Convolutional neural networks (CNNs) are inherently subject to invariable filters that can only aggregate local inputs with the same topological structures. It causes that CNNs are allowed to manage data with Euclidean or grid-like structures (e.g., images), not ones with non-Euclidean or graph structures (e.g., traffic networks). To broaden the reach of CNNs, we develop structure-aware convolution to eliminate the invariance, yielding a unified mechanism of dealing with both Euclidean and non-Euclidean structured data. Technically, filters in the structure-aware convolution are generalized to univariate functions, which are capable of aggregating local inputs with diverse topological structures. Since infinite parameters are required to determine a univariate function, we parameterize these filters with numbered learnable parameters in the context of the function approximation theory. By replacing the classical convolution in CNNs with the structure-aware convolution, Structure-Aware Convolutional Neural Networks (SACNNs) are readily established. Extensive experiments on eleven datasets strongly evidence that SACNNs outperform current models on various machine learning tasks, including image classification and clustering, text categorization, skeleton-based action recognition, molecular activity detection, and taxi flow prediction.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Learning the flows of hybrid systems that have both continuous and discrete time dynamics is challenging. The existing method learns the dynamics in each discrete mode, which suffers from the combination of mode switching and discontinuities in the flows. In this work, we propose CHyLL (Continuous Hybrid System Learning in Latent Space), which learns a continuous neural representation of a hybrid system without trajectory segmentation, event functions, or mode switching. The key insight of CHyLL is that the reset map glues the state space at the guard surface, reformulating the state space as a piecewise smooth quotient manifold where the flow becomes spatially continuous. Building upon these insights and the embedding theorems grounded in differential topology, CHyLL concurrently learns a singularity-free neural embedding in a higher-dimensional space and the continuous flow in it. We showcase that CHyLL can accurately predict the flow of hybrid systems with superior accuracy and identify the topological invariants of the hybrid systems. Finally, we apply CHyLL to the stochastic optimal control problem.

Deep learning models have achieved remarkable success across various domains, yet their learned representations and decision-making processes remain largely opaque and hard to interpret. This work introduces HOLE (Homological Observation of Latent Embeddings), a method for analyzing and interpreting deep neural networks through persistent homology. HOLE extracts topological features from neural activations and presents them using a suite of visualization techniques, including Sankey diagrams, heatmaps, dendrograms, and blob graphs. These tools facilitate the examination of representation structure and quality across layers. We evaluate HOLE on standard datasets using a range of discriminative models, focusing on representation quality, interpretability across layers, and robustness to input perturbations and model compression. The results indicate that topological analysis reveals patterns associated with class separation, feature disentanglement, and model robustness, providing a complementary perspective for understanding and improving deep learning systems.

Persistence diagrams serve as a core tool in topological data analysis, playing a crucial role in pathological monitoring, drug discovery, and materials design. However, existing quantum topological algorithms, such as the LGZ algorithm, can only efficiently compute summary statistics like Betti numbers, failing to provide persistence diagram information that tracks the lifecycle of individual topological features, severely limiting their practical value. This paper proposes a novel quantum-classical hybrid algorithm that achieves, for the first time, the leap from "quantum computation of Betti numbers" to "quantum acquisition of practical persistence diagrams." The algorithm leverages the LGZ quantum algorithm as an efficient feature extractor, mining the harmonic form eigenvectors of the combinatorial Laplacian as well as Betti numbers, constructing specialized topological kernel functions to train a quantum support vector machine (QSVM), and learning the mapping from quantum topological features to persistence diagrams. The core contributions of this algorithm are: (1) elevating quantum topological computation from statistical summaries to pattern recognition, greatly expanding its application value; (2) obtaining more practical topological information in the form of persistence diagrams for real-world applications while maintaining the exponential speedup advantage of quantum computation; (3) proposing a novel hybrid paradigm of "classical precision guiding quantum efficiency." This method provides a feasible pathway for the practical implementation of quantum topological data analysis.

Taking into account the regional characteristics of intrinsic and extrinsic properties of space is an essential issue in architectural design and urban renewal, which is often achieved step by step using image and graph-based GANs. However, each model nesting and data conversion may cause information loss, and it is necessary to streamline the tools to facilitate architects and users to participate in the design. Therefore, this study hopes to prove that I2I GAN also has the potential to recognize topological relationships autonomously. Therefore, this research proposes a method for quickly detecting the ability of pix2pix to learn topological relationships, which is achieved by adding two Grasshopper-based detection modules before and after GAN. At the same time, quantitative data is provided and its learning process is visualized, and changes in different input modes such as greyscale and RGB affect its learning efficiency. There are two innovations in this paper: 1) It proves that pix2pix can automatically learn spatial topological relationships and apply them to architectural design. 2) It fills the gap in detecting the performance of Image-based Generation GAN from a topological perspective. Moreover, the detection method proposed in this study takes a short time and is simple to operate. The two detection modules can be widely used for customizing image datasets with the same topological structure and for batch detection of topological relationships of images. In the future, this paper may provide a theoretical foundation and data support for the application of architectural design and urban renewal that use GAN to preserve spatial topological characteristics.