Coordinate-free interpolation on a Riemannian manifold Let $(M,g)$ be a Riemannian manifold, and let $\{(p_i,y_i):p_i \in M, y_i \in \mathbb{R}\}$ be a collection of real values assigned to distinct points of $M$.   Is there any good coordinate independent way to interpolate and find a function $y:M\rightarrow \mathbb{R}$ such that $y(p_i) = y_i$?
When $(M,g)$ is flat and $p_i$ gives a triangulation, we can interpolate linearly on each triangle (unambiguously, using the metric), giving us a piecewise-linear $y(p)$ on the union of the triangles.  For example, on $M = \mathbb{R}$ this is just piecewise linear interpolation.  The same procedure works on $\mathbb{R}^n$.  What if instead $M$ was the sphere?  The "linear" requirement probably doesn't make sense, but is there some weaker condition we can have?
 A: One option is to take the weighted average of the $y_i$’s.  You can weight them in various ways; e.g., you might assign each $y_i$ a weight of 
$$\frac{1}{d(p,p_i)}$$
or you might assign each $y_i$ a weight of 
$$\frac{1}{e^{d(p,p_i)}-1}$$
where $d$ is the distance.  Since the weights don’t usually sum to one, you’ll have to divide by their sum to normalize.
Anything reasonably called interpolation will be of this general form; the usual 1-d interpolation corresponds to weighting $y_i$ by
$$\frac{f(p,p_i)}{d(p,p_i)}$$
where $f$ is $0$ if some $p_j$ is properly between $p$ and $p_i$, and $1$ otherwise.
A: Any answer to to your question depends on what you mean by a "good" way to interpolate. Not knowing more there are four straightforward approaches 


*

*Choose the function with is equal to $y_i$ at $p_i$ and zero elsewhere on $M$.

*If $M$ is homeomorphic/diffeomorphic to $\mathbb{R}^n$ solve the problem in $\mathbb{R}^n$ and pull back the solution to $M$.

*You can always embed $M$ in a suitable $\mathbb{R}^n$, solve the problem there with your favorite interpolation technique for euclidean space and restrict the interpolant to your embedded manifold. With a bit more detail: After embedding $M$ with a map $J$ say, you have points $J(p_i)$ in $\mathbb{R}^n$ and values $y_i$. You determine an interpolating function $\hat{g}:\mathbb{R}^n\rightarrow \mathbb{R}$ such that $\hat{g}(J(p_i)) = y_i$. Then you define $g$ on $M$ as $g(p) = \hat{g}(J(p)).$

*Nearest neighbour should also work. For any point $p$ you find the $p_i$ with minimal distance to $p$ and assign the value $y_i$ to $p$.

A: Another solution that comes from semi-supervised learning is Laplacian regularization; in a precise sense (minimization of the Dirichlet energy), you extend the data from labeled vertices to unlabeled vertices of your manifold in the "smoothest" way.
If $\mathcal{O}$ designates the observation set (vertices at which a label  - it being discrete or continuous - is available) whereas $\mathcal{U}$ designates the set at which no labels are available, then we require the minimization of
$$
\text{E}(y) = \frac{1}{2} \sum_{ p,q \in \mathcal{O} \cup \mathcal{U}}{ w(p,q) ( y(p) - y(q))^2 }
$$
with the boundary condition
$$
y(p_i) = y_i \ \text{, for all } p_i \in \mathcal{O}
$$
where $w$ is a weight function, typically chosen to be Gaussian. This is the affinity function that defines non-negative edge weights.
See for instance Deep Neural Nets with Interpolating Function as Output Activation and The  game  theoretic  p-laplacian  and  semi-supervised  learning  with  few  labels.
