论文标题
具有最佳功能恢复中应用的有限帧的建设性子采样
Constructive subsampling of finite frames with applications in optimal function recovery
论文作者
论文摘要
在本文中，我们提出了随机和确定性的新的建设性方法，以在$ \ Mathbb c^m $中对有限帧的有效子采样。基于合适的随机亚采样策略，我们能够从任何给定的框架中提取$ 0 <a \ le b <\ infty $（和条件$ b/a $）类似条件的重新定制的副帧，仅由$ \ nathcal {o}（o}（o}（m \ log log m）组成。此外，利用基于Batson，Spielman和Srivastava制定的原理的确定性亚采样方法来控制Hermitian Rank-1矩阵总和的频谱，我们能够将元素数量减少到$ \ Mathcal {O}（O}（O}（M）$）（并不断接近一个）。通过通过预处理步骤控制权重，我们可以保留在未加权情况下绑定的较低框架。这允许推导新的准最佳未加权（左）Marcinkiewicz-Zygmund不等式的$ l_2（d，ν）$的不等式，其大小为$ \ MATHCAL {O}（O}（M）$的可构造节点集，用于$ M $ $ m $ $ m $的功能。这些可以应用，例如对于（纯）最小二乘采样的函数重建，我们获得了新的准最佳结果，避免了kadison-Serger定理。数值实验表明我们的结果的适用性。
In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in $\mathbb C^m$. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds $0<A\le B<\infty$ (and condition $B/A$) a similarly conditioned reweighted subframe consisting of merely $\mathcal{O}(m\log m)$ elements. Further, utilizing a deterministic subsampling method based on principles developed by Batson, Spielman, and Srivastava to control the spectrum of sums of Hermitian rank-1 matrices, we are able to reduce the number of elements to $\mathcal{O}(m)$ (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This permits the derivation of new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for $L_2(D,ν)$ with constructible node sets of size $\mathcal{O}(m)$ for $m$-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.
