Search

Training of generative models especially Generative Adversarial Networks can easily diverge in low-data setting. To mitigate this issue, we propose a novel implicit data augmentation approach which facilitates stable training and synthesize high-quality samples without need of label information. Specifically, we view the discriminator as a metric embedding of the real data manifold, which offers proper distances between real data points. We then utilize information in the feature space to develop a fully unsupervised and data-driven augmentation method. Experiments on few-shot generation tasks show the proposed method significantly improve results from strong baselines with hundreds of training samples., Comment: ECCV'22. Code available at https://github.com/dzld00/Adaptive-Feature-Interpolation-for-Low-Shot-Image-Generation

The study of multidimensional discriminator (critic) output for Generative Adversarial Networks has been underexplored in the literature. In this paper, we generalize the Wasserstein GAN framework to take advantage of multidimensional critic output and explore its properties. We also introduce a square-root velocity transformation (SRVT) block which favors training in the multidimensional setting. Proofs of properties are based on our proposed maximal p-centrality discrepancy, which is bounded above by p-Wasserstein distance and fits the Wasserstein GAN framework with multidimensional critic output n. Especially when n = 1 and p = 1, the proposed discrepancy equals 1-Wasserstein distance. Theoretical analysis and empirical evidence show that high-dimensional critic output has its advantage on distinguishing real and fake distributions, and benefits faster convergence and diversity of results., Comment: Frontiers in Adversarial Machine Learning ICML 2022

Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -- and indeed, linear -- algebra. However, matrices in this domain are typically large, with rows and columns numbered in billions. Low-rank approximation of such arrays typically destroys essential information; thus, new mathematical and computational paradigms are needed for very large, sparse matrices. We present the U-match matrix factorization scheme to address this challenge. U-match has two desirable features. First, it admits a compressed storage format that reduces the number of nonzero entries held in computer memory by one or more orders of magnitude over other common factorizations. Second, it permits direct solution of diverse problems in linear and homological algebra, without decompressing matrices stored in memory. These problems include look-up and retrieval of rows and columns; evaluation of birth/death times, and extraction of generators in persistent (co)homology; and, calculation of bases for boundary and cycle subspaces of filtered chain complexes. Such bases are key to unlocking a range of other topological techniques for use in TDA, and U-match factorization is designed to make such calculations broadly accessible to practitioners. As an application, we show that individual cycle representatives in persistent homology can be retrieved at time and memory costs orders of magnitude below current state of the art, via global duality. Moreover, the algebraic machinery needed to achieve this computation already exists in many modern solvers., Comment: 50 pages, 4 appendices; updated to fix typographical and metadata errors

As the aerospace industry becomes increasingly demanding for stronger lightweight materials, the 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 the 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 $V(X,\theta)$, 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 $0-40\%$, and quantitative TDA analysis was conducted based on the SEM images taken at random positions. The results showed high consistency ($R^2=0.975$) compared to the Herman's orientation factors derived from the 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 in 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.

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 high-dimensional 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. Experiments and analysis demonstrate the feasibility and effectiveness of the proposed framework., Comment: ICCV 2021. Code available at https://github.com/dzld00/pytorch-manifold-matching.git

This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both $\pi_0$ and $H_n$ information into a single data structure that distinguishes filtrations that merge trees and persistent homology cannot distinguish alone. Three variants on DMTs, which emphasize category theory, representation theory and persistence barcodes, respectively, offer different advantages in terms of theory and computation. Two notions of distance -- an interleaving distance and bottleneck distance -- for DMTs are defined and a hierarchy of stability results that both refine and generalize existing stability results is proved here. To overcome some of the computational complexity inherent in these distances, we provide a novel use of Gromov-Wasserstein couplings to compute optimal merge tree alignments for a combinatorial version of our interleaving distance which can be tractably estimated. We introduce computational frameworks for generating, visualizing and comparing decorated merge trees derived from synthetic and real data. Example applications include comparison of point clouds, interpretation of persistent homology of sliding window embeddings of time series, visualization of topological features in segmented brain tumor images and topology-driven graph alignment., Comment: Update: streamlined and improved exposition

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. 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.

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 multi-scale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel., Comment: 25 pages, 1 figure

In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a group of data for each orientation reversing periodic map such that two periodic maps with the same data must be conjugate to each other. On the other hand, we give the criterion to judge when two different groups of data correspond to the same conjugacy class. As an application of the results of this paper, we shall show that a given orientation reversing periodic map on $\Sigma_g$ with period larger than or equal to $3g$ must be conjugate to the power of a list of particular types of periodic maps.