A Stable Multi-Scale Kernel for Topological Machine Learning

2014/12/21 by Reininghaus, Jan, Huber, Stefan, Bauer, Ulrich +1 · 10 citations
#Algebraic Topology (math.AT) #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML)

paper · doi:10.48550/arxiv.1412.6821

Abstract

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.

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• From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis
• Prototype Selection Using Topological Data Analysis
• Through the Grapevine: Vineyard Distance as a Measure of Topological Dissimilarity
• Connectome-Guided Automatic Learning Rates for Deep Networks
• TACO-Net: Topological Signatures Triumph in 3D Object Classification
• Wasserstein Stability of Persistence Landscapes and Barcodes
• Persistence Spheres: Bi-continuous Representations of Persistence Diagrams
• Extracting Interpretable Higher-Order Topological Features across Multiple Scales for Alzheimer's Disease Classification
• Data-Efficient Neural Training with Dynamic Connectomes
• TopoImages: Incorporating Local Topology Encoding into Deep Learning Models for Medical Image Classification
• Path representations in multiparameter persistent homology

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