Discrete transforms of quantized persistence diagrams

• URL: http://arxiv.org/abs/2312.17093v3
• Date: Mon, 21 Oct 2024 10:50:05 GMT
• Title: Discrete transforms of quantized persistence diagrams
• Authors: Michael Etienne Van Huffel, Olympio Hacquard, Vadim Lebovici, Matteo Palo,
• Abstract summary: We introduce Qupid, a novel and simple method for vectorizing persistence diagrams. Key features are the choice of log-scaled grids that emphasize information contained near the diagonal in persistence diagrams. We conduct an in-depth experimental analysis of Qupid, showing that the simplicity of our method results in very low computational costs.
• Score: 0.5249805590164902
• License:
• Abstract: Topological data analysis leverages topological features to analyze datasets, with applications in diverse fields like medical sciences and biology. A key tool of this theory is the persistence diagram, which encodes topological information but poses challenges for integration into standard machine learning pipelines. We introduce Qupid (QUantized Persistence and Integral transforms of Diagrams), a novel and simple method for vectorizing persistence diagrams. First, Qupid uses a binning procedure to turn persistence diagrams into finite measures on a grid and then applies discrete transforms to these measures. Key features are the choice of log-scaled grids that emphasize information contained near the diagonal in persistence diagrams, combined with the use of discrete transforms to enhance and efficiently encode the obtained topological information. We conduct an in-depth experimental analysis of Qupid, showing that the simplicity of our method results in very low computational costs while preserving highly competitive performances compared to state-of-the-art methods across numerous classification tasks on both synthetic and real-world datasets. Finally, we provide experimental evidence that our method is robust to a decrease in the grid resolution used.

Related papers

• Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation [78.54656076915565]
  Topological correctness plays a critical role in many image segmentation tasks. Most networks are trained using pixel-wise loss functions, such as Dice, neglecting topological accuracy. We propose a novel, graph-based framework for topologically accurate image segmentation.
  arXiv  Detail & Related papers  (2024-11-05T16:20:14Z)
• Learning signals defined on graphs with optimal transport and Gaussian process regression [1.1062090350704616]
  In computational physics, machine learning has emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. We propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty and active learning.
  arXiv  Detail & Related papers  (2024-10-21T07:39:44Z)
• Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
  The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure. We introduce a new class of manifold, named soft manifold, that can solve this situation. Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
  arXiv  Detail & Related papers  (2023-11-29T12:48:33Z)
• Bures-Wasserstein Means of Graphs [60.42414991820453]
  We propose a novel framework for defining a graph mean via embeddings in the space of smooth graph signal distributions. By finding a mean in this embedding space, we can recover a mean graph that preserves structural information. We establish the existence and uniqueness of the novel graph mean, and provide an iterative algorithm for computing it.
  arXiv  Detail & Related papers  (2023-05-31T11:04:53Z)
• ShapeWordNet: An Interpretable Shapelet Neural Network for Physiological Signal Classification [16.82411861562806]
  We propose a more effective and interpretable scheme tailored for the physiological signal classification task. We exploit the time series shapelets to extract prominent local patterns and perform interpretable sequence discretization. We name our method as ShapeWordNet and conduct extensive experiments on three real-world datasets to investigate its effectiveness.
  arXiv  Detail & Related papers  (2023-02-10T02:30:31Z)
• Approximating Persistent Homology for Large Datasets [0.0]
  Persistent homology produces a statistical summary in the form of a persistence diagram. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. We show that the mean of the persistence diagrams of subsamples is a valid approximation of the true persistent homology of the larger dataset.
  arXiv  Detail & Related papers  (2022-04-19T23:07:27Z)
• A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces [7.717214217542406]
  We propose a persistence diagram hashing framework that learns a binary code representation of persistence diagrams. This framework is built upon a generative adversarial network (GAN) with a diagram distance loss function to steer the learning process. Our proposed method is directly applicable to various datasets without the need of retraining the model.
  arXiv  Detail & Related papers  (2021-05-25T20:44:28Z)
• Smart Vectorizations for Single and Multiparameter Persistence [8.504400925390296]
  We introduce two new topological summaries for single and multi-persistence persistence, namely, saw functions and multi-persistence grid functions. These new topological summaries can be regarded as the complexity measures of the evolving subspaces determined by the filtration. We derive theoretical guarantees on the stability of the new saw and multi-persistence grid functions and illustrate their applicability for graph classification tasks.
  arXiv  Detail & Related papers  (2021-04-10T15:09:31Z)
• Learning the Implicit Semantic Representation on Graph-Structured Data [57.670106959061634]
  Existing representation learning methods in graph convolutional networks are mainly designed by describing the neighborhood of each node as a perceptual whole. We propose a Semantic Graph Convolutional Networks (SGCN) that explores the implicit semantics by learning latent semantic-paths in graphs.
  arXiv  Detail & Related papers  (2021-01-16T16:18:43Z)
• A Fast and Robust Method for Global Topological Functional Optimization [70.11080854486953]
  We introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. This scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.
  arXiv  Detail & Related papers  (2020-09-17T18:46:16Z)
• A Trainable Optimal Transport Embedding for Feature Aggregation and its Relationship to Attention [96.77554122595578]
  We introduce a parametrized representation of fixed size, which embeds and then aggregates elements from a given input set according to the optimal transport plan between the set and a trainable reference. Our approach scales to large datasets and allows end-to-end training of the reference, while also providing a simple unsupervised learning mechanism with small computational cost.
  arXiv  Detail & Related papers  (2020-06-22T08:35:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.