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.

Empirically multidimensional discriminator (critic) output can be advantageous, while a solid explanation for it has not been discussed. In this paper, (i) we rigorously prove that high-dimensional critic output has advantage on distinguishing real and fake distributions; (ii) we also introduce an square-root velocity transformation (SRVT) block which further magnifies this advantage. The proof is based on our proposed maximal p-centrality discrepancy which is bounded above by p-Wasserstein distance and perfectly fits the Wasserstein GAN framework with high-dimensional critic output n. We have also showed when n = 1, the proposed discrepancy is equivalent to 1-Wasserstein distance. The SRVT block is applied to break the symmetric structure of high-dimensional critic output and improve the generalization capability of the discriminator network. In terms of implementation, the proposed framework does not require additional hyper-parameter tuning, which largely facilitates its usage. Experiments on image generation tasks show performance improvement on benchmark datasets.

Leaf shape is a key plant trait that varies enormously. The diversity of leaf shape, and the range of applications for data on this trait, requires frequent methodological developments so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 469 leaves produced by Ginkgo biloba, and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo. We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth, and the angle of the two lobes at the base of the leaf that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. Both methods indicate that there is greater diversity in the shape of fossil leaves compared to modern leaves. The two approaches we have described can be applied to modern and fossil material, and are complementary: identifying similar primary patterns of variation, but revealing some different aspects of morphological variation.

Leaf shape is a key plant trait that varies enormously. The range of applications for data on this trait requires frequent methodological development so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 468 leaves produced by Ginkgo biloba , and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo . We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth and the angle of the two lobes at the leaf base that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. We have applied both methods to modern and fossil material: the methods are complementary, identifying similar primary patterns of variation, but also revealing different aspects of morphological variation. Our topological approach distinguishes long-shoot leaves from short-shoot leaves, both methods indicate that leaf shape influences or is at least related to leaf area, and both could be applied in palaeoclimatic and evolutionary studies of leaf shape.

We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions of persistence sheaves. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.