Topologically Regularized Data Embeddings

• URL: http://arxiv.org/abs/2301.03338v2
• Date: Tue, 7 Nov 2023 17:06:22 GMT
• Title: Topologically Regularized Data Embeddings
• Authors: Edith Heiter, Robin Vandaele, Tijl De Bie, Yvan Saeys, Jefrey Lijffijt
• Abstract summary: We introduce a generic approach based on algebraic topology to incorporate topological prior knowledge into low-dimensional embeddings. We show that jointly optimizing an embedding loss with such a topological loss function as a regularizer yields embeddings that reflect not only local proximities but also the desired topological structure. We empirically evaluate the proposed approach on computational efficiency, robustness, and versatility in combination with linear and non-linear dimensionality reduction and graph embedding methods.
• Score: 15.001598256750619
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Unsupervised representation learning methods are widely used for gaining insight into high-dimensional, unstructured, or structured data. In some cases, users may have prior topological knowledge about the data, such as a known cluster structure or the fact that the data is known to lie along a tree- or graph-structured topology. However, generic methods to ensure such structure is salient in the low-dimensional representations are lacking. This negatively impacts the interpretability of low-dimensional embeddings, and plausibly downstream learning tasks. To address this issue, we introduce topological regularization: a generic approach based on algebraic topology to incorporate topological prior knowledge into low-dimensional embeddings. We introduce a class of topological loss functions, and show that jointly optimizing an embedding loss with such a topological loss function as a regularizer yields embeddings that reflect not only local proximities but also the desired topological structure. We include a self-contained overview of the required foundational concepts in algebraic topology, and provide intuitive guidance on how to design topological loss functions for a variety of shapes, such as clusters, cycles, and bifurcations. We empirically evaluate the proposed approach on computational efficiency, robustness, and versatility in combination with linear and non-linear dimensionality reduction and graph embedding methods.

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)
• Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms [15.473123662393169]
  Deep neural networks (DNNs) show remarkable generalization properties. The source of these capabilities remains elusive, defying the established statistical learning theory. Recent studies have revealed that properties of training trajectories can be indicative of generalization.
  arXiv  Detail & Related papers  (2024-07-11T17:56:03Z)
• Towards a mathematical understanding of learning from few examples with nonlinear feature maps [68.8204255655161]
  We consider the problem of data classification where the training set consists of just a few data points. We reveal key relationships between the geometry of an AI model's feature space, the structure of the underlying data distributions, and the model's generalisation capabilities.
  arXiv  Detail & Related papers  (2022-11-07T14:52:58Z)
• Supervised Dimensionality Reduction and Classification with Convolutional Autoencoders [1.1164202369517053]
  A Convolutional Autoencoder is combined to simultaneously produce supervised dimensionality reduction and predictions. The resulting Latent Space can be utilized to improve traditional, interpretable classification algorithms. The proposed methodology introduces advanced explainability regarding, not only the data structure through the produced latent space, but also about the classification behaviour.
  arXiv  Detail & Related papers  (2022-08-25T15:18:33Z)
• Rethinking Persistent Homology for Visual Recognition [27.625893409863295]
  This paper performs a detailed analysis of the effectiveness of topological properties for image classification in various training scenarios. We identify the scenarios that benefit the most from topological features, e.g., training simple networks on small datasets.
  arXiv  Detail & Related papers  (2022-07-09T08:01:11Z)
• Topologically Regularized Data Embeddings [22.222311627054875]
  We introduce a new set of topological losses, and propose their usage as a way for topologically regularizing data embeddings to naturally represent a prespecified model. We include experiments on synthetic and real data that highlight the usefulness and versatility of this approach.
  arXiv  Detail & Related papers  (2021-10-18T11:25:47Z)
• Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms [71.62575565990502]
  We prove that the generalization error of an optimization algorithm can be bounded on the complexity' of the fractal structure that underlies its generalization measure. We further specialize our results to specific problems (e.g., linear/logistic regression, one hidden/layered neural networks) and algorithms.
  arXiv  Detail & Related papers  (2021-06-09T08:05:36Z)
• Joint Network Topology Inference via Structured Fusion Regularization [70.30364652829164]
  Joint network topology inference represents a canonical problem of learning multiple graph Laplacian matrices from heterogeneous graph signals. We propose a general graph estimator based on a novel structured fusion regularization. We show that the proposed graph estimator enjoys both high computational efficiency and rigorous theoretical guarantee.
  arXiv  Detail & Related papers  (2021-03-05T04:42:32Z)
• 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 Topological Framework for Deep Learning [0.7310043452300736]
  We show that the classification problem in machine learning is always solvable under very mild conditions. In particular, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task.
  arXiv  Detail & Related papers  (2020-08-31T15:56:42Z)
• Self-Learning with Rectification Strategy for Human Parsing [73.06197841003048]
  We propose a trainable graph reasoning method to correct two typical errors in the pseudo-labels. The reconstructed features have a stronger ability to represent the topology structure of the human body. Our method outperforms other state-of-the-art methods in supervised human parsing tasks.
  arXiv  Detail & Related papers  (2020-04-17T03:51:30Z)

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.