Continuous Shearlet Tight Frames

Based on the shearlet transform we present a general construction of continuous tight frames for L ²(ℝ²) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform.     

Construction of Compactly Supported Shearlet Frames

Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital settings. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.     

Global Symmetric Approximation of Frames

Journal of Fourier Analysis and Applications - We solve the problem of best approximation by Parseval frames to an arbitrary frame in a subspace of an infinite dimensional Hilbert space. We...     

Approximation by Partial Isometries and Symmetric Approximation of Finite Frames

We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize all the solutions. In particular, this allows us to give a simple necessary and sufficient condition for uniqueness. We then apply these results to solve the global problem of approximation by partial isometries, and to extend the notion of symmetric approximation of frames introduced in Frank et al. (Trans Am Math Soc 354: 777–793, 2002). In addition, we characterize symmetric approximations of frames belonging to a prescribed subspace.     

Nonlinear Frames and Sparse Reconstructions in Banach Spaces

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement \(F(x)+\epsilon \) may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union \(\mathbf{A}\) of closed linear subspaces of a Hilbert space \(\mathbf{H}\) from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple \((\mathbf{A}, \mathbf{M}, \mathbf{H})\), and show that the minimizer

$$\begin{aligned} x^*=\mathrm{argmin}_{\hat{x}\in {\mathbf M}\ \mathrm{with} \ \Vert F(\hat{x})-F(x^0)\Vert \le \epsilon } \Vert \hat{x}\Vert _{\mathbf M} \end{aligned}$$

provides a suboptimal approximation to the original sparse signal \(x^0\in \mathbf{A}\) when the measurement map F has the sparse Riesz property and the almost linear property on \({\mathbf A}\). The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property.

     

Sparse Approximation of Singularity Functions

We are concerned with the sparse approximation of functions on the d-dimensional unit cube [0,1]^d, which contain powers of distance functions to lower-dimensional k-faces (corners, edges, etc.). These functions arise, e.g., from corners, edges, etc., of domains in solutions to elliptic PDEs. Usually, they deteriorate the rate of convergence of numerical algorithms to approximate these solutions. We show that functions of this type can be approximated with respect to the H¹ norm by sparse grid wavelet spaces V_L, (V_L) = N_L, of biorthogonal spline wavelets of degree p essentially at the rate p: \[ \|u - P_Lu\|_{H^1([0,1]^d)} \leq CN_L^{-p}\,(\log_2 N_L)^s \|u\|, \qquad s = s(p,d), \] where || · || is a weighted Sobolev norm and P_Lu \in V_L.     

Dualizable algebras with parallelogram terms

We prove that if A is a finite algebra with a parallelogram term that satisfies the split centralizer condition, then A is dualizable. This yields yet another proof of the dualizability of any finite algebra with a near unanimity term, but more importantly proves that every finite module, group or ring in a residually small variety is dualizable.     

Approximation of the Inverse Frame Operator and Applications to Gabor Frames

A frame allows every element in a Hilbert space to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculation of the frame coefficients requires inversion of an operator S on . We show how the inverse of S can be approximated as close as we like using finite-dimensional linear algebra. In contrast with previous methods, our approximation can be used for any frame. Various consequences for approximation of the frame coefficients or approximation of the solution to a moment problem are discussed. We also apply the results to Gabor frames and frames consisting of translates of a single function.     

Sparse Dual Frames and Dual Gabor Functions of Minimal Time and Frequency Supports

Exploitation of the optimality of (non-exact) frames from a sparse dual point of view is presented. Sparse dual frames and dual Gabor functions of the minimal time and/or frequency supports are studied and constructed through the notion of sparse representations. Conditions on the sparsest dual frames and the dual Gabor functions of the minimal time and/or frequency supports are discussed. Algorithms and examples are provided.     

A Note on the Approximation Properties of Frames of General Form

Mathematical Notes -     

Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

Journal of Fourier Analysis and Applications - In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the...     

Shearlet Smoothness Spaces

The shearlet representation has gained increasingly more prominence in recent years as a flexible and efficient mathematical framework for the analysis of anisotropic phenomena. This is achieved by combining traditional multiscale analysis with a superior ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.     

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

One of the open problems in the field of forward uncertainty quantification(UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in [1] to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned l1 minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus.     

Irregular Wavelet Frames and Gabor Frames

Given g∈L²(R ⁿ ), we consider irregular wavelet for the form\(\left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j \) > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L²(R ⁿ ) are given. For a class of functions g∈L²²(R ⁿ ) we prove that certain growth conditions on {λ _j } will frames, and that some other types of sequences exclude the frame property. We also give a sufficient condition for a Gabor system\(\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} \)to be a frame.     

Irregular Wavelet Frames and Gabor Frames

Given g { l^\fracn2 g( l_j x - kb ) }_jezjezⁿ ,where  l_j \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L ²(R ⁿ ) are given. For a class of functions g{ e^{zrib( j,x )} g( x - l_k ) }_{jezⁿ ,kez}\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame.     

基于新的Hessian近似矩阵的稀疏重构算法

一般来说,基于二次近似模型的优化算法具有良好的数值表现.然而,当基于二次近似模型的优化算法求解大规模优化问题时,若使用稠密矩阵近似目标函数在迭代点的Hessian矩阵,需要花费大量的计算成本和存储成本,因此设计Hessian矩阵合适的标量近似矩阵特别重要.对于正则化模型,利用最近三次迭代的信息,设计粗糙的标量矩阵,使用拟牛顿公式进行更新,结合近似最优梯度法的思想和梯度法的延迟策略,构造Hessian矩阵新的含有更多二阶信息的标量近似矩阵.结合非单调线搜索,提出基于新的Hessian近似矩阵的稀疏重构算法,并进行收敛性分析.实验结果表明,与经典稀疏重构算法算法相比,基于新的Hessian近似矩阵的稀疏重构算法在重构效果相似的情况下能较大地减少迭代次数和较快地重构信号.     

广义Frame与广义Frame的商

本文将frame、frame同态、商frame与核的概念在范畴意义下作推广,并且证明Frame范畴是广义Frame范畴的反射子范畴.进而讨论了一个广义frame A的商与A上核函子之间的关系.特别地,我们证明了A上全部核函子所构成的范畴N(A)是一个广义frame.     

Shearlet coorbit spaces and associated Banach frames

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.     

Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

A dimension reduction method based on the “Nonlinear Level set Learning” (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.