Understand the concepts of functional distance and geometric distance in support vector machines

How to understand the functional distance and geometric distance in support vector machine

SVM divides samples into two categories through hyperplanes.

When the hyperplane is determined, the distance of a point from the hyperplane can be expressed relatively. For a two-class classification problem, if the point is on the positive side of the hyperplane, it is judged as 1; otherwise, it is judged as -1.

If the classification result is considered correct, it is correct; otherwise, it is wrong. Moreover, the larger the value, the higher the confidence of the classification result. vice versa.

So the functional interval between the sample point and the hyperplane is defined as

However, there is a problem with this definition: when we shrink or enlarge the hyperplane M times at the same time, the function interval changes. To solve this problem, we need to fix the size of the hyperplane, i.e., so that the function margin remains constant. In this way, we can get the geometric interval.

The geometric interval is defined as follows

In fact, the geometric interval is the distance from the point to the hyperplane. Imagine the distance formula from a point to a straight line learned in middle school

In multi-dimensional space, the geometric interval refers to the distance from the point to the hyperplane. Functional distance is the numerator in the unnormalized distance formula.

Define the minimum geometric distance from the training set to the hyperplane is

The method of SVM training classifier is to find the hyperplane so that the positive and negative samples are on both sides of the hyperplane, and the geometric distance between the sample and the hyperplane is the largest.

So SVM can be expressed as solving the following optimization problem

The above content is explained in detail in "Statistical Learning Methods".

What is the unit of distance in drawing geometry

How to learn drawing geometry: 1. Pay attention to the analysis of spatial geometric relationships and the correspondence between spatial geometric prototypes and plane graphics. This process of repeated research and thinking of "from space to plane, and then from plane to space" is the most basic and most effective learning method in this course.

Some beginners ignore the analysis of spatial geometric relationships and the correspondence between spatial geometric prototypes and plane patterns, and just try to use some conclusions in books to solve problems. There are also some beginners who only pay attention to the spatial geometric relationships, and put aside the projection rules that have been summarized in books. Whenever they solve a specific problem, they use their own models to compare the spatial conditions to directly obtain the answer. This kind of theory that is divorced from reality and ignores theoretical learning methods will bring difficulties to learning.

2. According to the nature of the course, this course is a basic technical course, and the practice of drawing and viewing pictures is very important. To this end, in the learning process: ① Focus on studying various illustrations. When reviewing, you should not just stop at reading. Instead, you should describe the drawing process of the illustrations on paper while reading. In this way, not only will it be easy to understand the content of the textbook, but you will also be able to truly grasp the principles of projection and its specific applications. ② Make systematic summaries frequently. For each chapter learned, a certain number of exercises must be completed to consolidate it. ③ Consciously cultivate a serious, meticulous and patient work style, and develop the habit of accurate drawing and neat drawings.

3. Drawing geometry and engineering drawings are closely related. Drawing geometry provides basic principles and methods for using two-dimensional graphics to express parts and related diagramming methods in engineering drawings. In line with the principle of integrating theory with practice, attention should be paid to the connection and cooperation between drawing geometry and engineering drawings during study.

In the 21st century, computer technology has penetrated into all areas of human society. The development of computer-aided design (CAD) and computer graphics (CG) technology has successfully solved many problems of drawing geometric diagrams and illustrations. Compared with the traditional painting-based geometric problem-solving method, the purpose of the two is the same, but the means (the use of tools) are different. As a tool in the hands of humans, computers have an absolute advantage in work efficiency and accuracy. Traditional drawing geometry theory and computer graphics theory together form the graphics foundation of CAD and CG technology. From the perspective of tool use, it is more necessary to master the basic theory of drawing geometry.

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