Exploring the History and Matrix of Artificial Intelligence: Artificial Intelligence Tutorial (2)

In the first article of this series, we discussed the connections and differences between artificial intelligence, machine learning, deep learning, data science and other fields. We also made some hard choices about the programming languages, tools, and more that the entire series would use. Finally, we also introduced a little bit of matrix knowledge. In this article, we will discuss in depth the matrix, the core of artificial intelligence. But before that, let’s first understand the history of artificial intelligence

Why do we need to understand the history of artificial intelligence? There have been many AI booms in history, but in many cases the huge expectations for AI's potential failed to materialize. Understanding the history of artificial intelligence can help us see whether this wave of artificial intelligence will create miracles or is just another bubble about to burst.

When did our understanding of the origin of artificial intelligence begin? Was it after the invention of the digital computer? Or earlier? I believe the pursuit of an omniscient being goes back to the beginning of civilization. For example, Delphi in ancient Greek mythology was a prophet who could answer any question. The search for creative machines that surpass human intelligence has also fascinated us since ancient times. There have been several failed attempts to build chess machines throughout history. Among them is the infamous Mechaturk, which is not a real robot but is controlled by a chess player hidden inside. The logarithms invented by John Napier, Blaise Pascal's calculator, and Charles Babbage's Analytical Engine all played a key role in the development of artificial intelligence

So, the development of artificial intelligence so far What are the milestones? As mentioned earlier, the invention of the digital computer is the most important event in the history of artificial intelligence research. Unlike electromechanical devices, whose scalability depends on power requirements, digital devices benefit from technological advances, such as from vacuum tubes to transistors to integrated circuits and now VLSI.

Another important milestone in the development of artificial intelligence is Alan Turing’s first theoretical analysis of artificial intelligence. He proposed the famous Turing Test

In the late 1950s, John McCarthy

By the 1970s and 1980s, algorithms played a major role in this period. During this time, many new efficient algorithms were proposed. In the late 1960s, Donald Knuth (I strongly recommend you to get to know him, in the computer science world, he is equivalent to Gauss or Euler in the mathematics world) famous "The Art of Computer Programming" The publication of the first volume of Programming marked the beginning of the algorithm era. During these years, many general-purpose algorithms and graph algorithms were developed. In addition, programming based on artificial neural networks also emerged at this time. Although as early as the 1940s, Warren S. McCulloch and Walter Pitts

artificial intelligence had at least two promising opportunities in the digital age, Both opportunities fell short of expectations. Is the current wave of artificial intelligence similar to this? This question is difficult to answer. However, I personally believe that artificial intelligence will have a huge impact this time (LCTT translation annotation: This article was published in June 2022, ChatGTP was launched half a year later). Why do I have such a prediction? First, high-performance computing equipment is now cheap and readily available. In the 1960s or 1980s, there were only a few such powerful computing devices, whereas now we have millions or even billions of them. Second, there is now a vast amount of data available for training artificial intelligence and machine learning programs. Imagine how many digital images engineers who were engaged in digital image processing in the 1990s could use to train algorithms? Maybe thousands or tens of thousands. Now, the data science platform Kaggle (a subsidiary of Google) alone has more than 10,000 data sets. The vast amount of data generated by the Internet every day makes it easier to train algorithms. Third, high-speed Internet connections make it easier to work with large institutions. In the first decade of the 21st century, collaboration among computer scientists was difficult. However, the speed of the Internet now makes collaboration with artificial intelligence projects such as Google Colab, Kaggle, and Project Jupiter a reality. Based on these three factors, I believe that artificial intelligence will exist forever this time, and there will be many excellent applications

More matrix knowledge

Figure 1: Matrix A, B, C, D

After understanding the history of artificial intelligence, it is now time to return to the topic of matrices and vectors. I have briefly introduced them in previous articles. This time, we'll delve deeper into the world of the Matrix. First, please look at Figure 1 and Figure 2, which show a total of 8 matrices from A to H. Why are so many matrices needed in artificial intelligence and machine learning tutorials? First of all, as mentioned before, matrices are the core of linear algebra, and linear algebra is not the brain of machine learning, but it is the core of machine learning. Secondly, in the following discussion, each matrix has a specific purpose

Figure 2: Matrices E, F, G, H

Let Let's look at how matrices are represented and how to get their details. Figure 3 shows how to represent matrix A using NumPy. Although matrices and arrays are not exactly the same, in practical applications we often use them as synonyms

Figure 3: Representing matrix A in NumPy

I strongly It is recommended that you carefully learn how to use NumPy's array function to create a matrix. Although NumPy also provides the matrix function to create two-dimensional arrays and matrices. But it will be deprecated in the future, so its use is no longer recommended. Some details of matrix A are also shown in Figure 3 . A.size tells us the number of elements in the array. In our case it's 9. Code A.nidm represents the dimension of the array. It is easy to see that matrix A is two-dimensional. A.shape represents the order of matrix A. The order of matrix is ​​the number of rows and columns of the matrix. While I won't explain it further, you need to be aware of the size, dimension, and order of your matrices when using the NumPy library. Figure 4 shows why the size, dimension, and order of a matrix should be carefully identified. Small differences in how an array is defined can result in differences in its size, dimensionality, and order. Therefore, programmers should pay special attention to these details when defining matrices.

Figure 4: Array size, dimension and order

Now let’s do some basic matrix operations. Figure 5 shows how matrices A and B are added. NumPy provides two methods for adding matrices, the add function and the operator. Note that only matrices of the same order can be added. For example, two 4 × 3 matrices can be added, but a 3 × 4 matrix and a 2 × 3 matrix cannot be added. However, since programming is different from mathematics, NumPy does not actually follow this rule. Figure 5 also shows adding matrices A and D. Remember, this kind of matrix addition is mathematically illegal. One is called broadcasting broadcasting

Re-expression: Figure 5: Matrix summation

Re-expression: Figure 5: Matrix summation

A.shape == B.shape

The broadcast mechanism is not omnipotent. If you try to add matrices D and H, an operation error will occur.

当然除了矩阵加法外还有其它矩阵运算。图 6 展示了矩阵减法和矩阵乘法。它们同样有两种形式，矩阵减法可以由 subtract 函数或减法运算符 - 来实现，矩阵乘法可以由 matmul 函数或矩阵乘法运算符 @ 来实现。图 6 还展示了 逐元素乘法element-wise multiplication 运算符 * 的使用。请注意，只有 NumPy 的 matmul 函数和 @ 运算符执行的是数学意义上的矩阵乘法。在处理矩阵时要小心使用 * 运算符。

图 6：更多矩阵运算

对于一个 m x n 阶和一个 p x q 阶的矩阵，当且仅当 n 等于 p 时它们才可以相乘，相乘的结果是一个 m x q 阶矩的阵。图 7 显示了更多矩阵相乘的示例。注意 E@A 是可行的，而 A@E 会导致错误。请仔细阅读对比 D@G 和 G@D 的示例。使用 shape 属性，确定这 8 个矩阵中哪些可以相乘。虽然根据严格的数学定义，矩阵是二维的，但我们将要处理更高维的数组。作为例子，下面的代码创建一个名为 T 的三维数组。

图 7：更多矩阵乘法的例子

T = np.array([[[11,22], [33,44]], [[55,66], [77,88]]])

Pandas

到目前为止，我们都是通过键盘输入矩阵的。如果我们需要从文件或数据集中读取大型矩阵并处理，那该怎么办呢？这时我们就要用到另一个强大的 Python 库了——Pandas。我们以读取一个小的 CSV （逗号分隔值comma-separated value）文件为例。图 8 展示了如何读取 cricket.csv 文件，并将其中的前三行打印到终端上。在本系列的后续文章中将会介绍 Pandas 的更多特性。

图 8：用 Pandas 读取 CSV 文件

图 8：用 Pandas 读取 CSV 文件

矩阵的秩

需要进行改写的内容是：矩阵的秩

Figure 9: Finding the rank of the matrix

This is the end of this content. In the next article, we will expand the library of tools so that they can be used to develop artificial intelligence and machine learning programs. We will also discuss neural network, supervised learning, unsupervised learning in more detail

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