From Linear to Manifold Interpolation: Exemplifying the paradigm shift through interpolation

Abstract

The advent of artificial intelligence, specifically neural networks, has marked a significant turning point in the field of computation. During such transformative times, we are often faced with a dearth of appropriate vocabulary, which forces us to rely on existing terms, regardless of their inadequacy. This paper argues that the term "interpolation," typically used in deep learning (DL), is a prime example of this phenomenon. It is not uncommon for beginners to misunderstand its meaning, as DL pioneer Francois Chollet (2017) has noted. This misreading is especially true in the discipline of architecture, and this study aims to demonstrate how the meaning of "interpolation" has evolved in the second digital turn. We begin by illustrating, using 2D data, the difference between linear interpolation in the context of topological figures and its use in DL algorithms. We then demonstrate how 3DGANs can be employed to interpolate across different topologies in complex 3D space, highlighting the distinction between linear and manifold interpolation. In both 2D and 3D examples, our results indicate that the process does not involve continuous morphing but instead resembles the piecing together of a jigsaw puzzle to form many parts of a larger ambient space. Our study reveals how previous architectural research on DL has employed the term "interpolation" without clarifying the crucial differences from its use in the first digital turn. We demonstrate the new possibilities that manifold interpolation offers for architecture, which extend well beyond parametric variations of the same topology.