared with existing methods, it can provide more precise sparse solutions with higher robustness to measurement noises, and address the issue of underestimating on the amplitude of local impact forces, which is expected to enhance the performance and in-situ applicability of MFI.

Approximate Morse-Sard type results for non-separable Banach spaces

For the Banach spaces E=c0(Γ),ℓp(Γ), where Γ is an arbitrary infinite set and 1<p<∞, we show that for every (non-zero) quotient F of E, every continuous function f:E→F can be uniformly approximated by smooth functions with no critical points, that is for every continuous function ε:E→(0,∞), there exists a Ck smooth function g:E→F such that g′(x) is surjective and ‖f(x)−g(x)‖≤ε(x) for all x∈E (k=∞ if E=c0(Γ) or p is an even integer, k=p−1 if p is an odd integer and k=[p] otherwise). Moreover, for a wide class of (not necessarily separable) infinite dimensional Banach spaces E and a suitable class of quotients F of E, every continuous function f:E→F can be uniformly approximated by Ck smooth functions with no critical points (k depending on the properties of the smoothness of the space E). In particular, for every Banach space E with Ck smooth partitions of unity (k∈N∪{∞}) and an infinite dimensional separable complemented subspace with a Ck smooth and LUR norm, we show that every continuous function f:E→Rn (n∈N) can be uniformly approximated by C1 smooth functions with no critical points.

Counterexamples in rotundity of norms in Banach spaces

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense.In the last part of this note, we solve a fourth question posed by the same three authors by constructing a C∞-smooth norm in c0 whose dual norm is not strictly convex.

Rotund Gâteaux smooth norms which are not locally uniformly rotund

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gâteaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed in a recent monograph by A.J. Guirao, V. Montesinos, and V. Zizler.

Seismic Thermography

ABSTRACT Variations in temperature within the Earth are of great interest because they indicate the thickness and, consequently, mechanical strength of the lithosphere and density variations and convection patterns in the sublithospheric mantle. Seismic tomography maps seismic velocity variations in the mantle, which strongly depend on temperature. Temperatures are, thus, often inferred from tomography. Tomographic models, however, are nonunique solutions of inverse problems, regularized to ensure model smoothness or small model norm, not plausible temperature distributions. For example, lithospheric geotherms computed from seismic velocity models typically display unrealistic oscillations, with improbable temperature decreases with depth within shallow mantle lithosphere. The errors due to the intermediate-model nonuniqueness are avoided if seismic data are inverted directly for temperature. The recently developed thermodynamic inversion methods use computational petrology and thermodynamic databases to jointly invert seismic and other data for temperature and composition. Because seismic velocity sensitivity to composition is much weaker than to temperature, we can invert seismic data primarily for temperature, with reasonable assumptions on composition and other relevant properties and with additional inversion parameters such as anisotropy. Here, we illustrate thus-defined seismic thermography with thermal imaging of the lithosphere and asthenosphere using surface waves. We show that the accuracy of the models depends critically on the accuracy of the extraction of structural information from the seismic data. Random errors have little effect but correlated errors of even a small portion of 1% can affect the models strongly. We invert data with different noise characteristics and test a simple method to estimate phase velocity errors. Seismic thermography builds on the techniques of seismic tomography and relies on computational petrology, but it is emerging as a field with its scope of goals, technical challenges, and methods. It produces increasingly accurate models of the Earth, with important inferences on its dynamics and evolution.

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely:λ1(β,Ω)=minψ∈W1,p(Ω)∖{0}⁡∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN−1∫Ω|ψ|pdx, where p∈]1,+∞[, Ω is a bounded, anisotropic mean convex domain in RN, νΩ is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on RN. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ1 in the case β>0, and a upper bound in the case β<0. As a consequence, we prove, for β>0, a lower bound for λ1(β,Ω) in terms of the anisotropic inradius of Ω and, for β<0, an upper bound of λ1(β,Ω) in terms of β.

Real Aperture Radar Angular Super-Resolution Imaging Using Modified Smoothed L0 Norm with a Regularization Strategy

Restricted by the ill-posed antenna measurement matrix, the conventional smoothed L0 norm algorithm (SL0) fails to enable direct real aperture radar angular super-resolution imaging. This paper proposes a modified smoothed L0 norm (MSL0) algorithm to address this issue. First, as the pseudo-inverse of the ill-posed antenna measurement matrix is required to set the initial values and calculate the gradient projection, a regularization strategy is employed to relax the ill-posedness. Based on the regularization strategy, the proposed MSL0 algorithm can avoid noise amplification when faced with the ill-posed antenna measurement matrix of real aperture radar. Additionally, to prevent local minima problems, we introduce a hard thresholding operator, based on which the proposed MSL0 algorithm can accurately reconstruct sparse targets. Simulations and experimental results verify the performance of the proposed MSL0 algorithm.

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Recovery performance improvement of image compressive sensing using complex‐valued Vandermonde matrix

AbstractHere, a novel image‐based quantized compressive sensing (QCS) framework based on complex‐valued Vandermonde (Vander) matrix is proposed. In the proposed QCS framework, a discrete wavelet transform (DWT) serves as a sparse basis and a partial complex‐valued Vander matrix serves as a measurement matrix. The theoretical analysis based on mutual coherence metric of compressive sensing (CS) theory shows that the proposed Vander measurement matrix has the best reconstruction performance among other conventional measurement matrices. The simulation results also show that the recovery quality using the proposed measurement matrix can be greatly improved compared with the other existing real‐valued measurement matrices. In particular, the experiment results also show that under the same measurement matrix, the reconstruction performance of Smoothed l0 norm (SL0) algorithm is better than that of Orthogonal Matching Pursuit (OMP) algorithm, sparsity adaptive matching pursuit (SAMP) algorithm and approximate message passing (AMP) algorithm. In addition, a sparse measurement matrix scheme is further proposed to achieve a trade‐off between recovery performance and computational complexity. The theoretical analysis and simulation results both show the proposed image‐based QCS is efficient.

Fast and Robust Non-Rigid Registration Using Accelerated Majorization-Minimization

Non-rigid 3D registration, which deforms a source 3D shape in a non-rigid way to align with a target 3D shape, is a classical problem in computer vision. Such problems can be challenging because of imperfect data (noise, outliers and partial overlap) and high degrees of freedom. Existing methods typically adopt the lp type robust norm to measure the alignment error and regularize the smoothness of deformation, and use a proximal algorithm to solve the resulting non-smooth optimization problem. However, the slow convergence of such algorithms limits their wide applications. In this paper, we propose a formulation for robust non-rigid registration based on a globally smooth robust norm for alignment and regularization, which can effectively handle outliers and partial overlaps. The problem is solved using the majorization-minimization algorithm, which reduces each iteration to a convex quadratic problem with a closed-form solution. We further apply Anderson acceleration to speed up the convergence of the solver, enabling the solver to run efficiently on devices with limited compute capability. Extensive experiments demonstrate the effectiveness of our method for non-rigid alignment between two shapes with outliers and partial overlaps, with quantitative evaluation showing that it outperforms state-of-the-art methods in terms of registration accuracy and computational speed. The source code is available at https://github.com/yaoyx689/AMM_NRR.

Talagrand’s influence inequality revisited

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies $$\mathrm{Var}_{\sigma_n}(f) \leq C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i f\|_{L_1(\sigma_n)}\big)}.$$ In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach space-valued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman (2002) on the action of negative powers of the hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$ has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube $\mathscr{C}_n$, thus deriving new nonembeddability results for these finite metrics.

Renormings preserving local geometry at countably many points in spheres of Banach spaces and applications

We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [7] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a Ck-smooth norm an equivalent norm which is Ck-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in c0(Γ) for any infinite Γ a C∞-smooth norm whose ball is dentable but whose sphere lacks any extreme points.

What does Birkhoff--James orthogonality know about the norm?

It is shown that Birkhoff-James orthogonality knows everything about the smooth norms in reflexive Banach spaces and can also compute the dimensions of the underlying normed spaces.

Smooth and Polyhedral Norms via Fundamental Biorthogonal Systems

Abstract Let $\mathcal {X}$ be a Banach space with a fundamental biorthogonal system, and let $\mathcal {Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal {Y}$ admits a $C^\infty $-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal {Y}$ admits locally finite, $\sigma $-uniformly discrete $C^\infty $-smooth and LFC partitions of unity and a $C^1$-smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelöf determined Banach space (hence, all reflexive ones), $L_1(\mu )$ for every measure $\mu $, $\ell _\infty (\Gamma )$ spaces for every set $\Gamma $, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density $\omega _1$ are covered by our result.

MIMO Radar Imaging Based on Two-dimensional Compressive Sensing Recovery Algorithm

Conventional sparse signal recovery-based MIMO radar imaging method rearranges the received two-dimensional (2D) signals into a vector, However, transforming the two dimensional equation to one dimensional equation then solving linear equation based sparse signal recovery problem requires huge memory and computational budget. A two-dimensional compressed sensing MIMO radar imaging algorithm based on improved smooth L0 norm is proposed in this paper. Firstly, a two-dimensional MIMO radar imaging model is established to transform the imaging problem into a sparse optimization problem. Then, the two-dimensional negative exponential function sequence is used as the smoothed function sequence to approximate the L0 norm. The two-dimensional imaging of MIMO radar is realized by cyclic iteration and gradient projection algorithm. Simulation results show that the proposed algorithm has advantages over the traditional compressed sensing algorithms.

Improving the performance of time varying spherical radial basis functions in regional VTEC modeling with sparse data

Least-squares parameterization of retrieved Vertical Total Electron Content (VTEC) from permanent dual-frequency GPS stations in terms of Time Varying Spherical Radial Basis Functions (TV-SRBFs) with optimal shape parameter leads to proper regional VTEC reconstruction at Ionospheric Pierce Points (IPPs). However, in the area where there is no observation, the modeling result is not satisfactory. This feature is especially tangible when the number of IPPs in the study area is low with poor geometric distribution, which leads to a rank deficient problem. Whereas, if the global ionospheric maps (GIMs) are used as a background in estimating the optimal shape parameter of TV-SRBFs and coefficients of the expansion, the stated obstacle to produce accurate regional ionospheric maps will be removed. For this purpose, an objective function is defined and optimized. The objective function consists of three terms. First, L-2 norm of the differences between the reproduced VTECs and values obtained from the receivers’ observations. Second, L-2 norm of the differences between the reproduced values on a regular grid with the values obtained from the spatial interpolation of GIM at a specific moment in the modeling interval. Third, the discrete smoothing norm of the coefficients of TV-SRBFs. Since the obtained VTECs from GPS receivers are more accurate than the resulted VTECs from interpolation of GIMs and also a regularization parameter is needed due to the ill-posedness, the three objectives cannot be considered with the same weights in the multi-objective optimization. Hence the weighted sum method with the preference-based solution is applied to obtain the optimal relative weight of GIMs, the regularization parameter and the coefficients. Moreover, utilizing the least-squares collocation (LSC) can also improve the modeling accuracy to some extent. The proposed method is applied for hourly regional VTEC modeling over Iran from 3 May to 9 May 2016, which had both low and very high geomagnetic activities. A portion of the retrieved VTECs which is excluded from the modeling dataset, is considered as test data. The Root Mean Square (RMS) of the differences between obtained VTECs from regional constructed model and test data is 0.94 TECU in the study period. RMS of the differences between obtained VTECs from the regional model and VTECs of two control stations which are not involved in the modeling are 1.12 TECU and 1.24 TECU. Also, the regional model performs well in ionospheric delay estimation and the accuracy of single-frequency positioning at control stations in the study period is better than 0.7 m.

A Photoacoustic Imaging Algorithm Based on Regularized Smoothed L0 Norm Minimization.

The recently emerging technique of sparse reconstruction has received much attention in the field of photoacoustic imaging (PAI). Compressed sensing (CS) has large potential in efficiently reconstructing high-quality PAI images with sparse sampling signal. In this article, we propose a CS-based error-tolerant regularized smooth L0 (ReSL0) algorithm for PAI image reconstruction, which has the same computational advantages as the SL0 algorithm while having a higher degree of immunity to inaccuracy caused by noise. In order to evaluate the performance of the ReSL0 algorithm, we reconstruct the simulated dataset obtained from three phantoms. In addition, a real experimental dataset from agar phantom is also used to verify the effectiveness of the ReSL0 algorithm. Compared to three L0 norm, L1 norm, and TV norm-based CS algorithms for signal recovery and image reconstruction, experiments demonstrated that the ReSL0 algorithm provides a good balance between the quality and efficiency of reconstructions. Furthermore, the PSNR of the reconstructed image calculated by the introduced method was better than the other three methods. In particular, it can notably improve reconstruction quality in the case of noisy measurement.

On the smoothing of the norm objective penalty function for two-cardinality sparse constrained optimization problems

This paper studies a smoothing norm objective penalty function for two-cardinality sparse constrained optimization problems. Good properties are proved for the smoothing norm objective penalty functions in solving two-cardinality sparse constrained optimization problems, after which an algorithm is presented. Furthermore, its convergence is also proved under some conditions. The proposed algorithm can solve a satisfactory approximate optimal solution in a numerical example.

Uniqueness of minimal projections in smooth expanded matrix spaces

Let us consider the space M(n, m) of all real or complex matrices on n rows and m columns. In 2000 Lesław Skrzypek proved the uniqueness of minimal projection of this space onto its subspace M(n,1)+M(1,m) which consists of all sums of matrices with constant rows and matrices with constant columns. We generalize this result using some new methods proved by Lewicki and Skrzypek (J Approx Theory 148:71–91, 2007). Let S be a space of all functions from Xtimes Y times Z into {mathbb {R}} or {mathbb {C}}, where X, Y, Z are finite sets. It could be interpreted as a space of three-dimensional matrices M(n, m, r). Let T be a subspace of S consisting of all sums of functions which depend on one variable. Let S be equipped with a smooth norm Vert .Vert . We show that there exists the unique minimal projection of S onto its subspace T.

An improved regularization-based method of blur kernel estimation for blind motion deblurring

In engineering application, image blur is a troublesome problem due to camera shaking or relative motion. To address it, this paper proposes an efficient regularization method for blind motion deblurring of image. In the process of estimating blur kernel. The optimization energy function is improved. Smooth L0 norm is introduced instead of L0 norm to facilitate the optimization of energy function. Besides, a regular term is reduced so that the corresponding parameter is saved as well. In addition, an optimization algorithm is presented to automatically solve the optimal values of parameters through simulated annealing. Experimental results show that the proposed method is able to accurately estimates blur kernel and generate satisfactory retrieved images. Compared with the deep learning-based methods, the proposed method is simpler, faster and less dependent on hardware, which is easier to be implemented on embedded devices so that can be better applied to areas such as UAV photography.