We propose novel finite-dimensional spaces of well-behaved ℝn → ℝn transformations. The latter are obtained by (fast and highly-accurate) integration of continuous piecewise-affine velocity fields. The proposed method is simple yet highly expressive, effortlessly handles optional constraints (e.g., volume preservation and/or boundary conditions), and supports convenient modeling choices such as smoothing priors and coarse-to-fine analysis. Importantly, the proposed approach, partly due to its rapid likelihood evaluations and partly due to its other properties, facilitates tractable inference over rich transformation spaces, including using Markov-Chain Monte-Carlo methods. Its applications include, but are not limited to: monotonic regression (more generally, optimization over monotonic functions); modeling cumulative distribution functions or histograms; time-warping; image warping; image registration; real-time diffeomorphic image editing; data augmentation for image classifiers. Our GPU-based code is publicly available.
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机译：我们提出了表现良好的s n →ℝ n 变换的新型有限维空间。后者是通过（快速且高精度）连续分段仿射速度场积分获得的。所提出的方法是简单而又富有表现力的，不费力地处理可选约束（例如，体积保持和/或边界条件），并支持方便的建模选择，例如平滑先验和粗到精分析。重要的是，提出的方法部分是由于其快速的似然评估，部分是由于其其他属性，有助于在丰富的变换空间上进行易于推断的推断，包括使用马尔可夫链蒙特卡洛方法。它的应用包括但不限于：单调回归（更一般而言，对单调函数的优化）；以及模拟累积分布函数或直方图；时间扭曲图像变形；图像配准；实时衍射图像编辑图像分类器的数据增强。我们基于GPU的代码是公开可用的。
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