  Goodness-of-fit tests are crucial tools for assessing the validity of
statistical models. In this paper, we introduce a novel approach, the Spectral
Smooth Test (SST), that generalizes Neyman's smooth test to high-dimensional
data settings. While conventional goodness-of-fit tests for univariate data are
well-established, extending them to high dimensions, such as images,
trajectories, and SNPs, poses significant challenges. Our proposed SST
leverages spectral bases, which adapt naturally to the geometry of feature
spaces, to model multivariate distributions. Unlike traditional orthogonal
bases, these spectral bases are tailored to the data distribution, enabling
more effective function modeling. The SST framework offers a principled way to
estimate the underlying model, thereby providing actionable insights even when
the null hypothesis is rejected. We present experimental results demonstrating
the robustness of SST across various tuning parameter choices and compare its
performance against other goodness-of-fit tests. Furthermore, we apply SST to
the MNIST dataset as a real-world example, showcasing its effectiveness in
high-dimensional scenarios.
