Kernel
A kernel is a mathematical function used to compute similarity between data points. In algorithms like Support Vector Machines (SVMs), kernels project input data into a higher-dimensional feature space, making it easier to separate classes that are not linearly separable in the original space.
Background
The kernel trick avoids explicit computation of high-dimensional mappings. Instead, it directly computes inner products in the transformed space, enabling efficient handling of non-linear problems.
Examples
- SVM classification with radial basis function (RBF) kernels.
- Kernel PCA for nonlinear dimensionality reduction.
- Text similarity analysis using string kernels.
Strengths and challenges
- ✅ Enables complex, nonlinear classification and regression.
- ✅ Offers flexibility with multiple kernel choices.
- ❌ Kernel selection is often empirical and problem-dependent.
- ❌ Computationally demanding for very large datasets.
A kernel can be thought of as a similarity engine: it tells us how alike two data points are, not just in their raw form but in a richer, often hidden feature space. This makes kernels especially valuable in cases where linear separation is impossible in the original data representation.
One of the most powerful aspects is the kernel trick. Instead of explicitly mapping data into a high-dimensional space—which could be computationally prohibitive—the kernel computes the inner product as if the mapping had been done. This elegant shortcut unlocks nonlinear classification and regression without exploding the cost of computation.
Beyond SVMs, kernels appear in Gaussian Processes, where they act as covariance functions, shaping the model’s assumptions about smoothness and similarity. They also underpin kernelized clustering and even play a role in signal processing. The challenge, however, remains in selecting the right kernel: too simple and it may underfit, too complex and it risks overfitting or becoming inefficient on large datasets.
📚 Further Reading
- Scholkopf, B., Smola, A. J. (2002). Learning with Kernels.
