Towards structure-preserving quantum encodings

Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such encodings may vary substantially from one task to another, and there exist only a few cases where structure has provided insight into their design and implementation, such as symmetry in geometric quantum learning. Here, we propose the perspective that category theory offers a natural mathematical framework for analyzing encodings that respect structure inherent in datasets and learning tasks. We illustrate this with pedagogical examples, which include geometric quantum machine learning, quantum metric learning, topological data analysis, and more. Moreover, our perspective provides a language in which to ask meaningful and mathematically precise questions for the design of quantum encodings and circuits for quantum machine learning tasks.

A Mapper Algorithm with implicit intervals and its optimization

The Mapper algorithm is an essential tool for visualizing complex, high dimensional data in topology data analysis (TDA) and has been widely used in biomedical research. It outputs a combinatorial graph whose structure implies the shape of the data. However,the need for manual parameter tuning and fixed intervals, along with fixed overlapping ratios may impede the performance of the standard Mapper algorithm. Variants of the standard Mapper algorithms have been developed to address these limitations, yet most of them still require manual tuning of parameters. Additionally, many of these variants, including the standard version found in the literature, were built within a deterministic framework and overlooked the uncertainty inherent in the data. To relax these limitations, in this work, we introduce a novel framework that implicitly represents intervals through a hidden assignment matrix, enabling automatic parameter optimization via stochastic gradient descent. In this work, we develop a soft Mapper framework based on a Gaussian mixture model(GMM) for flexible and implicit interval construction. We further illustrate the robustness of the soft Mapper algorithm by introducing the Mapper graph mode as a point estimation for the output graph. Moreover, a stochastic gradient descent algorithm with a specific topological loss function is proposed for optimizing parameters in the model. Both simulation and application studies demonstrate its effectiveness in capturing the underlying topological structures. In addition, the application to an RNA expression dataset obtained from the Mount Sinai/JJ Peters VA Medical Center Brain Bank (MSBB) successfully identifies a distinct subgroup of Alzheimer's Disease.

Prompt-based Unifying Inference Attack on Graph Neural Networks

Graph neural networks (GNNs) provide important prospective insights in applications such as social behavior analysis and financial risk analysis based on their powerful learning capabilities on graph data. Nevertheless, GNNs' predictive performance relies on the quality of task-specific node labels, so it is common practice to improve the model's generalization ability in the downstream execution of decision-making tasks through pre-training. Graph prompting is a prudent choice but risky without taking measures to prevent data leakage. In other words, in high-risk decision scenarios, prompt learning can infer private information by accessing model parameters trained on private data (publishing model parameters in pre-training, i.e., without directly leaking the raw data, is a tacitly accepted trend). However, myriad graph inference attacks necessitate tailored module design and processing to enhance inference capabilities due to variations in supervision signals. In this paper, we propose a novel Prompt-based unifying Inference Attack framework on GNNs, named ProIA. Specifically, ProIA retains the crucial topological information of the graph during pre-training, enhancing the background knowledge of the inference attack model. It then utilizes a unified prompt and introduces additional disentanglement factors in downstream attacks to adapt to task-relevant knowledge. Finally, extensive experiments show that ProIA enhances attack capabilities and demonstrates remarkable adaptability to various inference attacks.

Can LLMs Convert Graphs to Text-Attributed Graphs?

Graphs are ubiquitous data structures found in numerous real-world applications, such as drug discovery, recommender systems, and social network analysis. Graph neural networks (GNNs) have become a popular tool to learn node embeddings through message passing on these structures. However, a significant challenge arises when applying GNNs to multiple graphs with different feature spaces, as existing GNN architectures are not designed for cross-graph feature alignment. To address this, recent approaches introduce text-attributed graphs, where each node is associated with a textual description, enabling the use of a shared textual encoder to project nodes from different graphs into a unified feature space. While promising, this method relies heavily on the availability of text-attributed data, which can be difficult to obtain in practice. To bridge this gap, we propose a novel method named Topology-Aware Node description Synthesis (TANS), which leverages large language models (LLMs) to automatically convert existing graphs into text-attributed graphs. The key idea is to integrate topological information with each node's properties, enhancing the LLMs' ability to explain how graph topology influences node semantics. We evaluate our TANS on text-rich, text-limited, and text-free graphs, demonstrating that it enables a single GNN to operate across diverse graphs. Notably, on text-free graphs, our method significantly outperforms existing approaches that manually design node features, showcasing the potential of LLMs for preprocessing graph-structured data, even in the absence of textual information. The code and data are available at https://github.com/Zehong-Wang/TANS.

When Dimensionality Reduction Meets Graph (Drawing) Theory: Introducing a Common Framework, Challenges and Opportunities

In the vast landscape of visualization research, Dimensionality Reduction (DR) and graph analysis are two popular subfields, often essential to most visual data analytics setups. DR aims to create representations to support neighborhood and similarity analysis on complex, large datasets. Graph analysis focuses on identifying the salient topological properties and key actors within networked data, with specialized research on investigating how such features could be presented to the user to ease the comprehension of the underlying structure. Although these two disciplines are typically regarded as disjoint subfields, we argue that both fields share strong similarities and synergies that can potentially benefit both. Therefore, this paper discusses and introduces a unifying framework to help bridge the gap between DR and graph (drawing) theory. Our goal is to use the strongly math-grounded graph theory to improve the overall process of creating DR visual representations. We propose how to break the DR process into well-defined stages, discussing how to match some of the DR state-of-the-art techniques to this framework and presenting ideas on how graph drawing, topology features, and some popular algorithms and strategies used in graph analysis can be employed to improve DR topology extraction, embedding generation, and result validation. We also discuss the challenges and identify opportunities for implementing and using our framework, opening directions for future visualization research.

Dynamic Entity-Masked Graph Diffusion Model for histopathological image Representation Learning

Significant disparities between the features of natural images and those inherent to histopathological images make it challenging to directly apply and transfer pre-trained models from natural images to histopathology tasks. Moreover, the frequent lack of annotations in histopathology patch images has driven researchers to explore self-supervised learning methods like mask reconstruction for learning representations from large amounts of unlabeled data. Crucially, previous mask-based efforts in self-supervised learning have often overlooked the spatial interactions among entities, which are essential for constructing accurate representations of pathological entities. To address these challenges, constructing graphs of entities is a promising approach. In addition, the diffusion reconstruction strategy has recently shown superior performance through its random intensity noise addition technique to enhance the robust learned representation. Therefore, we introduce H-MGDM, a novel self-supervised Histopathology image representation learning method through the Dynamic Entity-Masked Graph Diffusion Model. Specifically, we propose to use complementary subgraphs as latent diffusion conditions and self-supervised targets respectively during pre-training. We note that the graph can embed entities' topological relationships and enhance representation. Dynamic conditions and targets can improve pathological fine reconstruction. Our model has conducted pretraining experiments on three large histopathological datasets. The advanced predictive performance and interpretability of H-MGDM are clearly evaluated on comprehensive downstream tasks such as classification and survival analysis on six datasets. Our code will be publicly available at https://github.com/centurion-crawler/H-MGDM.

Big data approach to Kazhdan-Lusztig polynomials

We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of up to 11 strands.

Preserving Information: How does Topological Data Analysis improve Neural Network performance?

Artificial Neural Networks (ANNs) require significant amounts of data and computational resources to achieve high effectiveness in performing the tasks for which they are trained. To reduce resource demands, various techniques, such as Neuron Pruning, are applied. Due to the complex structure of ANNs, interpreting the behavior of hidden layers and the features they recognize in the data is challenging. A lack of comprehensive understanding of which information is utilized during inference can lead to inefficient use of available data, thereby lowering the overall performance of the models. In this paper, we introduce a method for integrating Topological Data Analysis (TDA) with Convolutional Neural Networks (CNN) in the context of image recognition. This method significantly enhances the performance of neural networks by leveraging a broader range of information present in the data, enabling the model to make more informed and accurate predictions. Our approach, further referred to as Vector Stitching, involves combining raw image data with additional topological information derived through TDA methods. This approach enables the neural network to train on an enriched dataset, incorporating topological features that might otherwise remain unexploited or not captured by the network's inherent mechanisms. The results of our experiments highlight the potential of incorporating results of additional data analysis into the network's inference process, resulting in enhanced performance in pattern recognition tasks in digital images, particularly when using limited datasets. This work contributes to the development of methods for integrating TDA with deep learning and explores how concepts from Information Theory can explain the performance of such hybrid methods in practical implementation environments.

TDAvec: Computing Vector Summaries of Persistence Diagrams for Topological Data Analysis in R and Python

Persistent homology is a widely-used tool in topological data analysis (TDA) for understanding the underlying shape of complex data. By constructing a filtration of simplicial complexes from data points, it captures topological features such as connected components, loops, and voids across multiple scales. These features are encoded in persistence diagrams (PDs), which provide a concise summary of the data's topological structure. However, the non-Hilbert nature of the space of PDs poses challenges for their direct use in machine learning applications. To address this, kernel methods and vectorization techniques have been developed to transform PDs into machine-learning-compatible formats. In this paper, we introduce a new software package designed to streamline the vectorization of PDs, offering an intuitive workflow and advanced functionalities. We demonstrate the necessity of the package through practical examples and provide a detailed discussion on its contributions to applied TDA. Definitions of all vectorization summaries used in the package are included in the appendix.

Differentiable Topology Estimating from Curvatures for 3D Shapes

In the field of data-driven 3D shape analysis and generation, the estimation of global topological features from localized representations such as point clouds, voxels, and neural implicit fields is a longstanding challenge. This paper introduces a novel, differentiable algorithm tailored to accurately estimate the global topology of 3D shapes, overcoming the limitations of traditional methods rooted in mesh reconstruction and topological data analysis. The proposed method ensures high accuracy, efficiency, and instant computation with GPU compatibility. It begins with an efficient calculation of the self-adjoint Weingarten map for point clouds and its adaptations for other modalities. The curvatures are then extracted, and their integration over tangent differentiable Voronoi elements is utilized to estimate key topological invariants, including the Euler number and Genus. Additionally, an auto-optimization mechanism is implemented to refine the local moving frames and area elements based on the integrity of topological invariants. Experimental results demonstrate the method's superior performance across various datasets. The robustness and differentiability of the algorithm ensure its seamless integration into deep learning frameworks, offering vast potential for downstream tasks in 3D shape analysis.