# Manifold Laplacian-Beltrami estimation with uniform sample

Given a measure on a smooth, compact manifold $M \subset \mathbb{R}^N$ we construct the corresponding operator

$$L^tf(x)=f(x)\int_{M} e^-\frac{||x-y||^2}{4t}d\nu(y)-\int_{M}f(y)e^-\frac{||x-y||^2}{4t}d\nu(y).$$

Let data points $x_1, \dots, x_n$ be sampled from a uniform distribution on this manifold $M \subset \mathbb{R}^N$ and the following be an empirical estimate of $L^tf(x)$:

$$L^t_nf(x)=f(x)\frac{1}{n}\sum_{j}e^\frac{-{\langle x , x_j\rangle}}{4t}-\frac{1}{n}\sum_{j}f(x_j)e^\frac{-\langle x , x_j \rangle}{4t}.$$

How can I measure/quantify the quality of this empirical estimate in estimating the above operator $L^tf(x)$? Ex: Do I use the Hoeffding's inequality? Wondering-what would be a good way to do this?

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