Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation
Provides a robust LID estimation method for high-dimensional data, addressing a key bottleneck in diffusion model analysis and related fields.
Existing LID estimation methods fail in high-dimensional spaces due to noise from normal directions. LHSD uses spectral filtering on the log-density Hessian to robustly estimate intrinsic dimension, achieving linear scalability with dimension D and demonstrating superior robustness on synthetic and real data, including detecting memorization in large-scale diffusion models.
While diffusion models enable new approaches for estimating Local Intrinsic Dimension (LID), existing methods fail in high-dimensional spaces where noise from vast normal directions overwhelms the tangent signal. We propose Local Hessian Spectral Dimension (LHSD), which resolves this by applying spectral filtering to the log-density Hessian, explicitly cutting off large eigenvalues associated with normal directions to count zero-curvature tangent directions. Implemented using Stochastic Lanczos Quadrature (SLQ), LHSD avoids full Hessian construction, achieving linear scalability with dimension $D$. Experiments on synthetic and real data confirm LHSD's superior robustness and its utility in detecting memorization in large-scale diffusion models.