44 research outputs found
Compact exact Lagrangian intersections in cotangent bundles via sheaf quantization
We show that the cardinality of the transverse intersection of two compact
exact Lagrangian submanifolds in a cotangent bundle is bounded from below by
the dimension of the Hom space of sheaf quantizations of the Lagrangians in
Tamarkin's category. Our sheaf-theoretic method can also deal with clean and
degenerate Lagrangian intersections.Comment: 36 pages, 4 figures, with an appendix by Tomohiro Asan
Microlocal Lefschetz classes of graph trace kernels
In this paper, we define the notion of graph trace kernels as a
generalization of trace kernels. We associate a microlocal Lefschetz class with
a graph trace kernel and prove that this class is functorial with respect to
the composition of kernels. We apply graph trace kernels to the microlocal
Lefschetz fixed point formula for constructible sheaves.Comment: 18 pages, revised, to appear in Publ. RIM
A note on Hamiltonian stability for sheaves
We show a strong Hamiltonian stability result for a simpler and larger
distance on the Tamarkin category. We also give a stability result with support
conditions.Comment: 11 page
Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds
Machine learning for point clouds has been attracting much attention, with
many applications in various fields, such as shape recognition and material
science. To enhance the accuracy of such machine learning methods, it is known
to be effective to incorporate global topological features, which are typically
extracted by persistent homology. In the calculation of persistent homology for
a point cloud, we need to choose a filtration for the point clouds, an
increasing sequence of spaces. Because the performance of machine learning
methods combined with persistent homology is highly affected by the choice of a
filtration, we need to tune it depending on data and tasks. In this paper, we
propose a framework that learns a filtration adaptively with the use of neural
networks. In order to make the resulting persistent homology
isometry-invariant, we develop a neural network architecture with such
invariance. Additionally, we theoretically show a finite-dimensional
approximation result that justifies our architecture. Experimental results
demonstrated the efficacy of our framework in several classification tasks.Comment: 17 pages with 4 figure
PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures
Persistence diagrams, the most common descriptors of Topological Data
Analysis, encode topological properties of data and have already proved pivotal
in many different applications of data science. However, since the (metric)
space of persistence diagrams is not Hilbert, they end up being difficult
inputs for most Machine Learning techniques. To address this concern, several
vectorization methods have been put forward that embed persistence diagrams
into either finite-dimensional Euclidean space or (implicit) infinite
dimensional Hilbert space with kernels. In this work, we focus on persistence
diagrams built on top of graphs. Relying on extended persistence theory and the
so-called heat kernel signature, we show how graphs can be encoded by
(extended) persistence diagrams in a provably stable way. We then propose a
general and versatile framework for learning vectorizations of persistence
diagrams, which encompasses most of the vectorization techniques used in the
literature. We finally showcase the experimental strength of our setup by
achieving competitive scores on classification tasks on real-life graph
datasets