# Kernel Matrix

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A Kernel Matrix is a symmetric and positive semidefinite matrix that encodes the relative positions of all points.

## References

### 2018b

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Kernel_method#Mathematics:_the_kernel_trick Retrieved:2018-8-10.
• Theoretically, a Gram matrix $\mathbf{K} \in \mathbb{R}^{n \times n}$ with respect to $\{\mathbf{x}_1, \dotsc, \mathbf{x}_n\}$ (sometimes also called a "kernel matrix" ), where $K_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$ , must be positive semi-definite (PSD). Empirically, for machine learning heuristics, choices of a function $k$ that do not satisfy Mercer's condition may still perform reasonably if $k$ at least approximates the intuitive idea of similarity. Regardless of whether $k$ is a Mercer kernel, $k$ may still be referred to as a "kernel". If the kernel function $k$ is also a covariance function as used in Gaussian processes, then the Gram matrix $\mathbf{K}$ can also be called a covariance matrix.

### 2007

• (Nguyen & Ho, 2007) ⇒ Canh Hao Nguyen, and Tu Bao Ho (2007, January). "Kernel Matrix Evaluation". In IJCAI (pp. 987-992).
• QUOTE: Training example set $\{x_i\}_{i=1,\cdots, n} \subset X$ with the corresponding target vector $y=\{ y_i \}^T_{i=1, \cdots, n} \subset \{−1, 1\}^n$. Suppose that $y_1 = \cdots = y_{n_+} = 1$ and $y_{n_++1} = .. = y_{n_++n_−} = −1;\; n_+$ examples belong to class $1,\; n_−$ examples belong to class $−1,\; n_+ + n_− = n$. Under a feature map $\phi$, the kernel matrix is defined as:

$K = \{k_{ij} = \langle \phi(x_i), \phi(x_j)\rangle\}_{i=1, \cdots, n,\;j=1,\cdots, n}$

### 2004

1. Theorem 7.2.10 Let $v_1,\ldots,v_m$ be vectors in an inner product space with inner product $\langle{\cdot,\cdot}\rangle$ and let $G = [\langle{v_j,v_i}\rangle]_{i,j=1}^m \in M_m$ . Then
(a) is Hermitian and positive-semidefinite
(b) is positive-definite if and only if the vectors $v_1,\ldots,v_m$ are linearly-independent.
(c) $\operatorname{rank}G=\dim\operatorname{span}\{v_1,\ldots,v_m\}$