Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimization via gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice. Focusing on the central case of topological optimization for point clouds, we propose in this work to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space, with quantifiable Lipschitz constants. In particular, we show that our approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on a subsample can be used to update the coordinates of the full input object, allowing us to perform topological optimization on point clouds at an unprecedented scale. Finally, we also showcase the relevance of our approach for black-box autoencoder (AE) regularization, where we aim at enforcing topological priors on the latent spaces associated to fixed, pre-trained, black-box AE models, and where we show that learning a diffeomorphic flow can be done once and then re-applied to new data in linear time (while vanilla topological optimization has to be re-run from scratch). Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, yielding better interpretability of the model.

Topological Data Analysis (TDA) provides a pipeline to extract quantitative and powerful topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to which extent a given object exhibits some topological properties. One can then use these losses to perform topological optimization via gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends (locally) on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice. In this work, focusing on the central case of topological optimization for point clouds, we propose to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space. In particular, this approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on the subsample can be used to update the coordinates of the full and possibly large input object. We then illustrate the usefulness of our approach on black-box autoencoder (AE) regularization, where we aim at applying some topological priors on the latent spaces associated to fixed, black-box AE models without modifying their (unknown) architectures and parameters. We empirically show that, while vanilla topological optimization has to be re-run every time that new data comes out of the black-box models, learning a diffeomorphic flow can be done once and then re-applied to new data in linear time. Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, allowing for better interpretability of the model.

Topological Data Analysis (TDA) provides a pipeline to extract quantitative and powerful topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to which extent a given object exhibits some topological properties. One can then use these losses to perform topological optimization via gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends (locally) on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice. In this work, focusing on the central case of topological optimization for point clouds, we propose to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space.

Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimization via gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice. Focusing on the central case of topological optimization for point clouds, we propose in this work to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space, with quantifiable Lipschitz constants. In particular, we show that our approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on a subsample can be used to update the coordinates of the full input object, allowing us to perform topological optimization on point clouds at an unprecedented scale. Finally, we also showcase the relevance of our approach for black-box autoencoder (AE) regularization, where we aim at enforcing topological priors on the latent spaces associated to fixed, pre-trained, black-box AE models, and where we show that learning a diffeomorphic flow can be done once and then re-applied to new data in linear time (while vanilla topological optimization has to be re-run from scratch). Moreover, reverting the flow allows us to generate data by sampling the topologicallyoptimized latent space directly, yielding better interpretability of the model.

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.

This research underscores the efficacy of Fourier topological optimization in refining MRI imagery, thereby bolstering the classification precision of Alzheimer's Disease through convolutional neural networks. Recognizing that MRI scans are indispensable for neurological assessments, but frequently grapple with issues like blurriness and contrast irregularities, the deployment of Fourier topological optimization offered enhanced delineation of brain structures, ameliorated noise, and superior contrast. The applied techniques prioritized boundary enhancement, contrast and brightness adjustments, and overall image lucidity. Employing CNN architectures VGG16, ResNet50, InceptionV3, and Xception, the post-optimization analysis revealed a marked elevation in performance. Conclusively, the amalgamation of Fourier topological optimization with CNNs delineates a promising trajectory for the nuanced classification of Alzheimer's Disease, portending a transformative impact on its diagnostic paradigms.

Unsupervised feature learning often finds low-dimensional embeddings that capture the structure of complex data. For tasks for which expert prior topological knowledge is available, incorporating this into the learned representation may lead to higher quality embeddings. For example, this may help one to embed the data into a given number of clusters, or to accommodate for noise that prevents one from deriving the distribution of the data over the model directly, which can then be learned more effectively. However, a general tool for integrating different prior topological knowledge into embeddings is lacking. Although differentiable topology layers have been recently developed that can (re)shape embeddings into prespecified topological models, they have two important limitations for representation learning, which we address in this paper. First, the currently suggested topological losses fail to represent simple models such as clusters and flares in a natural manner. Second, these losses neglect all original structural (such as neighborhood) information in the data that is useful for learning. We overcome these limitations by introducing a new set of topological losses, and proposing their usage as a way for topologically regularizing data embeddings to naturally represent a prespecified model. We include thorough experiments on synthetic and real data that highlight the usefulness and versatility of this approach, with applications ranging from modeling high-dimensional single cell data, to graph embedding.

Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, 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.