Traditional machine learning often assumes that the observed data of interest are supported on a linear vector space Topological deep learning (TDL) is a rapidly and can be described by a set of feature vectors. However, evolving field that uses topological features to understand there is growing awareness that, in many cases, this viewpoint and design deep learning models. This is insufficient to describe several data within the real paper posits that TDL may complement graph representation world. For example, molecules may be described more appropriately learning and geometric deep learning by graphs than feature vectors. Other examples by incorporating topological concepts, and can include three-dimensional objects represented by meshes, thus provide a natural choice for various machine as encountered in computer graphics and geometry processing, learning settings. To this end, this paper discusses or data supported on top of a complex social network open problems in TDL, ranging from practical of interrelated actors. Hence, there has been an increased benefits to theoretical foundations. For each problem, interest in importing concepts from geometry and topology it outlines potential solutions and future research into the usual machine learning pipelines to gain further opportunities.

Institutional bias can impact patient outcomes, educational attainment, and legal system navigation. Written records often reflect bias, and once bias is identified; it is possible to refer individuals for training to reduce bias. Many machine learning tools exist to explore text data and create predictive models that can search written records to identify real-time bias. However, few previous studies investigate large language model embeddings and geometric models of biased text data to understand geometry's impact on bias modeling accuracy. To overcome this issue, this study utilizes the RedditBias database to analyze textual biases. Four transformer models, including BERT and RoBERTa variants, were explored. Post-embedding, t-SNE allowed two-dimensional visualization of data. KNN classifiers differentiated bias types, with lower k-values proving more effective. Findings suggest BERT, particularly mini BERT, excels in bias classification, while multilingual models lag. The recommendation emphasizes refining monolingual models and exploring domain-specific biases.

Learning disentangled representations is regarded as a fundamental task for improving the generalization, robustness, and interpretability of generative models. However, measuring disentanglement has been challenging and inconsistent, often dependent on an ad-hoc external model or specific to a certain dataset. To address this, we present a method for quantifying disentanglement that only uses the generative model, by measuring the topological similarity of conditional submanifolds in the learned representation. To illustrate the effectiveness and applicability of our method, we empirically evaluate several state-of-the-art models across multiple datasets. We find that our method ranks models similarly to existing methods. Figure 1: Factors in the dSprites dataset displaying topological similarity and semantic correspondence to respective latent dimensions in a disentangled generative model, as shown through Wasserstein RLT distributions of homology and latent interpolations along respective dimensions. Learning disentangled representations is important for a variety of tasks, including adversarial robustness, generalization to novel tasks, and interpretability (Stutz et al., 2019; Alemi et al., 2017; Ridgeway, 2016; Bengio et al., 2013). Recently, deep generative models have shown marked improvement in disentanglement across an increasing number of datasets and a variety of training objectives (Chen et al., 2016; Lin et al., 2020; Higgins et al., 2017; Kim and Mnih, 2018; Chen et al., 2018b; Burgess et al., 2018; Karras et al., 2019). Nevertheless, quantifying the extent of this disentanglement has remained challenging and inconsistent.

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and distance-bases filtrations. We present three novel applications: the topological layer can (i) serve as a regularizer directly on data or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained with persistence features. The code is publicly available and we hope its availability will facilitate the use of persistent homology in deep learning and other gradient based applications.

Deep neural networks [10] are a powerful and fascinating methodology for solving problems with large and complex data sets. They use directed graphs as a template for very large computations, and have demonstrated a great deal of success in the study of various kinds of data, including images, text, time series, and many others. One issue that restricts their applicability, however, is the fact that it is not understood in any kind of detail how they work. A related problem is that there is often a certain kind of overfitting to particular data sets, which results in the possibility of so-called adversarial behavior, where they can be made to fail by making very small changes to image data that is almost imperceptible to a human. For these reasons, it is very desirable to develop methods for gaining understanding of the internal states of the neural networks.

This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods that, based on the dissimilarity structure, output hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter. Our construction of hierarchical clustering methods is built around the concept of admissible methods, which are those that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less. Two particular methods, termed reciprocal and nonreciprocal clustering, are shown to provide upper and lower bounds in the space of admissible methods. Furthermore, alternative clustering methodologies and axioms are considered. In particular, modifying the axiom of value such that clustering in two-node networks occurs at the minimum of the two dissimilarities entails the existence of a unique admissible clustering method.

This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation. We leverage the fact that, for asymmetric networks, every admissible method must be contained between reciprocal and nonreciprocal clustering, and describe three families of intermediate methods. Grafting methods exchange branches between dendrograms generated by different admissible methods. The convex combination family combines admissible methods through a convex operation in the space of dendrograms, and thirdly, the semi-reciprocal family clusters nodes that are related by strong cyclic influences in the network. Algorithms for the computation of hierarchical clusters generated by reciprocal and nonreciprocal clustering as well as the grafting, convex combination, and semi-reciprocal families are derived using matrix operations in a dioid algebra. Finally, the introduced clustering methods and algorithms are exemplified through their application to a network describing the interrelation between sectors of the United States (U.S.) economy.

This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter, induced by the given dissimilarity structures. Our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less. Several admissible methods are constructed and two particular methods, termed reciprocal and nonreciprocal clustering, are shown to provide upper and lower bounds in the space of admissible methods. Alternative clustering methodologies and axioms are further considered. Allowing the outcome of hierarchical clustering to be asymmetric, so that it matches the asymmetry of the original data, leads to the inception of quasi-clustering methods. The existence of a unique quasi-clustering method is shown. Allowing clustering in a two-node network to proceed at the minimum of the two dissimilarities generates an alternative axiomatic construction. There is a unique clustering method in this case too. The paper also develops algorithms for the computation of hierarchical clusters using matrix powers on a min-max dioid algebra and studies the stability of the methods proposed. We proved that most of the methods introduced in this paper are such that similar networks yield similar hierarchical clustering results. Algorithms are exemplified through their application to networks describing internal migration within states of the United States (U.S.) and the interrelation between sectors of the U.S. economy.

This paper introduces hierarchical quasi-clustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data. We show that this output structure is equivalent to a finite quasi-ultrametric space and study admissibility with respect to two desirable properties. We prove that a modified version of single linkage is the only admissible quasi-clustering method. Moreover, we show stability of the proposed method and we establish invariance properties fulfilled by it. Algorithms are further developed and the value of quasi-clustering analysis is illustrated with a study of internal migration within United States.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.