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数学代写|凸优化代写Convex Optimization代考|EECS559 Application to signal recovery

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数学代写|凸优化代写Convex Optimization代考|Application to signal recovery

Application to signal recovery. Proposition $4.37$ allows us to build computation-based lower risk bounds in the signal recovery problem considered in Section 4.2, in particular, the problem where one wants to recover the linear image $B x$ of an unknown signal $x$ known to belong to a given ellitope
$$\mathcal{X}=\left{x \in \mathbf{R}^n: \exists t \in \mathcal{T}: x^T S_{\ell} x \leq t_{\ell}, \ell \leq L\right}$$
(with our usual restriction on $S_{\ell}$ and $\mathcal{T}$ ) via observation
$$\omega=A x+\sigma \xi, \xi \sim \mathcal{N}\left(0, I_m\right)$$
and the risk of a candidate estimate, as in Section 4.2, is defined according to (4.113). ${ }^{22}$ It is convenient to assume that the matrix $B$ (which in our general setup can be an arbitrary $\nu \times n$ matrix) is a nonsingular $n \times n$ matrix. ${ }^{23}$ Under this assumption, setting
$$\mathcal{Y}=B^{-1} \mathcal{X}=\left{y \in \mathbf{R}^n: \exists t \in \mathcal{T}: y^T\left[B^{-1}\right]^T S_{\ell} B^{-1} y \leq t_{\ell}, \ell \leq L\right}$$
and $\bar{A}=A B^{-1}$, we lose nothing when replacing the sensing matrix $A$ with $\bar{A}$ and treating as our signal $y \in \mathcal{Y}$ rather than $\mathcal{X}$. Note that in our new situation $A$ is replaced with $\bar{A}, \mathcal{X}$ with $\mathcal{Y}$, and $B$ is the unit matrix $I_n$. For the sake of simplicity, we assume from now on that $A$ (and therefore $\bar{A}$ ) has trivial kernel. Finally, let $\tilde{S}{\ell} \succeq S{\ell}$ be close to $S_k$ positive definite matrices, e.g., $\tilde{S}{\ell}=S{\ell}+10^{-100} I_n$. Setting $\bar{S}{\ell}=\left[B^{-1}\right]^T \tilde{S}{\ell} B^{-1}$ and
$$\overline{\mathcal{Y}}=\left{y \in \mathbf{R}^n: \exists t \in \mathcal{T}: y^T \bar{S}{\ell} y \leq t{\ell}, \ell \leq L\right}$$

数学代写|凸优化代写Convex Optimization代考|Lower-bounding

Lower-bounding Risk $\mathrm{opt}$. In order to make the bounding scheme just outlined give its best, we need a mechanism which allows us to generate $k$-dimensional “disks” $\Theta \subset \overline{\mathcal{Y}}$ along with associated quantities $r, \gamma$. In order to design such a mechanism, it is convenient to represent $k$-dimensional linear subspaces of $\mathbf{R}^n$ as the image spaces of orthogonal $n \times n$ projectors $P$ of rank $k$. Such a projector $P$ gives rise to the disk $\Theta_P$ of the radius $r=r_P$ contained in $\overline{\mathcal{Y}}$, where $r_P$ is the largest $\rho$ such that the set $\left{y \in \operatorname{Im} P: y^T P y \leq \rho^2\right}$ is contained in $\overline{\mathcal{Y}}$ (“condition $\left.\mathcal{C}(r)^{\prime \prime}\right)$, and we can equip the disk with $\gamma$ satisfying (ii) if and only if
$$\operatorname{Tr}\left(P \bar{A}^T \bar{A} P\right) \leq \gamma$$
or, which is the same (recall that $P$ is an orthogonal projector)
$$\operatorname{Tr}\left(\bar{A} P \bar{A}^T\right) \leq \gamma$$
(“condition $\mathcal{D}(\gamma)$ “). Now, when $P$ is a nonzero orthogonal projector, the simplest sufficient condition for the validity of $\mathcal{C}(r)$ is the existence of $t \in \mathcal{T}$ such that
$$\forall\left(y \in \mathbf{R}^n, \ell \leq L\right): y^T P \bar{S}{\ell} P y \leq t{\ell} r^{-2} y^T P y$$
or, which is the same,
$$\exists s: r^2 s \in \mathcal{T} \& P \bar{S}{\ell} P \preceq s{\ell} P, \ell \leq L$$

数学代写|凸优化代写Convex Optimization代考|Application to signal recovery

《left 缺少或无法识别的分隔符
(我们通常限制 $S_{\ell}$ 和 $\mathcal{T}$ ) 通过观察
$$\omega=A x+\sigma \xi, \xi \sim \mathcal{N}\left(0, I_m\right)$$

〈left 缺少或无法识别的分隔符

、left 缺少或无法识别的分隔符

数学代写|凸优化代写Convex Optimization代考|Lower-bounding

$$\operatorname{Tr}\left(P \bar{A}^T \bar{A} P\right) \leq \gamma$$

$$\operatorname{Tr}\left(\bar{A} P \bar{A}^T\right) \leq \gamma$$
(“健康) 状兄 $\mathcal{D}(\gamma)$ “) 。现在，当 $P$ 是一个非零正交投影仪，是有效的最简单的充分条件 $\mathcal{C}(r)$ 是存在的 $t \in \mathcal{T}$ 这样
$$\forall\left(y \in \mathbf{R}^n, \ell \leq L\right): y^T P \bar{S} \ell P y \leq t \ell r^{-2} y^T P y$$

$$\exists s: r^2 s \in \mathcal{T} \& P \bar{S} \ell P \preceq s \ell P, \ell \leq L$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。