We propose a manifold matching approach to generative models which includes a distribution generator (or data generator) and a metric generator. In our framework, we view the real data set as some manifold embedded in a highdimensional Euclidean space. The distribution generator aims at generating samples that follow some distribution condensed around the real data manifold. It is achieved by matching two sets of points using their geometric shape descriptors, such as centroid and p-diameter, with learned distance metric; the metric generator utilizes both real data and generated samples to learn a distance metric which is close to some intrinsic geodesic distance on the real data manifold. The produced distance metric is further used for manifold matching. The two networks are learned simultaneously during the training process. We apply the approach on both unsupervised and supervised learning tasks: in unconditional image generation task, the proposed method obtains competitive results compared with existing generative models; in super-resolution task, we incorporate the framework in perception-based models and improve visual qualities by producing samples with more natural textures. Both theoretical analysis and real data experiments guarantee the feasibility and effectiveness of the proposed framework.

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We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties of the shape of a signal, eliminating otherwise highly persistent homology classes that may exist simply because of the nature of the domain on which the signal is defined. We investigate the stability of these invariants using metrics that downplay regions where signals are weak. The distance between two signals is small if they exhibit high similarity in regions where they are strong, regardless of the nature of their full domains, in particular allowing different homotopy types. Consistency and estimation of persistent homology of metric measure spaces from data are studied within this framework. We also apply the methodology to the construction of multiscale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel.

As the aerospace industry is increasingly demanding stronger, lightweight materials, ultra-strong carbon nanotube (CNT) composites with highly aligned CNT network structures could be the answer. In this work, a novel methodology applying topological data analysis (TDA) to scanning electron microscope (SEM) images was developed to detect CNT orientation. The CNT bundle extensions in certain directions were summarized algebraically and expressed as visible barcodes. The barcodes were then calculated and converted into the total spread function, V(X, θ), from which the alignment fraction and the preferred direction could be determined. For validation purposes, the random CNT sheets were mechanically stretched at various strain ratios ranging from 0 to 40%, and quantitative TDA was conducted based on the SEM images taken at random positions. The results showed high consistency (R2 = 0.972) compared to Herman’s orientation factors derived from polarized Raman spectroscopy and wide-angle X-ray scattering analysis. Additionally, the TDA method presented great robustness with varying SEM acceleration voltages and magnifications, which might alter the scope of alignment detection. With potential applications in nanofiber systems, this study offers a rapid and simple way to quantify CNT alignment, which plays a crucial role in transferring the CNT properties into engineering products.

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