### 统计代写|统计推断代写Statistical inference代考|MAST30020

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计推断代写Statistical inference代考|Validity of ℓ1 minimization in the noiseless case

The minimal requirement on sensing matrix $A$ which makes $\ell_{1}$ minimization valid is to guarantee the correct recovery of exactly s-sparse signals in the noiseless case, and we start with investigating this property.
1.2.1.1 Notational convention
From now on, for a vector $x \in \mathbf{R}^{n}$

• $I_{x}=\left{j: x_{j} \neq 0\right}$ stands for the support of $x$; we also set
$$I_{x}^{+}=\left{j: x_{j}>0\right}, I_{x}^{-}=\left{j: x_{j}<0\right} \quad\left[\Rightarrow I_{x}=I_{x}^{+} \cup I_{x}^{-}\right]$$
• for a subset $I$ of the index set ${1, \ldots, n}, x_{I}$ stands for the vector obtained from $x$ by zeroing out entries with indices not in $I$, and $I^{o}$ for the complement of $I$ :
$$I^{o}={i \in{1, \ldots, n}: i \notin I}$$
• for $s \leq n, x^{s}$ stands for the vector obtained from $x$ by zeroing out all but the $s$
• entries largest in magnitude. ${ }^{5}$ Note that $x^{s}$ is the best $s$-sparse approximation of $x$ in all $\ell_{p}$ norms, $1 \leq p \leq \infty$;
• for $s \leq n$ and $p \in[1, \infty]$, we set
$$|x|_{s, p}=\left|x^{s}\right|_{p}$$
note that $|\cdot|_{s, p}$ is a norm.
$1.2 .1 .2 \mathrm{~s}$-Goodness
Definition of $s$-goodness. Let us say that an $m \times n$ sensing matrix $A$ is $s$-good if whenever the true signal $x$ underlying noiseless observations is $s$-sparse, this signal will be recovered exactly by $\ell_{1}$ minimization. In other words, $A$ is $s$-good if whenever $y$ in (1.4) is of the form $y=A x$ with s-sparse $x, x$ is the unique optimal solution to (1.4).

Nullspace property. There is a simply-looking necessary and sufficient condition for a sensing matrix $A$ to be $s$-good-the nullspace property originating from $[70]$. After this property is guessed, it is easy to see that it indeed is necessary and sufficient for $s$-goodness; we, however, prefer to derive this condition from the “first principles,” which can be easily done via Convex Optimization. Thus, in the case in question, as in many other cases, there is no necessity to be smart to arrive at the truth via a “lucky guess”; it suffices to be knowledgeable and use the standard tools.

## 统计代写|统计推断代写Statistical inference代考|Imperfect ℓ1 minimization

We have found a necessary and sufficient condition for $\ell_{1}$ minimization to recover exactly s-sparse signals in the noiseless case. More often than not, both these assumptions are violated: instead of $s$-sparse signals, we should speak about “nearly $s$-sparse” ones, quantifying the deviation from sparsity by the distance from the signal $x$ underlying the observations to its best $s$-sparse approximation $x^{s}$. Similarly, we should allow for nonzero observation noise. With noisy observations and/or imperfect sparsity, we cannot hope to recover the signal exactly. All we may hope for, is to recover it with some error depending on the level of observation noise and “deviation from s-sparsity,” and tending to zero as the level and deviation tend to 0 . We are about to quantify the nullspace property to allow for instructive “error analysis.”

By itself, the nullspace property says something about the signals from the kernel of the sensing matrix. We can reformulate it equivalently to say something important about all signals. Namely, observe that given sparsity $s$ and $\kappa \in(0,1 / 2)$, the nullspace property
$$|w|_{s, 1} \leq \kappa|w|_{1} \forall w \in \operatorname{Ker} A$$
is satisfied if and only if for a properly selected constant $C$ one has ${ }^{6}$
$$|w|_{s, 1} \leq C|A w|_{2}+\kappa|w|_{1} \forall w .$$
Indeed, (1.10) clearly implies (1.9); to get the inverse implication, note that for every $h$ orthogonal to Ker $A$ it holds
$$|A h|_{2} \geq \sigma|h|_{2},$$
where $\sigma>0$ is the minimal positive singular value of $A$. Now, given $w \in \mathbf{R}^{n}$, we can decompose $w$ into the sum of $\tilde{w} \in \operatorname{Ker} A$ and $h \in(\operatorname{Ker} A)^{\perp}$, so that
\begin{aligned} &|w|_{s, 1} \leq|\bar{w}|_{s, 1}+|h|_{s, 1} \leq \kappa|\bar{w}|_{1}+\sqrt{s}|h|_{s, 2} \leq \kappa\left[|w|_{1}+|h|_{1}\right]+\sqrt{s}|h|_{2} \ &\leq \kappa|w|_{1}+[\kappa \sqrt{n}+\sqrt{s}]|h|_{2} \leq \underbrace{\sigma^{-1}[\kappa \sqrt{n}+\sqrt{s}]}{C} \underbrace{|A h|{2}}{-|A w|{2}}+\kappa|w|_{1}, \end{aligned}
as required in (1.10).

## 统计代写|统计推断代写Statistical inference代考|Regular ℓ1 recovery

Given the observation scheme (1.1) with an $m \times n$ sensing matrix $A$, we define the regular $\ell_{1}$ recovery of $x$ via observation $y$ as
$$\widehat{x}{\text {reg }}(y) \in \underset{u}{\operatorname{Argmin}}\left{|u|{1}:\left|H^{T}(A u-y)\right| \leq \rho\right},$$
where the contrast matrix $H \in \mathbf{R}^{m \times N}$, the norm $|\cdot|$ on $\mathbf{R}^{N}$ and $\rho>0$ are parameters of the construction.
The role of $\mathbf{Q}$-conditions we have introduced is clear from the following
Theorem 1.3. Let $s$ be a positive integer, $q \in[1, \infty]$ and $\kappa \in(0,1 / 2)$. Assume that a pair $(H,|\cdot|)$ satisfies the condition $\mathbf{Q}{q}(s, \kappa)$ associated with $A$, and let $$\Xi{\rho}=\left{\eta:\left|H^{T} \eta\right| \leq \rho\right} .$$
Then for all $x \in \mathbf{R}^{n}$ and $\eta \in \Xi_{\rho}$ one has
$$\left|\widehat{x}{\text {reg }}(A x+\eta)-x\right|{p} \leq \frac{4(2 s)^{\frac{1}{p}}}{1-2 \kappa}\left[\rho+\frac{\left|x-x^{s}\right|_{1}}{2 s}\right], 1 \leq p \leq q .$$
The above result can be slightly strengthened by replacing the assumption that $(H,|\cdot|)$ satisfies $\mathbf{Q}{q}(s, \kappa)$ with some $\kappa<1 / 2$, with a weaker-by observation $\mathbf{A}$ from Section 1.2.2.1 – assumption that $(H,|\cdot|)$ satisfies $\mathbf{Q}{1}(s, \varkappa)$ with $\varkappa<1 / 2$ and satisfies $\mathbf{Q}_{q}(s, \kappa)$ with some (perhaps large) $\kappa$ :

Theorem 1.4. Given $A$, integer $s>0$, and $q \in[1, \infty]$, assume that $(H,|\cdot|)$ satisfies the condition $\mathbf{Q}{1}(s, \varkappa)$ with $\varkappa<1 / 2$ and the condition $\mathbf{Q}{q}(s, \kappa)$ with some $\kappa \geq \varkappa$, and let $\Xi_{\rho}$ be given by (1.14). Then for all $x \in \mathbf{R}^{n}$ and $\eta \in \Xi_{\rho}$ it holds:
$$\left|\widehat{x}{\text {reg }}(A x+\eta)-x\right|{p} \leq \frac{4(2 s)^{\frac{1}{p}}[1+\kappa-x]^{\frac{q(p-1)}{p(q-1)}}}{1-2 \varkappa}\left[\rho+\frac{\left|x-x^{s}\right|_{1}}{2 s}\right], 1 \leq p \leq q$$
For proofs of Theorems $1.3$ and 1.4, see Section 1.5.1.
Before commenting on the above results, let us present their alternative versions.

## 统计代写|统计推断代写Statistical inference代考|Validity of ℓ1 minimization in the noiseless case

1.2.1.1 符号约定

• I_{x}=\left{j: x_{j} \neq 0\right}I_{x}=\left{j: x_{j} \neq 0\right}代表支持X; 我们还设置
I_{x}^{+}=\left{j: x_{j}>0\right}, I_{x}^{-}=\left{j: x_{j}<0\right} \quad\左[\Rightarrow I_{x}=I_{x}^{+} \cup I_{x}^{-}\right]I_{x}^{+}=\left{j: x_{j}>0\right}, I_{x}^{-}=\left{j: x_{j}<0\right} \quad\左[\Rightarrow I_{x}=I_{x}^{+} \cup I_{x}^{-}\right]
• 对于一个子集我索引集的1,…,n,X我代表从获得的向量X通过将索引不在的条目清零我， 和我○为补我 :
我○=一世∈1,…,n:一世∉我
• 为了s≤n,Xs代表从获得的向量X通过清零除s
• 数量级最大的条目。5注意Xs是最好的s-稀疏近似X在所有ℓp规范，1≤p≤∞;
• 为了s≤n和p∈[1,∞]， 我们设置
|X|s,p=|Xs|p
注意|⋅|s,p是一种规范。
1.2.1.2 s-善良
的定义s-善良。让我们说一个米×n传感矩阵一个是s-只要有真实信号就好了X基本的无噪声观察是s-sparse，这个信号将完全恢复ℓ1最小化。换句话说，一个是s- 好，如果任何时候是(1.4) 中的形式为是=一个Xs-稀疏的X,X是 (1.4) 的唯一最优解。

## 统计代写|统计推断代写Statistical inference代考|Imperfect ℓ1 minimization

|在|s,1≤ķ|在|1∀在∈克尔⁡一个

|在|s,1≤C|一个在|2+ķ|在|1∀在.

|一个H|2≥σ|H|2,

|在|s,1≤|在¯|s,1+|H|s,1≤ķ|在¯|1+s|H|s,2≤ķ[|在|1+|H|1]+s|H|2 ≤ķ|在|1+[ķn+s]|H|2≤σ−1[ķn+s]⏟C|一个H|2⏟−|一个在|2+ķ|在|1,

## 统计代写|统计推断代写Statistical inference代考|Regular ℓ1 recovery

\widehat{x}{\text {reg }}(y) \in \underset{u}{\operatorname{Argmin}}\left{|u|{1}:\left|H^{T}(A uy )\对| \leq \rho\right},\widehat{x}{\text {reg }}(y) \in \underset{u}{\operatorname{Argmin}}\left{|u|{1}:\left|H^{T}(A uy )\对| \leq \rho\right},

\Xi{\rho}=\left{\eta:\left|H^{T} \eta\right| \leq \rho\right} 。\Xi{\rho}=\left{\eta:\left|H^{T} \eta\right| \leq \rho\right} 。

|X^注册 (一个X+这)−X|p≤4(2s)1p1−2ķ[ρ+|X−Xs|12s],1≤p≤q.

|X^注册 (一个X+这)−X|p≤4(2s)1p[1+ķ−X]q(p−1)p(q−1)1−2ε[ρ+|X−Xs|12s],1≤p≤q

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## MATLAB代写

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