Topology Applied to Machine Learning: From Global to Local

Abstract

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.

On the effectiveness of persistent homology

“…Subsequently the discussion has refined when it was shown that short and medium-length persistence intervals have the most distinguishing power for specific types of applications [8,81]. The current understanding is roughly that long intervals reflect the topological signal, and (many) short intervals can help in detecting geometric features [3]. We believe that our work brings new insight into this discussion.…”

“…In the first decade after the introduction of PH, it was seen primarily as the descriptor of global topology. Recently, there have been many discussions and greater understanding that PH also captures local geometry [3]. However, it is still suggested that the long persistence intervals capture topology (as was the case with the detection of holes in Section 3), and many-even too many for the human eye to count-short persistence intervals capture geometrical properties (as was the case with curvature prediction in Section 4).…”

“…We will look into scenarios in which data contains noise, and therefore, instead of the standard distance function, we consider Distance-to-Measure (DTM) as the filtration function. We extract 1-dimensional PDs, that are then transformed to PIs, PLs, or a simple signature consisting only of lifespans l = d − b of the 10 most persisting cycles, 3 and classified with an SVM. We consider a "PH simple" pipeline, which relies on the 10 lifespans, and a "PH" pipeline wherein grid search is employed to choose the best out of the three aforementioned persistent signatures and the values of their parameters.…”

A new construction for sublevel set persistence

“…The homology groups in (1) have several advantages over distance-based persistent homology, but are generally difficult to compute, particularly in higher dimensions. One application of sublevel set persistence is to topological data analysis, which involves computing the superlevel set persistence of a suitable density estimator ρ (or the sublevel set persistence of − log(ρ)).…”

“…The higher dimensional case has been studied using the Morse-Smale complex [11] in the context of grayscale images, which has also been applied to topics such as energy landscapes in particle systems [14]. For other applications and references, see also [1,2,3,4,5,10,22].…”