Conformal prediction in manifold learning

The paper presents a geometrically motivated view on conformal prediction applied to nonlinear multi-output regression tasks for obtaining valid measure of accuracy of Manifold Learning Regression algorithms. A considered regression task is to estimate an unknown smooth mapping f from q-dimensional inputs x ∈ X to m-dimensional outputs y = f(x) based on training dataset Z(n) consisting of “input-output” pairs {Zi = (xi,yi = f(xi)) , i = 1, 2, . . . , n}. Manifold Learning Regression (MLR) algorithm solves this task using Manifold learning technique. At first, unknown q-dimensional Regression manifold M(f) = {(x, f(x)) ∈ R : x ∈ X ⊂ R}, embedded in ambient (q + m)-dimensional space, is estimated from the training data Z(n), sampled from this manifold. The constructed estimator MMLR, which is also q-dimensional manifold embedded in ambient space R, is close to M in terms of Hausdorff distance. After that, an estimator fMLR of the unknown function f , mapping arbitrary input x ∈ X to output fMLR(x), is constructed as the solution to the equation M(fMLR) = MMLR. Conformal prediction allows constructing a prediction region for an unknown output y = f(x) at Out-of-Sample input point x for a given confidence level using given nonconformity measure, characterizing to which extent an example Z = (x,y) is different from examples in the known dataset Z(n). The paper proposes a new nonconformity measure based on MLR estimators using an analog of Bregman distance.