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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

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