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Kernel functions enable SVM classifiers to operate in high-dimensional feature spaces without explicitly computing the transformed features. This page documents the base kernel interface and common kernel types.

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KernelFunction

Type-erased value-semantic kernel handle that provides copyable, movable, and polymorphic kernel functionality.

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Constructor

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impl
std::unique_ptr<Kernel>
Unique pointer to a concrete kernel implementation

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Methods

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operator()

Evaluates the kernel function on two input vectors.
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x
const Vector&
First input vector
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y
const Vector&
Second input vector
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return
double
Kernel evaluation result k(x, y)

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valid()

Checks if the kernel function is initialized.
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return
bool
True if the kernel contains a valid implementation, false otherwise

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LinearKernel

Linear kernel corresponding to the identity feature map in ℝⁿ. Formula: k(x, y) = ⟨x, y⟩ The linear kernel computes the standard dot product between two vectors. It is the simplest kernel and corresponds to no transformation of the feature space.

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Constructor

Default constructor with no parameters.

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Methods

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operator()

Computes the dot product between two vectors.
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x
const Vector&
First input vector
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y
const Vector&
Second input vector
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return
double
Dot product ⟨x, y⟩

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Example

#include <mlpp/classifiers/kernel/rkhs_kernels.hpp>

using namespace mlpp::classifiers::kernel;

// Create a linear kernel
LinearKernel linear;

// Create kernel function wrapper
KernelFunction kernel(std::make_unique<LinearKernel>());

// Evaluate on two vectors
Vector x = {1.0, 2.0, 3.0};
Vector y = {4.0, 5.0, 6.0};

double result = kernel(x, y);  // Returns 32.0 (1*4 + 2*5 + 3*6)

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PolynomialKernel

Polynomial kernel that generates a finite-dimensional RKHS with dimension growing combinatorially with the degree. Formula: k(x, y) = (γ ⟨x, y⟩ + c)^d

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Constructor

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gamma
double
Scale parameter γ for the dot product
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coef0
double
Independent coefficient c (offset term)
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degree
std::size_t
Polynomial degree d

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Methods

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operator()

Computes the polynomial kernel between two vectors.
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x
const Vector&
First input vector
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y
const Vector&
Second input vector
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return
double
Polynomial kernel value (γ ⟨x, y⟩ + c)^d

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Example

#include <mlpp/classifiers/kernel/rkhs_kernels.hpp>

using namespace mlpp::classifiers::kernel;

// Create a polynomial kernel of degree 3
// with gamma=1.0 and offset=1.0
PolynomialKernel poly(1.0, 1.0, 3);

KernelFunction kernel(std::make_unique<PolynomialKernel>(1.0, 1.0, 3));

Vector x = {1.0, 2.0};
Vector y = {3.0, 4.0};

double result = kernel(x, y);  // (1*3 + 2*4 + 1)^3 = 12^3 = 1728

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RBFKernel

Radial Basis Function (Gaussian) kernel that induces an infinite-dimensional, separable RKHS. Universally consistent under mild conditions. Formula: k(x, y) = exp(-γ ||x - y||²) The RBF kernel is one of the most popular kernels in practice due to its smoothness properties and universal approximation capabilities.

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Constructor

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gamma
double
Bandwidth parameter γ controlling the kernel width. Larger values lead to more localized influence.

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Methods

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operator()

Computes the RBF kernel between two vectors.
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x
const Vector&
First input vector
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y
const Vector&
Second input vector
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return
double
RBF kernel value exp(-γ ||x - y||²)

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Example

#include <mlpp/classifiers/kernel/rkhs_kernels.hpp>

using namespace mlpp::classifiers::kernel;

// Create an RBF kernel with gamma=0.5
RBFKernel rbf(0.5);

KernelFunction kernel(std::make_unique<RBFKernel>(0.5));

Vector x = {1.0, 2.0, 3.0};
Vector y = {1.5, 2.5, 3.5};

double result = kernel(x, y);  // exp(-0.5 * 0.75) ≈ 0.687

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Kernel base class

Abstract base class for all kernel implementations.

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Methods

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operator()

Pure virtual method for kernel evaluation.
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x
const Vector&
First input vector
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y
const Vector&
Second input vector
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return
double
Kernel evaluation k(x, y)

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clone()

Creates a deep copy of the kernel.
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return
std::unique_ptr<Kernel>
Unique pointer to a cloned kernel instance

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Type aliases

  • Vector = std::vector<double> - Standard vector type used throughout the kernel API

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