Recipes for Stable Linear Embeddings From Hilbert Spaces to  {mathbb {R}}^{m}

Gilles Puy; Mike E. Davies; Rémi Gribonval

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Recipes for Stable Linear Embeddings From Hilbert Spaces to {mathbb {R}}^{m}

机译：从希尔伯特空间到{mathbb {R}} ^ {m}的稳定线性嵌入的食谱

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We consider the problem of constructing a linear map from a Hilbert space H (possibly infinite dimensional) to ℝm that satisfies a restricted isometry property (RIP) on an arbitrary signal model, i.e., a subset of H. We present a generic framework that handles a large class of low-dimensional subsets but also unstructured and structured linear maps. We provide a simple recipe to prove that a random linear map satisfies a general RIP with high probability. We also describe a generic technique to construct linear maps that satisfy the RIP. Finally, we detail how to use our results in several examples, which allow us to recover and extend many known compressive sampling results.

机译：我们考虑在任意信号模型（即H的一个子集）上构造从希尔伯特空间H（可能是无穷大维）到ℝm满足受限等距特性（RIP）的线性映射的问题。我们提出了一个处理一大类低维子集，但也包括非结构化和结构化的线性图。我们提供了一个简单的方法来证明随机线性映射以高概率满足一般RIP。我们还描述了一种通用技术来构造满足RIP的线性映射。最后，我们在几个示例中详细说明了如何使用我们的结果，这些示例使我们可以恢复和扩展许多已知的压缩采样结果。

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来源
《Information Theory, IEEE Transactions on》 |2017年第4期|2171-2187|共17页
作者
Gilles Puy; Mike E. Davies; Rémi Gribonval;
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作者单位

Technicolor, 975 Avenue des Champs Blancs, Cesson- Sévigné, FR;

Digital Communications, The University of Edinburgh, U.K.;

INRIA Rennes-Bretagne Atlantique, Campus de Beaulieu, Rennes Cedex, France;

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正文语种 eng
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Hilbert space; Linear matrix inequalities; Sensors; Manifolds; Extraterrestrial measurements; Europe; Probability distribution;

机译：希尔伯特空间线性矩阵不等式传感器歧管地外测量欧洲概率分布;

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1. Wiener–Hopf Operators on Spaces of Functions on ${mathbb{R}}^{+}$ with Values in a Hilbert Space [J] . Violeta Petkova Integral Equations and Operator Theory . 2007,第3期

机译：具有希尔伯特空间中的值的$ {mathbb {R}} ^ {+} $上的函数空间上的Wiener–Hopf运算符
2. Reproducing kernel Hilbert space embedding for adaptive estimation of nonlinearities in piezoelectric systems [J] . Paruchuri Sai Tej, Guo Jia, Kurdila Andrew Nonlinear dynamics . 2020,第2期

机译：再现内核HILBERT空间嵌入压电系统非线性的自适应估计
3. Embeddings of mathbbC*{mathbb{C}^*}-surfaces into weighted projective spaces [J] . Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg manuscripta mathematica . 2010,第1a2期

机译：mathbbC * {mathbb {C} ^ *}-表面到加权投影空间的嵌入
4. Linear embeddings of low-dimensional subsets of a Hilbert space to R^m [C] . Puy Gilles, Davies Mike, Gribonval Remi European Signal Processing Conference . 2015

机译：Hilbert空间的低维子集到R ^{m 的线性嵌入}
5. Linear operators on some non-archimedean Hilbert spaces and their spectral theory. [D] . Attimu, Dodzi K. 2008

机译：一些非档案的希尔伯特空间上的线性算子及其谱理论。
6. Hybrid proximal linearized algorithm for the split DC program in infinite-dimensional real Hilbert spaces [O] . Chih-Sheng Chuang, Pei-Jung Yang -1

机译：无限维实希尔伯特空间中分裂DC程序的混合近端线性化算法
7. Recipes for stable linear embeddings from Hilbert spaces to $\mathbb{R}^m$ [O] . Puy, Gilles, Davies, Mike,, Gribonval, Rémi 2017

机译：从希尔伯特空间到$ \ mathbb {R} ^ m $的稳定线性嵌入的食谱
8. Uniform Embeddings of Hyperbolic Groups in Hilbert Spaces [R] . Sela, Z. 1991

机译：Hilbert空间中双曲群的一致嵌入

Recipes for Stable Linear Embeddings From Hilbert Spaces to {mathbb {R}}^{m}

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