$$k(\mathbf{x},\mathbf{x}') = e^{-\frac{(\mathbf{x}-\mathbf{x}')^2}{2 \sigma^2}} \approx z(\mathbf{x})^Tz(\mathbf{x}')$$

$$z(\mathbf{x}): \mathbb{R}^{D} \rightarrow \mathbb{R}^{Q}$$

$$z(\mathbf{x}) = \sqrt{\frac{2}{Q}} [ cos(\mathbf{w}_1^{T}\mathbf{x} + b_1),...,cos(\mathbf{w}_Q^{T}\mathbf{x} + b_Q)]$$

Where $\mathbf{w}_1,...,\mathbf{w}_Q \in \mathbb{R}^D$ is Q i.i.d samples from $p(\mathbf{w}) = \frac{1}{2\pi} \int e^{-j\mathbf{w}'\delta}k(\delta)d\triangle$, and $b_1,...,b_Q \in \mathbb{R}$ is Q samples from $\mathcal{N}(0,2\pi)$

For now $$p(\mathbf{w}) = (2\pi)^{\frac{2}{Q}}e^{-\frac{|\mathbf{w}|_2^2}{2}}$$

CONVOLUTION

Properties of translation equivariance and notions of locality.

$\mathbf{F}_l \in \mathbb{R}^{H\times W\times C}$

$\varphi: \mathbb{R}^{H\times W\times C} \rightarrow \mathbb{R}^{H_o\times W_o\times C_o}$

$\sigma \in \mathbb{R}^{+}$ Scale

$$\mathbf{F}l = \phi(\mathbf{F}{l-1})= cos(\frac{\mathbf{W}l}{\sigma}\otimes\mathbf{F}{l-1}+\mathbf{b}_l)$$

pip install -U git+https://github.com/aguirrejuan/ConvRFF.git