Related papers: Extracting Complex Topology from Multivariate Func…

First steps towards a mathematical theory of deep convolutional neural networks for feature extraction were made---for the continuous-time case---in Mallat, 2012, and Wiatowski and B\"olcskei, 2015. This paper considers the discrete case,…

Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of interest. We present a…

Representation learning on graphs is a fundamental problem that can be crucial in various tasks. Graph neural networks, the dominant approach for graph representation learning, are limited in their representation power. Therefore, it can be…

Many scientific areas are faced with the challenge of extracting information from large, complex, and highly structured data sets. A great deal of modern statistical work focuses on developing tools for handling such data. This paper…

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as…

Complex, high-dimensional data is used in a wide range of domains to explore problems and make decisions. Analysis of high-dimensional data, however, is vulnerable to the hidden influence of confounding variables, especially as users apply…

A proper form of data characterization can guide the process of learning-algorithm selection and model-performance estimation. The field of meta-learning has provided a rich body of work describing effective forms of data characterization…

Template 3D shapes are useful for many tasks in graphics and vision, including fitting observation data, analyzing shape collections, and transferring shape attributes. Because of the variety of geometry and topology of real-world shapes,…

To incorporate spatial (neighborhood) and bidirectional hierarchical relationships as well as features and priors of the samples into their classification, we formulated the classification problem on three variants of multiresolution…

Multimodal relation extraction is an essential task for knowledge graph construction. In this paper, we take an in-depth empirical analysis that indicates the inaccurate information in the visual scene graph leads to poor modal alignment…

Multimodal evidence is critical in computational pathology: gigapixel whole slide images capture tumor morphology, while patient-level clinical descriptors preserve complementary context for prognosis. Integrating such heterogeneous signals…

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and…

The relation between two Morse functions defined on a common domain can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the functions are aligned. Both the Jacobi set itself as…

Many problems in real-world applications involve predicting several random variables which are statistically related. Markov random fields (MRFs) are a great mathematical tool to encode such relationships. The goal of this paper is to…

Unsupervised feature learning often finds low-dimensional embeddings that capture the structure of complex data. For tasks for which prior expert topological knowledge is available, incorporating this into the learned representation may…

The starting point for much of multivariate analysis (MVA) is an $n\times p$ data matrix whose $n$ rows represent observations and whose $p$ columns represent variables. Some multivariate data sets, however, may be best conceptualized not…

In the realm of deep learning, spatial attention mechanisms have emerged as a vital method for enhancing the performance of convolutional neural networks. However, these mechanisms possess inherent limitations that cannot be overlooked.…

Knowledge graphs have recently become the state-of-the-art tool for representing the diverse and complex knowledge of the world. Examples include the proprietary knowledge graphs of companies such as Google, Facebook, IBM, or Microsoft, but…

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations,…