# Mercel Kernel

### Description

A Mercer Kernel is a Kernel that is positive semi-definite. When a kernel is positive semi-definite, one may exploit the *kernel trick* [1], the idea of mapping data to a high-dimensional feature space where some linear algorithm is applied that works exclusively with inner products. Assume we have some mapping from an input space to a feature space , then a kernel function (or kernel)

may be used to define the inner product in feature space . Figure 1 shows that the application of a linear algorithm in feature space could correspond to a nonlinear estimate in input space .

Positive definiteness in the context of kernel functions also implies that a kernel matrix created using a particular kernel is positive semi-definite. A matrix is positive semi-definite if its associated eigenvalues are nonnegative.

### Refinement of

Kernel

### Associated types

### Notation

### Definitions

### Valid Expressions

### Expression Semantics

### Complexity Guarantees

### Invariants

### Models

### Notes

### See also

### References

[1]

Mark Aizerman, Emmanuil Braverman, and Lev Rozonoèr. Theoretical foundations of the potential function method in pattern recognition learning. *Automation and Remote Control*, 25:821–837, 1964.