Карта особливостей ядра Гаусса

У SVM ядро ​​Гаусса визначається як:

$$K(x,y)=\exp\left({-\frac{\|x-y\|_2^2}{2\sigma^2}}\right)=\phi(x)^T\phi(y)$$ де$x, y\in \mathbb{R^n}$. Я не знаю явного рівняння$\phi$. Я хочу це знати.

Я також хотів би знати , є чи

$$\sum_ic_i\phi(x_i)=\phi \left(\sum_ic_ix_i \right)$$ , де $c_i\in \mathbb R$ . Зараз я думаю, що це не дорівнює, тому що за допомогою ядра обробляється ситуація, коли лінійний класифікатор не працює. Я знаю $\phi$ проектує х у нескінченний простір. Тож якщо вона все ще залишається лінійною, незалежно від того, скільки її розмірів, svm все одно не може зробити хорошої класифікації.

$\phi$$e^x$. For notational simplicity, assume $x\in \mathbb{R}^1$:

$$\phi(x) = e^{-x^2/2\sigma^2} \Big[ 1, \sqrt{\frac{1}{1!\sigma^2}}x,\sqrt{\frac{1}{2!\sigma^4}}x^2,\sqrt{\frac{1}{3!\sigma^6}}x^3,\ldots\Big]^T$$

This is also discussed in more detail in these slides by Chih-Jen Lin of NTU (slide 11 specifically). Note that in the slides $\gamma=\frac{1}{2\sigma^2}$ is used as kernel parameter.

The equation in the OP only holds for the linear kernel.

For any valid psd kernel $k : \mathcal X \times \mathcal X \to \mathbb R$, there exists a feature map $\varphi : \mathcal X \to \mathcal H$ such that $k(x, y) = \langle \varphi(x), \varphi(y) \rangle_{\mathcal H}$. The space $\mathcal H$ and embedding $\varphi$ in fact need not be unique, but there is an important unique pair $(\mathcal H, \varphi)$ known as the reproducing kernel Hilbert space (RKHS).

The RKHS is discussed by: Steinwart, Hush and Scovel, An Explicit Description of the Reproducing Kernel Hilbert Spaces of Gaussian RBF Kernels, IEEE Transactions on Information Theory 2006 (doi, free citeseer pdf).

It's somewhat complicated, but it boils down to this: define $e_n : \mathbb C \to \mathbb C$ as

$$e_n(z) := \sqrt{\frac{(2 \sigma^2)^n}{n!}} z^n e^{-\sigma^2 z^2} .$$

Let $n : \mathbb{N}_0 \to \mathbb{N}_0^d$ be a sequence ranging over all $d$-tuples of nonnegative integers; if $d = 3$, perhaps $n(0) = (0, 0, 0)$, $n(1) = (0, 0, 1)$, $n(2) = (0, 1, 1)$, and so on. Denote the $j$th component of the $i$th tuple by $n_{ij}$.

Then the $i$th component of $\varphi(x)$ is $\prod_{j=1}^d e_{n_{ij}}(x_j)$. So $\varphi$ maps vectors in $\mathbb R^d$ to infinite-dimensional complex vectors.

The catch to this is that we further have to define norms for these infinite-dimensional complex vectors in a special way; see the paper for details.

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Steinwart et al. also give a more straightforward (to my thinking) embedding into $L_2(\mathbb R^d)$, the Hilbert space of square-integrable functions from $\mathbb R^d \to \mathbb R$:

$$\Phi_\sigma(x) = \frac{(2 \sigma)^{\frac{d}{2}}}{\pi^{\frac{d}{4}}} e^{- 2 \sigma^2 \lVert x - \cdot \rVert_2^2} .$$ Note that $\Phi_\sigma(x)$ is itself a function from $\mathbb R^d$ to $\mathbb R$. It's basically the density of a $d$-dimensional Gaussian with mean $x$ and covariance $\frac{1}{4 \sigma^2} I$; only the normalizing constant is different. Thus when we take $$\langle \Phi(x), \Phi(y) \rangle_{L_2} = \int [\Phi(x)](t) \; [\Phi(y)](t) \,\mathrm d t ,$$ we're taking the product of Gaussian density functions, which is itself a certain constant times a Gaussian density functions. When you do that integral by $t$, then, the constant that falls out ends up being exactly $k(x, y)$.

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These are not the only embeddings that work.

Another is based on the Fourier transform, which the celebrated paper of Rahimi and Recht (Random Features for Large-Scale Kernel Machines, NIPS 2007) approximates to great effect.

You can also do it using Taylor series: effectively the infinite version of Cotter, Keshet, and Srebro, Explicit Approximations of the Gaussian Kernel, arXiv:1109.4603.

It seems to me that your second equation will only be true if $\phi$ is a linear mapping (and hence $K$ is a linear kernel). As the Gaussian kernel is non-linear, the equality will not hold (except perhaps in the limit as $\sigma$ goes to zero).