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# 计算机代写|机器学习代写Machine Learning代考|CS234

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## 计算机代写|机器学习代写Machine Learning代考|Balanced Decomposition Schemes

Balanced decomposition schemes are a subclass of decomposition schemes that are based on balanced binary class partitions. The concept of balanced binary class partitions is provided in Definition 3.5 given below.

Definition 3.5. (Balanced Binary Class Partitions) If the number $K$ of classes in a class set $Y$ is even, then a binary class partition $P(Y)=\left{Y^{-}, Y^{+}\right}$is said to be balanced iff $\left|Y^{-}\right|=\left|Y^{+}\right|$.

The number of all balanced binary class partitions is provided in Corollary 3.4 given below.

Corollary 3.4. The number of all balanced binary class partitions $P(Y)$ equals $\frac{K !}{2\left(\frac{K}{2} !\right)^2}$.
Proof. The number of all balanced binary class partitions $P(Y)$ is $\frac{1}{2}\left(\frac{K}{\frac{K}{2}}\right)=\frac{K !}{2\left(\frac{K}{2} !\right)^2}$
Given the concept of balanced binary class partitions we define the concept of balanced decomposition schemes in Definition 3.6 below.

Definition 3.6. (Balanced Decomposition Schemes) A decomposition scheme denoted as $S P(Y)$ is said to be balanced iff each binary class partition $P(Y) \in S P(Y)$ is balanced.

Theorem 3.2, given below, states that the class of balanced decomposition schemes is non-empty. More precisely we show that by using balanced binary class partitions we can design a decomposition scheme.

## 计算机代写|机器学习代写Machine Learning代考|Minimally-Sized Balanced Decomposition Schemes

Minimally-sized balanced decomposition schemes (MBDSs) are balanced decomposition schemes of minimal size. This type of decomposition schemes was suggested by Mayoraz \& Moreira [14] but was never studied in detail. This subsection provides the definition of MBDSs and their properties.

Definition 3.7. (Minimally-Sized Balanced Decomposition Schemes) Given the set $S P^M(Y)$ of all balanced binary class partitions, a balanced decomposition scheme $S P(Y) \subseteq S P^M(Y)$ is said to be minimally-sized iff there does not exist another balanced decomposition scheme $S P^{\prime}(Y) \subseteq S P^M(Y)$ such that $\left|S P^{\prime}(Y)\right|<|S P(Y)|$.

Notation 1. A minimally-sized balanced decomposition scheme is denoted by $S P^m(Y)$.

Theorem 3.3 below determines the size of MBDSs as a function of the number $K$ of classes.

Theorem 3.3. A balanced decomposition scheme $S P(Y)$ is minimally-sized iff the size of $\operatorname{SP}(Y)$ equals $\left\lceil\log _2(K)\right\rceil$.

Proof. $(\rightarrow$ ) Consider a minimally-sized balanced decomposition scheme $S P(Y)$. Any decomposition matrix $M$ of $S P(Y)$ represents a minimally-sized binary code for $K$ classes. The size of that code is $\left\lceil\log _2(K)\right\rceil$.Thus, the size of $\operatorname{SP}(Y)$ equals $\left\lceil\log _2(K)\right\rceil$.
$(\leftarrow)$ Consider a balanced decomposition scheme $S P(Y)$ with size of $\left\lceil\log _2(K)\right\rceil$. Any decomposition matrix $M$ of $S P(Y)$ represents a binary code for $K$ classes and the size of this code is $\left\lceil\log _2(K)\right\rceil$. Thus, the code is minimally-sized. This implies that $\operatorname{SP}(Y)$ is minimally-sized.

## 计算机代写|机器学习代写Machine Learning代考|Balanced Decomposition Schemes

3.6.定义(平衡分解方案)如果每个二元类分区$P(Y) \in S P(Y)$是平衡的，那么表示为$S P(Y)$的分解方案就是平衡的。

## 计算机代写|机器学习代写Machine Learning代考|Minimally-Sized Balanced Decomposition Schemes

$(\leftarrow)$考虑一个大小为$\left\lceil\log _2(K)\right\rceil$的平衡分解方案$S P(Y)$。$S P(Y)$的任何分解矩阵$M$表示$K$类的二进制代码，该代码的大小为$\left\lceil\log _2(K)\right\rceil$。因此，代码是最小大小的。这意味着$\operatorname{SP}(Y)$是最小大小的。

## 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。