We describe a framework for estimating Hilbert–Samuel multiplicities for pairs of projective varieties from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce to a point and to a curve . Next, we establish that equals the Euler characteristic (and hence the cardinality) of the complex link of in . Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of in ) to determine this Euler characteristic with high confidence.