Chapter 17: Score Function and Energy-Based Models. In the chapter mind map this icon labels Denoising Score Matching (DSM) objective. The discussion below is excerpted and lightly edited from § Denoising Score Matching (DSM) in Mathematics for AI and Machine Learning.

While score matching is theoretically elegant, it faces practical challenges in high dimensions:

• Manifold structure: Real data (e.g., natural images) lies on a low-dimensional manifold embedded in high-dimensional space. The true data distribution $p_{\text{data}}(\mathbf{x})$ has zero probability mass almost everywhere, making $\nabla \log p_{\text{data}}(\mathbf{x})$ undefined or unstable at most points.
• Sparse data: In high dimensions, data samples are sparse. The score function $\nabla \log p_{\text{data}}(\mathbf{x})$ is only well-defined near the data manifold, but we need to learn it everywhere for effective sampling.
• Numerical instability: Computing $\frac{\partial s_{\theta,i}(\mathbf{x})}{\partial x_i}$ (second derivatives) can be numerically unstable, especially when the learned distribution is sharp or multi-modal.