Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation

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.

Machine learning data, such as images, often live near much lower-dimensional structures hidden inside very high-dimensional spaces. Estimating this local dimension is important for understanding model behavior, detecting unusual examples, and studying when generative models memorize their training data. However, existing methods become unreliable in high dimensions because they mix together meaningful directions along the data structure with noisy directions pointing away from it. Our method, called Local Hessian Spectral Dimension (LHSD), separates these two kinds of directions. It does this by looking at how sharply the data density bends in different directions: directions that stay nearly flat are treated as part of the underlying structure, while strongly curved directions are filtered out as noise. We also use an efficient approximation method so that this can be done without building an enormous matrix. Experiments on synthetic and real data show that LHSD estimates local dimension more accurately in challenging high-dimensional settings. This makes it a practical tool for studying the geometry of modern machine learning data.