Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery
This addresses the problem of high-dimensional data analysis for researchers by demonstrating adaptivity to geometric structures, but it is incremental as it builds on existing manifold learning and deep network theories.
The paper tackles nonparametric regression of Hölder functions on low-dimensional manifolds using deep ReLU networks, achieving a mean squared error convergence rate of n^{-2(s+α)/(2(s+α)+d)} log^3 n, which depends on the intrinsic dimension d rather than the ambient dimension D.
Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of Hölder functions on low-dimensional manifolds using deep ReLU networks. Suppose $n$ training data are sampled from a Hölder function in $\mathcal{H}^{s,α}$ supported on a $d$-dimensional Riemannian manifold isometrically embedded in $\mathbb{R}^D$, with sub-gaussian noise. A deep ReLU network architecture is designed to estimate the underlying function from the training data. The mean squared error of the empirical estimator is proved to converge in the order of $n^{-\frac{2(s+α)}{2(s+α) + d}}\log^3 n$. This result shows that deep ReLU networks give rise to a fast convergence rate depending on the data intrinsic dimension $d$, which is usually much smaller than the ambient dimension $D$. It therefore demonstrates the adaptivity of deep ReLU networks to low-dimensional geometric structures of data, and partially explains the power of deep ReLU networks in tackling high-dimensional data with low-dimensional geometric structures.