How does SVM determine the position of a new point relative to the decision boundary?
Support Vector Machines (SVM) are a popular machine learning algorithm used for classification and regression tasks. SVMs are particularly effective when dealing with high-dimensional data and can handle both linear and non-linear decision boundaries. In this answer, we will focus on how SVM determines the position of a new point relative to the decision boundary.
To understand how SVM works, it is important to first grasp the concept of a decision boundary. In binary classification, the decision boundary is a hyperplane that separates the data points belonging to different classes. SVM aims to find the optimal decision boundary by maximizing the margin, which is the distance between the decision boundary and the nearest data points from each class. These nearest data points are called support vectors.
When a new point is introduced to the SVM model, the algorithm determines its position relative to the decision boundary by calculating its distance from the decision boundary. This distance is then used to classify the new point as belonging to one of the classes.
To calculate the distance between the new point and the decision boundary, SVM uses a concept known as the decision function. The decision function takes the new point as input and returns a signed distance value. The sign of the distance indicates which side of the decision boundary the point lies on, while the magnitude of the distance reflects the confidence of the classification.
The decision function for SVM can be expressed as:
f(x) = sign(w^T * x + b)
Here, x represents the new point, w is the weight vector, and b is the bias term. The weight vector and bias term are learned during the training phase of the SVM algorithm. The sign function returns +1 if the new point is on one side of the decision boundary (positive class) and -1 if it is on the other side (negative class).
The distance between the new point and the decision boundary can be calculated as the absolute value of the decision function divided by the norm of the weight vector:
distance = |f(x)| / ||w||
The norm of the weight vector ||w|| represents the length or magnitude of the weight vector. By dividing the decision function by the norm of the weight vector, we normalize the distance to ensure it is independent of the scale of the weight vector.
It is worth noting that the distance calculated using the decision function is not the geometric distance between the new point and the decision boundary. Instead, it represents the margin or confidence of the classification. A larger distance indicates a higher confidence in the classification, while a smaller distance suggests a lower confidence.
To summarize, SVM determines the position of a new point relative to the decision boundary by calculating its distance from the decision boundary using the decision function. The sign of the distance indicates the class of the new point, while the magnitude of the distance reflects the confidence of the classification.
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