Interpolation, Extrapolation, Hyperpolation: Generalising into new dimensions (2409.05513v2)

Abstract: This paper introduces the concept of hyperpolation: a way of generalising from a limited set of data points that is a peer to the more familiar concepts of interpolation and extrapolation. Hyperpolation is the task of estimating the value of a function at new locations that lie outside the subspace (or manifold) of the existing data. We shall see that hyperpolation is possible and explore its links to creativity in the arts and sciences. We will also examine the role of hyperpolation in machine learning and suggest that the lack of fundamental creativity in current AI systems is deeply connected to their limited ability to hyperpolate.

• The paper introduces "hyperpolation" as a third form of generalization beyond interpolation and extrapolation, applicable when estimating function values outside both the convex and affine hull of known data.
• Hyperpolation has significant theoretical and practical implications, particularly for machine learning and generative AI, offering a potential pathway to overcome out-of-distribution challenges and foster true creativity.
• The concept challenges traditional boundaries of generalization, suggesting new directions for AI research aimed at enabling systems to produce genuinely novel outputs beyond existing data distributions, analogous to biological evolution.

Hyperpolation: Expanding the Horizons of Generalization

This paper, "Interpolation, Extrapolation, Hyperpolation: Generalising into New Dimensions" by Toby Ord, presents an innovative concept termed "hyperpolation," which is posited as a third form of generalization alongside interpolation and extrapolation. Hyperpolation is characterized by the task of estimating function values beyond the constraints of both the convex and affine hull of existing data, effectively expanding the field of generalization into new dimensions.

Defining Hyperpolation

Function generalization is commonly divided into interpolation, where new points lie within the convex hull of known data, and extrapolation, where points reside outside the convex hull but within the affine span. Hyperpolation emerges when points exist outside both these confines. The paper provides formal definitions, grounding hyperpolation as a non-trivial extension of existing concepts and exploring its mathematical intricacies. An interesting nuance highlighted is the absence of a unique solution in hyperpolation, necessitating criteria akin to those applied in interpolation and extrapolation, such as function smoothness or simplicity, to find "reasonable" solutions.

Theoretical and Practical Implications

The paper explores several theoretical implications of hyperpolation. It challenges the traditional boundaries of generalization, proposing hyperpolation as a tool not only for mathematical abstraction but also for creative exploration in disciplines like AI, art, and science. The utility of hyperpolation is exemplified through mathematical constructs such as ripple patterns and conic sections, which reveal richer structures through the application of hyperpolation.

In machine learning, hyperpolation could play a pivotal role in enhancing the ability of algorithms to generalize beyond their trained data manifolds. This could be particularly impactful for addressing the "out-of-distribution" problem, where models struggle to predict accurately beyond the constrained dimensions of their training data. By potentially redefining the approach to generalization, hyperpolation offers a pathway to developing more robust AI systems capable of higher-level creativity.

Creativity and AI

The linkage between hyperpolation and creativity is a prominent theme. Current generative AI systems, although capable of impressive feats, tend to operate within the constraints of interpolation and extrapolation, often remixing known data rather than pioneering new intellectual territories. Hyperpolation suggests a framework for transcending these limitations, enabling the formation of genuinely novel, high-quality outputs that are orthogonal to existing data and styles—a vital component of true creativity.

The paper underscores the difficulties modern AI encounters in achieving hyperpolation, evidenced by the challenges in generative tasks that require not merely derivation from existing styles but the synthesis of unique forms wholly unrepresented in the training data. While some modern approaches begin to tackle such challenges in a limited scope, as seen in predictive video models, the full realization of hyperpolation's potential remains a largely open field.

Evolutionary Paradigms

The biological analogy offered in the paper highlights how evolutionary processes naturally embody a form of hyperpolation through genetic mutations that introduce phenotypic expressions beyond existing typologies. This underscores the viability of exploration off established manifolds, albeit consolidated over extensive temporal frameworks.

Conclusions and Future Directions

The proposal of hyperpolation as a third pillar of generalization has the potential to redefine boundaries in both theoretical and applied domains. The paper calls for further exploration into practical methods for achieving hyperpolation and evaluating the capabilities of various machine learning paradigms in this context. Moreover, the integration of hyperpolation into generative AI holds promise for unlocking new levels of creative potential, moving beyond the limitations that have so far defined AI's scope.

Future research directions could include the development of specific algorithms better suited to hyperpolation, fine-tuning existing architectures to enhance their extrapolation capacities, and exploring manifold-based interpretations, which may provide a richer context for tangible improvements in AI creativity and scientific discovery.