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计算机代写|机器学习代写Machine Learning代考|Linear Machines

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计算机代写|机器学习代写Machine Learning代考|Linear Machines

The natural generalization of a (two-category) TLU to an $R$-category classifier is the structure, shown in Fig. 4.8, called a linear machine. Here, to use more familiar notation, the $\mathbf{W}$ and $\mathbf{X}$ are meant to be augmented vectors (with an $(n+1)$-st component). Such a structure is also sometimes called a “competitive” net or a “winner-take-all” net. The output of the linear machine is one of the numbers, ${1, \ldots, R}$, corresponding to which dot product is largest. Note that when $R=2$, the linear machine reduces to a TLU with weight vector $\mathbf{W}=\left(\mathbf{W}_1-\mathbf{W}_2\right)$.

The diagram in Fig. 4.9 shows the character of the regions in a 2dimensional space created by a linear machine for $R=5$. In $n$ dimensions, every pair of regions is either separated by a section of a hyperplane or is non-adjacent.

To train a linear machine, there is a straightforward generalization of the 2-category error-correction rule. Assemble the patterns in the training set into a sequence as before.

1. If the machine classifies a pattern correctly, no change is made to any of the weight vectors.
2. If the machine mistakenly classifies a category $u$ pattern, $\mathbf{X}_i$, in category $v(u \neq v)$, then:
$$\mathbf{W}_u \longleftarrow \mathbf{W}_u+c_i \mathbf{X}_i$$

and
$$\mathbf{W}_v \longleftarrow \mathbf{W}_v-c_i \mathbf{X}_i$$
and all other weight vectors are not changed.
This correction increases the value of the $u$-th dot product and decreases the value of the $v$-th dot product. Just as in the 2-category fixed increment procedure, this procedure is guaranteed to terminate, for constant $c_i$, if there exists weight vectors that make correct separations of the training set. Note that when $R=2$, this procedure reduces to the ordinary TLU error-correction procedure. A proof that this procedure terminates is given in [Nilsson, 1990, pp. 88-90] and in [Duda \& Hart, 1973, pp. 174-177].

计算机代写|机器学习代写Machine Learning代考|Motivation and Examples

To classify correctly all of the patterns in non-linearly-separable training sets requires separating surfaces more complex than hyperplanes. One way to achieve more complex surfaces is with networks of TLUs. Consider, for example, the 2-dimensional, even parity function, $f=x_1 x_2+\overline{x_1} \overline{x_2}$. No single line through the 2-dimensional square can separate the vertices $(1,1)$ and $(0,0)$ from the vertices $(1,0)$ and $(0,1)$-the function is not linearly separable and thus cannot be implemented by a single TLU. But, the network of three TLUs shown in Fig. 4.10 does implement this function. In the figure, we show the weight values along input lines to each TLU and the threshold value inside the circle representing the TLU.

The function implemented by a network of TLUs depends on its topology as well as on the weights of the individual TLUs. Feedforward networks have no cycles; in a feedforward network no TLU’s input depends (through zero or more intermediate TLUs) on that TLU’s output. (Networks that are not feedforward are called recurrent networks). If the TLUs of a feedforward network are arranged in layers, with the elements of layer $j$ receiving inputs only from TLUs in layer $j-1$, then we say that the network is a layered, feedforward network. The network shown in Fig. 4.10 is a layered, feedforward network having two layers (of weights). (Some people count the layers of TLUs and include the inputs as a layer also; they would call this network a three-layer network.) In general, a feedforward, layered network has the structure shown in Fig. 4.11. All of the TLUs except the “output” units are called hidden units (they are “hidden” from the output).
Implementing DNF Functions by Two-Layer Networks
We have already defined $k$-term DNF functions-they are DNF functions having $k$ terms. A $k$-term DNF function can be implemented by a two-layer network with $k$ units in the hidden layer-to implement the $k$ terms-and one output unit to implement the disjunction of these terms. Since any Boolean function has a DNF form, any Boolean function can be implemented by some two-layer network of TLUs. As an example, consider the function $f=x_1 x_2+x_2 \overline{x_3}+x_1 \overline{x_3}$. The form of the network that implements this function is shown in Fig. 4.12. (We leave it to the reader to calculate appropriate values of weights and thresholds.) The 3-cube representation of the function is shown in Fig. 4.13. The network of Fig. 4.12 can be designed so that each hidden unit implements one of the planar boundaries shown in Fig. 4.13.

代写|机器学习代写Machine Learning代考|Linear Machines

$$\mathbf{W}_u \longleftarrow \mathbf{W}_u+c_i \mathbf{X}_i$$

$$\mathbf{W}_v \longleftarrow \mathbf{W}_v-c_i \mathbf{X}_i$$

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。