Update docs/components/spec.rst

docs/components/spec.rst

@@ -12,7 +12,7 @@ The second part of the specification is crucial for determining the model's posi
 
 The RKME specification assumes that each learnware is a well-performed model on its training data. The RKME specification is based on RKME :math:`\widetilde{\Phi}`, which aims to provide a good representation by constructing a reduced set to approximate the empirical KME :math:`\Phi=\int_{\mathcal{X}} k(\boldsymbol{x}, \cdot) \mathrm{d} P(\boldsymbol{x})` of the underlying distribution. Theoretically, when the kernel function satisfies :math:`k(\boldsymbol{x}, \boldsymbol{x}) \leq 1` for all :math:`x \in \mathcal{X}`, we have the guarantee that
 
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    \|\widetilde{\Phi}-\Phi\|_{\mathcal{H}} \leq 2 \sqrt{\frac{2}{n}}+\sqrt{\frac{1}{m}}+\sqrt{\frac{2 \log (1 / \delta)}{m}},
 
 with a probability of at least :math:`1-\delta`, where :math:`n, m` are the sizes of the RKME reduced set and the original data, respectively. It is known that when using characteristic kernels such as the Gaussian kernel, KME can capture all information about the distribution. Additionally, when the RKHS of the kernel function is finite-dimensional, RKME has a linear convergence rate :math:`O\left(e^{-n}\right)` to empirical KME; for infinite-dimensional RKHS, it has been proved constructively that RKME can enjoy :math:`O(\sqrt{d} / n)` convergence rate under :math:`L_{\infty}` measure, where :math:`d` is the dimension of the original data. Therefore, RKME is guaranteed to be a good estimation of KME and a valid representation for data distribution that encodes the ability of a trained model.