Differentiable Mapper For Topological Optimization Of Data Representation

• URL: http://arxiv.org/abs/2402.12854v1
• Date: Tue, 20 Feb 2024 09:33:22 GMT
• Title: Differentiable Mapper For Topological Optimization Of Data Representation
• Authors: Ziyad Oulhaj, Mathieu Carri\`ere and Bertrand Michel
• Abstract summary: We build on a recently proposed framework incorporating topology to provide the first filter optimization scheme for Mapper graphs. We demonstrate the usefulness of our approach by optimizing Mapper graph representations on several datasets.
• Score: 33.33724208084121
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Unsupervised data representation and visualization using tools from topology is an active and growing field of Topological Data Analysis (TDA) and data science. Its most prominent line of work is based on the so-called Mapper graph, which is a combinatorial graph whose topological structures (connected components, branches, loops) are in correspondence with those of the data itself. While highly generic and applicable, its use has been hampered so far by the manual tuning of its many parameters-among these, a crucial one is the so-called filter: it is a continuous function whose variations on the data set are the main ingredient for both building the Mapper representation and assessing the presence and sizes of its topological structures. However, while a few parameter tuning methods have already been investigated for the other Mapper parameters (i.e., resolution, gain, clustering), there is currently no method for tuning the filter itself. In this work, we build on a recently proposed optimization framework incorporating topology to provide the first filter optimization scheme for Mapper graphs. In order to achieve this, we propose a relaxed and more general version of the Mapper graph, whose convergence properties are investigated. Finally, we demonstrate the usefulness of our approach by optimizing Mapper graph representations on several datasets, and showcasing the superiority of the optimized representation over arbitrary ones.

Related papers

• A distribution-guided Mapper algorithm [0.3683202928838613]
  We introduce a distribution guided Mapper algorithm named D-Mapper. Our proposed algorithm is a probabilistic model-based approach, which could serve as an alternative to non-prababilistic ones. Our numerical experiments indicate that the D-Mapper outperforms the classical Mapper algorithm in various scenarios.
  arXiv  Detail & Related papers  (2024-01-19T17:07:05Z)
• $G$-Mapper: Learning a Cover in the Mapper Construction [0.7852714805965528]
  The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. We present an algorithm that optimize the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on $G$-means clustering which searches for the optimal number of clusters in $k$-means by iteratively applying the Anderson-Darling test.
  arXiv  Detail & Related papers  (2023-09-12T22:51:16Z)
• Graph Signal Sampling for Inductive One-Bit Matrix Completion: a Closed-form Solution [112.3443939502313]
  We propose a unified graph signal sampling framework which enjoys the benefits of graph signal analysis and processing. The key idea is to transform each user's ratings on the items to a function (signal) on the vertices of an item-item graph. For the online setting, we develop a Bayesian extension, i.e., BGS-IMC which considers continuous random Gaussian noise in the graph Fourier domain.
  arXiv  Detail & Related papers  (2023-02-08T08:17:43Z)
• GrannGAN: Graph annotation generative adversarial networks [72.66289932625742]
  We consider the problem of modelling high-dimensional distributions and generating new examples of data with complex relational feature structure coherent with a graph skeleton. The model we propose tackles the problem of generating the data features constrained by the specific graph structure of each data point by splitting the task into two phases. In the first it models the distribution of features associated with the nodes of the given graph, in the second it complements the edge features conditionally on the node features.
  arXiv  Detail & Related papers  (2022-12-01T11:49:07Z)
• Learning Optimal Graph Filters for Clustering of Attributed Graphs [20.810096547938166]
  Many real-world systems can be represented as graphs where the different entities in the system are presented by nodes and their interactions by edges. An important task in studying large datasets with graphical structure is graph clustering. We introduce a graph signal processing based approach, where we learn the parameters of Finite Impulse Response (FIR) and Autoregressive Moving Average (ARMA) graph filters optimized for clustering.
  arXiv  Detail & Related papers  (2022-11-09T01:49:23Z)
• SHGNN: Structure-Aware Heterogeneous Graph Neural Network [77.78459918119536]
  This paper proposes a novel Structure-Aware Heterogeneous Graph Neural Network (SHGNN) to address the above limitations. We first utilize a feature propagation module to capture the local structure information of intermediate nodes in the meta-path. Next, we use a tree-attention aggregator to incorporate the graph structure information into the aggregation module on the meta-path. Finally, we leverage a meta-path aggregator to fuse the information aggregated from different meta-paths.
  arXiv  Detail & Related papers  (2021-12-12T14:18:18Z)
• Auto-weighted Multi-view Feature Selection with Graph Optimization [90.26124046530319]
  We propose a novel unsupervised multi-view feature selection model based on graph learning. The contributions are threefold: (1) during the feature selection procedure, the consensus similarity graph shared by different views is learned. Experiments on various datasets demonstrate the superiority of the proposed method compared with the state-of-the-art methods.
  arXiv  Detail & Related papers  (2021-04-11T03:25:25Z)
• Building powerful and equivariant graph neural networks with structural message-passing [74.93169425144755]
  We propose a powerful and equivariant message-passing framework based on two ideas. First, we propagate a one-hot encoding of the nodes, in addition to the features, in order to learn a local context matrix around each node. Second, we propose methods for the parametrization of the message and update functions that ensure permutation equivariance.
  arXiv  Detail & Related papers  (2020-06-26T17:15:16Z)
• Graph Pooling with Node Proximity for Hierarchical Representation Learning [80.62181998314547]
  We propose a novel graph pooling strategy that leverages node proximity to improve the hierarchical representation learning of graph data with their multi-hop topology. Results show that the proposed graph pooling strategy is able to achieve state-of-the-art performance on a collection of public graph classification benchmark datasets.
  arXiv  Detail & Related papers  (2020-06-19T13:09:44Z)
• ShapeVis: High-dimensional Data Visualization at Scale [10.007129417823858]
  We present ShapeVis, a scalable visualization technique for point cloud data inspired from topological data analysis. Our method captures the underlying geometric and topological structure of the data in a compressed graphical representation.
  arXiv  Detail & Related papers  (2020-01-15T07:59:13Z)
• Homology-Preserving Multi-Scale Graph Skeletonization Using Mapper on Graphs [5.86893539706548]
  We propose to apply the mapper construction -- a popular tool in topological data analysis -- to graph visualization. We develop a variation of the mapper construction targeting weighted, undirected graphs, called mog, which generates homology-preserving skeletons of graphs. We provide a software tool that enables interactive explorations of such skeletons and demonstrate the effectiveness of our method for synthetic and real-world data.
  arXiv  Detail & Related papers  (2018-04-03T19:18:36Z)

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.