Support Vector Machine, or SVM, is a powerful supervised learning algorithm primarily used for classification and regression tasks. The core idea behind SVM is to find an optimal hyperplane that separates different classes of data points while maximizing the margin between them. This hyperplane serves as the decision boundary for classification.
Support vectors are the data points that lie closest to the hyperplane and are critical in determining its position. These points define the margin, which is the distance between the hyperplane and the nearest data points from each class. The optimal hyperplane is the one that maximizes this margin, providing the best separation between classes and ensuring good generalization to new data.
When data is not linearly separable in its original space, SVM uses the kernel trick to map the data into a higher-dimensional space where it becomes linearly separable. This allows SVM to create non-linear decision boundaries in the original space. Common kernels include the RBF kernel, polynomial kernel, and sigmoid kernel. The kernel trick is computationally efficient because it doesn't require explicit mapping to the higher-dimensional space.
The mathematical foundation of SVM involves solving an optimization problem. The primal problem minimizes the norm of the weight vector subject to the constraint that all data points are correctly classified with a margin of at least one. This is converted to a dual problem using Lagrange multipliers, which is easier to solve. The final decision function uses only the support vectors, those data points with non-zero alpha coefficients, making SVM computationally efficient.
SVM has found widespread applications across many domains. In text classification, it excels at sentiment analysis and document categorization. For computer vision, SVM is used in image recognition and object detection. In bioinformatics, it helps classify genes and proteins. Financial institutions use SVM for market prediction and risk assessment. The key advantages of SVM include its effectiveness in high-dimensional spaces, memory efficiency through support vectors, versatility with different kernels, and strong performance even with small datasets.