### 金融代写|金融计量经济学Financial Econometrics代考| Why EN and CLOT

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## 金融代写|金融计量经济学Financial Econometrics代考|Why EN and CLOT

Need to go beyond LASSO. In the previous section, we showed that if we need to select a single method for all the problems, then natural invariance requirements lead to LASSO, i.e., to bounds on the sum of the absolute values of the parameters. In some practical situations, this works, while in others, it does not lead to good results. To deal with such situations, instead of fixing a single method for all the problems, a natural idea is to select a family of methods, so that in each practical situation, we should select an appropriate method from this family. Let us analyze how we can do it both for probabilistic and for general constraints.

Probabilistic case. Constraints in the probabilistic case are described by the corresponding function $\psi(z)$. The LASSO case corresponds to a 2-parametric family $\psi(z)=c_{0}+c_{1} \cdot z$. In terms of the corresponding constraints, all these functions from this family are equivalent to $\psi(z)=z$.

To get a more general method, a natural idea is to consider a 3-parametric family, i.e., a family of the type $\psi(z)=c_{0}+c_{1} \cdot z+c_{2} \cdot f(z)$ for some function $f(z)$. Constraints related to this family are equivalent to using the functions $\psi(z)=z+$ $c \cdot f(z)$ for some function $f(z)$. Which family-i.e., which function $f(z)$-should we choose? A natural idea is to ágain use scale-invariancè and shift-invariance.
Definition 8 We say that functions $\psi_{1}(z)$ and $\psi_{2}(z)$ are constraint-equivalent if:

• for each $n$ and for each $c_{1}$, there exists a value $c_{2}$ such that the condition $\sum_{i=0}^{n} \psi_{1}\left(a_{i}\right)=c_{1}$ is equivalent to $\sum_{i=0}^{n} \psi_{2}\left(a_{i}\right)=c_{2}$, and
• for each $n$ and for each $c_{2}$, there exists a value $c_{1}$ such that the condition $\sum_{i=0}^{n} \psi_{2}\left(a_{i}\right)=c_{2}$ is equivalent to $\sum_{i=0}^{n} \psi_{1}\left(a_{i}\right)=c_{1}$.
Definition 9
• We say that a family ${z+c \cdot f(z)}_{c}$ is scale-invariant if for each $c$ and $\lambda$, there exists a value $c^{\prime}$ for which the re-scaled function $\lambda \cdot z+c \cdot f(\lambda \cdot z)$ is constraintequivalent to $z+c^{\prime} \cdot f(z)$.
• We say that a family ${z+c \cdot f(z)}_{c}$ is shift-invariant if for each $c$ and for each sufficiently small number $\varepsilon$, there exists a value $c^{\prime}$ for which the shifted function $z-\varepsilon+c \cdot f(z-\varepsilon)$ is constraint-equivalent to $z+c^{\prime} \cdot f(z)$.

Proposition 3 A family ${z+c \cdot f(z)}_{c}$ corresponding to a smooth function $f(z)$ is scale-and shift-invariant if and only if the function $f(z)$ is quadratic.

## 金融代写|金融计量经济学Financial Econometrics代考|Beyond EN and CLOT

Discussion. What if 1-parametric families like EN and CLOT are not sufficient? In this case, we need to consider families
$$F=\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{n} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{m}}$$
with more parameters.
Definition 10

• We say that a family $\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{m}}$ is scale-invariant if for each $c=\left(c_{1}, \ldots, c_{m}\right)$ and $\lambda$, there exists a tuple $c^{\prime}=\left(c_{1}^{\prime}, \ldots, c_{m}^{\prime}\right)$ for which the re-scaled function
$$\lambda \cdot z+c_{1} \cdot f_{1}(\lambda \cdot z)+\cdots+c_{m} \cdot f_{m}(\lambda \cdot z)$$
is constraint-equivalent to $z+c_{1}^{\prime} \cdot f_{1}(z)+\cdots+c_{m}^{\prime} \cdot f_{m}(z)$.
• We say that a family $\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{m}}$ is shift-invariant if for each tuple $c$ and for each sufficiently small number $\varepsilon$, there exists a tuple $c^{\prime}$ for which the shifted function

Why LASSO, EN, and CLOT: Invariance-Based Explanation
49
$$z-\varepsilon+c_{1} \cdot f_{1}(z-\varepsilon)+\cdots+c_{m} \cdot f_{m}(z-\varepsilon)$$
is constraint-equivalent to $z+c_{1}^{\prime} \cdot f_{1}(z)+\cdots+c_{m}^{\prime} \cdot f_{m}(z)$.
Proposition 4 A family $\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{m}}$ corresponding to a smooth functions $f_{i}(z)$ is scale- and shift-invariant if and only if all the functions $f_{i}(z)$ are polynomials of order $\leq m+1$.

Discussion. So, if EN and CLOT are not sufficient, our recommendation is to use a constraint $\sum_{i=0}^{n} \psi\left(\left|a_{i}\right|\right)=c$ for some higher order polynomial $\psi(z)$.

Proof of Proposition 4 is similar to the s of Proposition 3 , the only difference is that instead of a single differential equation, we will have a system of linear differential equations.

Comment. Similarly to the quadratic case, the resulting general expression $\psi(z)=$ $g_{0}+g_{1} \cdot z+\cdots+a_{m+1} \cdot z^{m+1}$ can be viewed as keeping the first few terms in the Taylor expansion of a general function $\psi(z)$.

Acknowledgements This work was supported by the Institute of Geodesy, Leibniz University of Hannover. It was also supported in part by the US National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science) and HRD-1242122 (Cyber-ShARE Center of Excellence).
This paper was written when V. Kreinovich was visiting Leibniz University of Hannover.

## 金融代写|金融计量经济学Financial Econometrics代考|Main Results

Consider the Bloch sphere below (Nielsen and Chuang 2010). The states of a single bit two-level $(|0\rangle,|1\rangle)$ quantum bit (qubit) are described by the Bloch sphere above with $0 \leq \theta \leq \pi, 0 \leq \varphi \leq 2 \pi$; qubit is just a quantum system.

A single qubit quantum state $\rho$ can be represented with below density matrix,
$$\rho=\frac{1}{2}\left(\begin{array}{cc} 1+\eta \cos \theta & \eta e^{-i \varphi} \sin \theta \ \eta e^{i \varphi} \sin \theta & 1-\eta \cos \theta \end{array}\right), \eta \in[0,1], \quad 0 \leq \theta \leq \pi, \text { and } 0 \leq \varphi \leq 2 \pi$$
Also, the density matrix can take below representation (Nielsen and Chuang 2010),
$$\rho=\frac{1}{2}[I+\bar{r} \cdot \bar{\sigma}]=\frac{1}{2}\left[\begin{array}{cc} 1+r_{z} & r_{x}-i r_{y} \ r_{x}+i r_{y} & 1-r_{z} \end{array}\right]$$
where $\bar{r}=\left[r_{x}, r_{y}, r_{z}\right]$ is the Bloch vector with $|\bar{r}| \leq 1$, and $\bar{\sigma}=\left[\sigma_{x}, \sigma_{y}, \sigma_{z}\right]$ for $\sigma_{x}, \sigma_{y}, \sigma_{z}$ being the Pauli matrices.
$$\sigma_{x}=\left(\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right), \quad \sigma_{y}=\left(\begin{array}{cc} 0 & -i \ i & 0 \end{array}\right), \quad \sigma_{z}=\left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right)$$
Let $T_{1}, T_{2}, T_{3}$ and $T_{\pm h}$ denote the Bit flip operation, Phase flip operation, Bit-Phase flip operation and Displacements operation on a Bloch sphere respectively, for $h \in$ ${x, y, z}$. Denote the Bloch sphere by $\mathscr{Q}$ and $\mathscr{Q}{T}$ be the deformation of the Bloch sphere after an operation $T$. Let $F(T)$ denote the fixed point set of the operation $T$. Proposition 2.1 Suppose $p \in[0,1]$ is the same for $T{1}, T_{2}$ and $T_{3} ; p_{T_{1}}=p_{T_{2}}=$ $p_{T_{3}}=p$. Then, the six different compositions obtained from the permutation of $T_{i}, i=1,2,3$ gives the same out put,

Proof Let $\rho$ be a qubit (quantum bit) state in/on the Bloch sphere. Suppose the general representation of $\rho$ using the density matrix is
$$\rho=\left(\begin{array}{ll} a & b \ c & d \end{array}\right)$$

## 金融代写|金融计量经济学Financial Econometrics代考|Why EN and CLOT

• 对于每个n并且对于每个C1, 存在一个值C2使得条件∑一世=0nψ1(一个一世)=C1相当于∑一世=0nψ2(一个一世)=C2， 和
• 对于每个n并且对于每个C2, 存在一个值C1使得条件∑一世=0nψ2(一个一世)=C2相当于∑一世=0nψ1(一个一世)=C1.
定义 9
• 我们说一个家庭和+C⋅F(和)C是尺度不变的，如果对于每个C和λ, 存在一个值C′重新缩放的函数λ⋅和+C⋅F(λ⋅和)是约束等价于和+C′⋅F(和).
• 我们说一个家庭和+C⋅F(和)C是移位不变的，如果对于每个C并且对于每个足够小的数字e, 存在一个值C′移位函数和−e+C⋅F(和−e)是约束等价于和+C′⋅F(和).

## 金融代写|金融计量经济学Financial Econometrics代考|Beyond EN and CLOT

F=\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{n} \cdot f_{m}(z)\right}_{c_{1}, \ldots,厘米}}F=\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{n} \cdot f_{m}(z)\right}_{c_{1}, \ldots,厘米}}

• 我们说一个家庭\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{米}}\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{米}}是尺度不变的，如果对于每个C=(C1,…,C米)和λ, 存在一个元组C′=(C1′,…,C米′)重新缩放的函数
λ⋅和+C1⋅F1(λ⋅和)+⋯+C米⋅F米(λ⋅和)
是约束等价于和+C1′⋅F1(和)+⋯+C米′⋅F米(和).
• 我们说一个家庭\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{米}}\left{z+c_{1} \cdot f_{1}(z)+\cdots+c_{m} \cdot f_{m}(z)\right}_{c_{1}, \ldots, c_{米}}如果对于每个元组是移位不变的C并且对于每个足够小的数字e, 存在一个元组C′移位函数

49

## 金融代写|金融计量经济学Financial Econometrics代考|Main Results

ρ=12(1+这因⁡θ这和−一世披罪⁡θ 这和一世披罪⁡θ1−这因⁡θ),这∈[0,1],0≤θ≤圆周率, 和 0≤披≤2圆周率

ρ=12[我+r¯⋅σ¯]=12[1+r和rX−一世r是 rX+一世r是1−r和]

σX=(01 10),σ是=(0−一世 一世0),σ和=(10 0−1)

ρ=(一个b Cd)

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