广义输入凸神经网络的学习不变性
Learning Invariances with Generalised Input-Convex Neural Networks
摘要：
考虑到从输入向量到连续目标的光滑映射，我们的目标是使输入域的子空间在这种映射下保持不变。因此，我们想要隐式地定义由水平集的流形。具体来说，这种表征应该是全局参数形式，这对于不同的知情数据探索任务特别有用，例如构建基于网格的近似，沿着水平曲线采样点，或在流形上找到轨迹。然而，全局参数化只能在水平集连接时存在。为了这个目的，我们引入了一种新的和灵活的类神经网络，广义输入凸网络。这些网络表示保证具有连通水平集的函数，这些连通水平集在输入空间上形成光滑流形。我们进一步表明，这些水平集的全局参数化总是可以有效地找到。最后，我们证明了我们用于表征不变性的新技术在现实世界的应用中是一个强大的生成数据探索工具，例如计算化学。
英文摘要：
Considering smooth mappings from input vectors to continuous targets, our goal is to characterise subspaces of the input domain, which are invariant under such mappings. Thus, we want to characterise manifolds implicitly defined by level sets. Specifically, this characterisation should be of a global parametric form, which is especially useful for different informed data exploration tasks, such as building grid-based approximations, sampling points along the level curves, or finding trajectories on the manifold. However, global parameterisations can only exist if the level sets are connected. For this purpose, we introduce a novel and flexible class of neural networks that generalise input-convex networks. These networks represent functions that are guaranteed to have connected level sets forming smooth manifolds on the input space. We further show that global parameterisations of these level sets can be always found efficiently. Lastly, we demonstrate that our novel technique for characterising invariances is a powerful generative data exploration tool in real-world applications, such as computational chemistry.
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