Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms ||⋅|| _Hs([0,1]) for s∈ (-0.824926,2.5) . In particular, the multiwavelets form Riesz bases for L ₂ ([0,1]) . February 2, 1998. Date revised: February 19, 1999. Date accepted: March 5, 1999.     

缺项算子矩阵的二阶代数(Ⅰ)

对于任意给定的二阶多项式ｐ（ｔ）;本文获得希尔伯特空间上形如（?）的缺项算子矩阵具有一个补Ｔ使得ｐ（Ｔ）＝０成立的充分必要条件以及使得ｐ（Ｔ）＝０且ｐ（Ｔ）的范数不大于事先给定常数的充分必要条件．进而还求出所有可能的二阶代数补,特别地,对有限维情形给出简洁的表示。     

On the eigenvalues of matrices with prescribed upper triangular part

Consider a matrix D=[D_i,j] partitioned in submatrices D_i,j where the submatrices D_i,iare square. This paper studies the possible eigenvalues of D, when the blocks D_i,j, with i≤j, are fixed, and the other blocks vary. The result obtained generalizes twotheorems already known.     

Extensions of valuations to rational function fields over completions

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of v to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.     

Remarks on derived complete modules and complexes

Let R be a commutative ring and $I\subset R$ a finitely generated ideal. We discuss two definitions of derived I-adically complete (also derived I-torsion) complexes of R-modules, which appear in the literature: the idealistic and the sequential ones. The two definitions are known to be equivalent for a weakly proregular ideal I; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider I-adically flat R-modules.     

Multi-view side information-incorporated tensor completion

Tensor completion originates in numerous applications where data utilized are of high dimensions and gathered from multiple sources or views. Existing methods merely incorporate the structure information, ignoring the fact that ubiquitous side information may be beneficial to estimate the missing entries from a partially observed tensor. Inspired by this, we formulate a sparse and low-rank tensor completion model named SLRMV. The ${\ell }_0$-norm instead of its relaxation is used in the objective function to constrain the sparseness of noise. The CP decomposition is used to decompose the high-quality tensor, based on which the combination of Schatten $p$-norm on each latent factor matrix is employed to characterize the low-rank tensor structure with high computation efficiency. Diverse similarity matrices for the same factor matrix are regarded as multi-view side information for guiding the tensor completion task. Although SLRMV is a nonconvex and discontinuous problem, the optimality analysis in terms of Karush-Kuhn-Tucker (KKT) conditions is accordingly proposed, based on which a hard-thresholding based alternating direction method of multipliers (HT-ADMM) is designed. Extensive experiments remarkably demonstrate the efficiency of SLRMV in tensor completion.     

Recovering orthogonal tensors under arbitrarily strong,but locally correlated,noise

We consider the problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude. We focus on the particular case where each mode in the decomposition is corrupted by noise vectors with components that are correlated locally, that is, with nearby components. We show that this deterministic tensor completion problem has the unusual property that it can be solved in polynomial time if the rank of the tensor is sufficiently large. This is the polar opposite of the low-rank assumptions of typical low-rank tensor and matrix completion settings. We show that our problem can be solved through a system of coupled Sylvester-like equations and show how to accelerate their solution by an alternating solver. This enables recovery even with a substantial number of missing entries, for instance for $n$-dimensional tensors of rank $n$ with up to $40\%$ missing entries.     

Nonconvex optimization for third-order tensor completion under wavelet transform

The main aim of this paper is to develop a nonconvex optimization model for third-order tensor completion under wavelet transform. On the one hand, through wavelet transform of frontal slices, we divide a large tensor data into a main part tensor and three detail part tensors, and the elements of these four tensors are about a quarter of the original tensors. Solving these four small tensors can not only improve the operation efficiency, but also better restore the original tensor data. On the other hand, by using concave correction term, we are able to correct for low rank of tubal nuclear norm (TNN) data fidelity term and sparsity of $l_1$-norm data fidelity term. We prove that the proposed algorithm can converge to some critical point. Experimental results on image, magnetic resonance imaging and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state-of-the-arts including the TNN and other methods.