Intrinsic persistent homology via density-based metric learning

We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov–Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain intrinsic persistence diagrams, which are less sensitive to the particular embedding of the manifold in the Euclidean space. We show that this approach is robust to outliers and deduce a method for pattern recognition in signals, with applications in real data.Fil: Borghini, Eugenio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Fernández, Ximena Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Groisman, Pablo Jose. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Mindlin, Bernardo Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentin

Approximating Persistent Homology in Euclidean Space Through Collapses

The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at smaller scales than what the application calls for. In this paper we propose two methods to approximate the \v{C}ech persistence module. Both are constructed on the level of spaces, i.e. as sequences of simplicial complexes induced by nerves. We also show how the bottleneck distance between such persistence modules can be understood by how tightly they are sandwiched on the level of spaces. In turn, this implies the correctness of our approximation methods. Finally, we implement our methods and apply them to some example point clouds in Euclidean space

Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study

Persistent homology is a common tool of topological data analysis, whose main descriptor, the persistence diagram, aims at computing and encoding the geometry and topology of given datasets. In this article, we present a novel application of persistent homology to characterize the spatial arrangement of immune and epithelial (tumor) cells within the breast cancer immune microenvironment. More specifically, quantitative and robust characterizations are built by computing persistence diagrams out of a staining technique (quantitative multiplex immunofluorescence) which allows us to obtain spatial coordinates and stain intensities on individual cells. The resulting persistence diagrams are evaluated as characteristic biomarkers of cancer subtype and prognostic biomarker of overall survival. For a cohort of approximately 700 breast cancer patients with median 8.5-year clinical follow-up, we show that these persistence diagrams outperform and complement the usual descriptors which capture spatial relationships with nearest neighbor analysis. This provides new insights and possibilities on the general problem of building (topology-based) biomarkers that are characteristic and predictive of cancer subtype, overall survival and response to therapy

The Density of Expected Persistence Diagrams and its Kernel Based Estimation

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 R^2 that can equivalently be seen as discrete measures in R^2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the Cech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R^2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams et al., 2017] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density

PLLay: Efficient Topological Layer based on Persistence Landscapes

29 pages, 7 figuresInternational audienceWe propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show differentiability with respect to layer inputs, for a general persistent homology with arbitrary filtration. Thus, our proposed layer can be placed anywhere in the network and feed critical information on the topological features of input data into subsequent layers to improve the learnability of the networks toward a given task. A task-optimal structure of PLLay is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide a novel adaptation for the DTM function-based filtration, and show that the proposed layer is robust against noise and outliers through a stability analysis. We demonstrate the effectiveness of our approach by classification experiments on various datasets

Persistent homology based characterization of the breast cancer immune microenvironment: a feasibility study

International audiencePersistent homology is a powerful tool in topological data analysis. The main output, persistence diagrams, encode the geometry and topology of given datasets. We present a novel application of persistent homology to characterize the biological environment surrounding breast cancers, known as the tumor microenvironment. Specifically, we will characterize the spatial arrangement of immune and malignant epithelial (tumor) cells within the breast cancer immune microenvironment. Quantitative and robust characterizations are built by computing persistence diagrams from quantitative multiplex immunofluorescence, which is a technology which allows us to obtain spatial coordinates and protein intensities on individual cells. The resulting persistence diagrams are evaluated as characteristic biomarkers predictive of cancer subtype and prognostic of overall survival. For a cohort of approximately 700 breast cancer patients with median 8.5-year clinical follow-up, we show that these persistence diagrams outperform and complement the usual descriptors which capture spatial relationships with nearest neighbor analysis. Our results thus suggest new methods which can be used to build topology-based biomarkers which are characteristic and predictive of cancer subtype and response to therapy as well as prognostic of overall survival

The search for the lost attractor

N-body systems characterized by inverse square attractive forces may display a self similar collapse known as the gravo-thermal catastrophe. In star clusters, collapse is halted by binary stars, and a large fraction of Milky Way clusters may have already reached this phase. It has been speculated -- with guidance from simulations -- that macroscopic variables such as central density and velocity dispersion are governed post-collapse by an effective, low-dimensional system of ODEs. It is still hard to distinguish chaotic, low dimensional motion, from high dimensional stochastic noise. Here we apply three machine learning tools to state-of-the-art dynamical simulations to constrain the post collapse dynamics: topological data analysis (TDA) on a lag embedding of the relevant time series, Sparse Identification of Nonlinear Dynamics (SINDY), and Tests of Accuracy with Random Points (TARP).Comment: Accepted by ML4PS workshop at NeurIPS 202