Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short lifetimes are informally considered to be "topological noise," and those with a long lifetime are considered to be "topological signal." In this paper, we bring some statistical ideas to persistent homology. In particular, we derive confidence sets that allow us to separate topological signal from topological noise.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1252 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory.

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The current system for evaluating prostate cancer architecture is the Gleason grading system which divides the morphology of cancer into five distinct architectural patterns, labeled 1 to 5 in increasing levels of cancer aggressiveness, and generates a score by summing the labels of the two most dominant patterns. The Gleason score is currently the most powerful prognostic predictor of patient outcomes; however, it suffers from problems in reproducibility and consistency due to the high intra-observer and inter-observer variability amongst pathologists. In addition, the Gleason system lacks the granularity to address potentially prognostic architectural features beyond Gleason patterns. We evaluate prostate cancer for architectural subtypes using techniques from topological data analysis applied to prostate cancer glandular architecture. In this work we demonstrate the use of persistent homology to capture architectural features independently of Gleason patterns. Specifically, using persistent homology, we compute topological representations of purely graded prostate cancer histopathology images of Gleason patterns 3,4 and 5, and show that persistent homology is capable of clustering prostate cancer histology into architectural groups through a ranked persistence vector. Our results indicate the ability of persistent homology to cluster prostate cancer histopathology images into unique groups with dominant architectural patterns consistent with the continuum of Gleason patterns. In addition, of particular interest, is the sensitivity of persistent homology to identify specific sub-architectural groups within single Gleason patterns, suggesting that persistent homology could represent a robust quantification method for prostate cancer architecture with higher granularity than the existing semi-quantitative measures. The capability of these topological representations to segregate prostate cancer by architecture makes them an ideal candidate for use as inputs to future machine learning approaches with the intent of augmenting traditional approaches with topological features for improved diagnosis and prognosis.

We define a topology-based distance metric between road networks embedded in the plane. This distance measure is based on local persistent homology, and employs a local distance signature that enables identification and visualization of local differences between the road networks. This paper is motivated by the need to recognize changes in road networks over time and to assess the quality of different map construction algorithms. One particular challenge is evaluating the results when no ground truth is known. However, we demonstrate that we can overcome this hurdle by using a statistical technique known as the bootstrap.