# Integrating Interpolation and Extrapolation: A Hybrid Predictive Framework for Supervised Learning
###### Abstract
[14] addressed the challenge of learning representations that facilitate extrapolation and proposed a novel visual analogy benchmark that enables a graded assessment of extrapolation based on the distance from the convex domain defined by the training dataset. [PERSON] et al. [15] systematically explored the extrapolation behavior of deep operator networks through rigorous quantification of extrapolation complexity and proposed a novel strategy involving bias-variance trade-off for extrapolation. Figure 1: An illustrative example of interpolation. The effectiveness of extrapolation relies on the assumption about the functional form [16]. In Figure 3, which illustrates three known data points (from \(x_{1}\) to \(x_{3}\)), the true curve is a third-order polynomial (solid black line), but the polynomial extrapolation wrongly assumes a quadratic polynomial curve (black dashed line). This underscores that extrapolation is inherently uncertain, with a heightened risk of yielding misleading results. Such issues are optimally mitigated when the functional forms assumed by the extrapolation technique closely mirror the underlying nature of the data. Interpolation and extrapolation can be viewed as linear approximation methods within the unit disk of the complex plane [17]. The most effective methods identified for interpolation and extrapolation include widely adopted techniques such as cubic spline interpolation and Gaussian processes regression [18]. [PERSON] et al. [19] provided a rigorous demonstration that extrapolation poses significantly greater computational challenges than interpolation based on reweighting of sub-group likelihoods, while the statistical complexity remains relatively unchanged. Figure 3: The effectiveness of extrapolation with different functional forms. Figure 2: Illustrative examples of extrapolation: (**a**) linear extrapolation; (**b**) polynomial extrapolation. Interpolation and extrapolation, while serving distinct roles, are both crucial for making predictions from data. Interpolation is primarily employed to fill gaps in existing records, acting as a bridge to seamlessly integrate missing data within known boundaries, and \(k\)NN serves as a predictive model with good interpolation abilities. On the other hand, extrapolation goes beyond these bounds, making predictions for entirely new observations based on the trends and patterns identified in the existing dataset, and linear regression serves as a predictive model with good extrapolation abilities. The accuracy and efficacy of these methods, however, are heavily influenced by the context in which they are used. When working with a univariate feature variable, classifying a new data point as either interior or exterior to the known dataset is relatively straightforward. If the point falls within the dataset's range, interpolation is the method of choice. Conversely, if it lies outside this range, extrapolation should be employed. However, the task becomes more difficult with a multivariate feature vector. In such cases, the task of determining whether a new data point is interior or exterior to the existing feature space grows complex. The presence of multiple dimensions can lead to scenarios where a point might be deemed interior in one feature dimension but exterior in another. Consequently, this complexity gives rise to a pressing research question: _How can the intricacies of multivariate data be effectively dealt with by leveraging the strengths of both interpolation and extrapolation while mitigating their limitations?_