Related papers: Topological Data Analysis of Decision Boundaries w…
Largest theoretical contribution to Neural Networks comes from VC Dimension which characterizes the sample complexity of classification model in a probabilistic view and are widely used to study the generalization error. So far in the…
The rapid progress of Artificial Intelligence research came with the development of increasingly complex deep learning models, leading to growing challenges in terms of computational complexity, energy efficiency and interpretability. In…
Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can…
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…
While deep learning models and techniques have achieved great empirical success, our understanding of the source of success in many aspects remains very limited. In an attempt to bridge the gap, we investigate the decision boundary of a…
In classical statistical learning theory, one of the most well studied problems is that of binary classification. The information-theoretic sample complexity of this task is tightly characterized by the Vapnik-Chervonenkis (VC) dimension. A…
Actin cytoskeleton networks generate local topological signatures due to the natural variations in the number, size, and shape of holes of the networks. Persistent homology is a method that explores these topological properties of data and…
This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level-set persistence is presented using sheaves and cosheaves.
Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description…
We show that unlike machine learning classifiers, there are no complex boundary structures in the decision boundaries for well-trained deep models. However, we found that the complicated structures do appear in training but they vanish…
This paper considers a finite sample perspective on the problem of identifying an LTI system from a finite set of possible systems using trajectory data. To this end, we use the maximum likelihood estimator to identify the true system and…
Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML)…
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to…
In this work, we present a deep learning framework for multi-class breast cancer image classification as our submission to the International Conference on Image Analysis and Recognition (ICIAR) 2018 Grand Challenge on BreAst Cancer…
Persistent Homology is a powerful tool in Topological Data Analysis (TDA) to capture topological properties of data succinctly at different spatial resolutions. For graphical data, shape, and structure of the neighborhood of individual data…
Persistent homology has been widely used to study the topology of point clouds in $\mathbb{R}^n$. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this…
Deep neural networks proved to be a very useful and powerful tool with many practical applications. They especially excel at learning from large data sets with labeled samples. However, in order to achieve good learning results, the network…
In this document, we propose a bridge between the graphs and the geometric realizations of their Vietoris Rips complexes, i.e. Graphs, with their canonical \v{C}ech closure structure, have the same homotopy type that the realization of…
The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most…
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower…