论文标题
样品复杂性和有效尺寸用于歧管的回归
Sample complexity and effective dimension for regression on manifolds
论文作者
论文摘要
我们使用繁殖的Hilbert空间方法考虑了在流形上的回归理论。歧管模型出现在各种现代机器学习问题中，我们的目标是帮助了解利用多种多样结构的各种隐式和明确的降维方法的有效性。我们的第一个关键贡献是从差异几何形状中建立一种新型的Weyl定律的非沉淀版本。由此，我们能够证明，在歧管上平滑函数的某些空间实际上是有限维度的，其复杂性根据多种多样的维度而不是任何环境数据维度来缩放。最后，我们表明给定的（可能嘈杂的）函数值在歧管上随机统一采取的核能回归估计量（源自歧管的光谱分解）产生的最小值 - 最佳误差界限由有效维度控制。
We consider the theory of regression on a manifold using reproducing kernel Hilbert space methods. Manifold models arise in a wide variety of modern machine learning problems, and our goal is to help understand the effectiveness of various implicit and explicit dimensionality-reduction methods that exploit manifold structure. Our first key contribution is to establish a novel nonasymptotic version of the Weyl law from differential geometry. From this we are able to show that certain spaces of smooth functions on a manifold are effectively finite-dimensional, with a complexity that scales according to the manifold dimension rather than any ambient data dimension. Finally, we show that given (potentially noisy) function values taken uniformly at random over a manifold, a kernel regression estimator (derived from the spectral decomposition of the manifold) yields minimax-optimal error bounds that are controlled by the effective dimension.
