More Like this

1. Adaptive Partitioning for Template Functions on Persistence Diagrams

   https://doi.org/10.1109/ICMLA.2019.00202  

   Tymochko, Sarah; Munch, Elizabeth; Khasawneh, Firas A. (December 2019, 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA))

   As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically persistence diagrams, as inputs to machine learning techniques requires some mathematical creativity. The space of persistence diagrams does not have the desirable properties for machine learning, thus methods such as kernel methods and vectorization methods have been developed. One such featurization of persistence diagrams by Perea, Munch and Khasawneh uses continuous, compactly supported functions, referred to as "template functions," which results in a stable vector representation of the persistence diagram. In this paper, we provide a method of adaptively partitioning persistence diagrams to improve these featurizations based on localized information in the diagrams. Additionally, we provide a framework to adaptively select parameters required for the template functions in order to best utilize the partitioning method. We present results for application to example data sets comparing classification results between template function featurizations with and without partitioning, in addition to other methods from the literature. 

   more » « less
2. Learning metrics for persistence-based summaries and applications for graph classification

   Zhao, Qi; Wang, Yusu (November 2019, 33rd Conf. Neural Information Processing Systems (NeuRIPS))

   Recently a new feature representation framework based on a topological tool called persistent homology (and its persistence diagram summary) has gained much momentum. A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools. In these approaches, the importance (weight) of different persistence features are usually pre-set. However often in practice, the choice of the weight-functions hould depend on the nature of the specific data at hand. It is thus highly desirable to learn a best weight-function (and thus metric for persistence diagrams) from labelled data. We study this problem and develop a new weighted kernel, called WKPI, for persistence summaries, as well as an optimization framework to learn the weight (and thus kernel). We apply the learned kernel to the challenging task of graph classification, and show that our WKPI-based classification framework obtains similar or (sometimes significantly) better results than the best results from a range of previous graph classification frameworks on benchmark datasets. 

   more » « less
3. Learning Hyperbolic Representations of Topological Features

   Panagiotis Kyriakis, Iordanis Fostiropoulos (May 2021, Proceedings of the International Conference on Learning Representations (ICLR))

   null (Ed.)

   Learning task-specific representations of persistence diagrams is an important problem in topological data analysis and machine learning. However, current state of the art methods are restricted in terms of their expressivity as they are focused on Euclidean representations. Persistence diagrams often contain features of infinite persistence (i.e., essential features) and Euclidean spaces shrink their importance relative to non-essential features because they cannot assign infinite distance to finite points. To deal with this issue, we propose a method to learn representations of persistence diagrams on hyperbolic spaces, more specifically on the Poincare ball. By representing features of infinite persistence infinitesimally close to the boundary of the ball, their distance to non-essential features approaches infinity, thereby their relative importance is preserved. This is achieved without utilizing extremely high values for the learnable parameters, thus the representation can be fed into downstream optimization methods and trained efficiently in an end-to-end fashion. We present experimental results on graph and image classification tasks and show that the performance of our method is on par with or exceeds the performance of other state of the art methods. 

   more » « less
4. A Comparative Study of Machine Learning Methods for Persistence Diagrams

   https://doi.org/10.3389/frai.2021.681174  

   Barnes, Danielle; Polanco, Luis; Perea, Jose A. (July 2021, Frontiers in Artificial Intelligence)

   Many and varied methods currently exist for featurization, which is the process of mapping persistence diagrams to Euclidean space, with the goal of maximally preserving structure. However, and to our knowledge, there are presently no methodical comparisons of existing approaches, nor a standardized collection of test data sets. This paper provides a comparative study of several such methods. In particular, we review, evaluate, and compare the stable multi-scale kernel, persistence landscapes, persistence images, the ring of algebraic functions, template functions, and adaptive template systems. Using these approaches for feature extraction, we apply and compare popular machine learning methods on five data sets: MNIST, Shape retrieval of non-rigid 3D Human Models (SHREC14), extracts from the Protein Classification Benchmark Collection (Protein), MPEG7 shape matching, and HAM10000 skin lesion data set. These data sets are commonly used in the above methods for featurization, and we use them to evaluate predictive utility in real-world applications. 

   more » « less
5. On the Estimation of Persistence Intensity Functions and Linear Representations of Persistence Diagrams

   Wu, Weichen; Kim, Jisu; Rinaldo, Alessandro (May 2024, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   Persistence diagrams are one of the most pop- ular types of data summaries used in Topological Data Analysis. The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which cap- tures both topological signal and noise. To that effect, Chazal and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it – called the persistence density function. We present a class of kernel- based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees. 

   more » « less