Abstract
Diffusion samplers can reduce computation by generating selected spectral coordinates and filling the remaining directions with noise. How many directions must they retain? We study this question for data with power-law covariance spectra. For Gaussian data compared to a smoothed target, we prove matching bounds on the required number of retained directions, provided that the ambient dimension is sufficiently large. The truncation error depends on the combined Wiener gains of the omitted directions, regardless of the accuracy of the sampler on the retained coordinates. Keeping only directions whose signal exceeds the output noise level can therefore leave a non-vanishing error: many individually weak directions remain significant in aggregate. Combining this characterization with a diffusion convergence bound yields sufficient sampling-step complexity under exact scores. The upper bounds also extend to estimated principal components and, componentwise, to Gaussian mixtures. The practical prescription is to select the retained subspace using an aggregate spectral-tail error budget, then to choose the diffusion noise level accordingly.
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Cite this article
APA 7
Karimian, R., Mohades, A., & Lederer, J. (2026). How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra. https://omanscience.com/en/articles/how-many-directions-must-a-truncated-diffusion-sampler-retain-matching-bounds-under-power-law-spectra
MLA 9
Karimian, Radmehr, et al. "How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra." https://omanscience.com/en/articles/how-many-directions-must-a-truncated-diffusion-sampler-retain-matching-bounds-under-power-law-spectra.
Chicago (author–date)
Karimian, Radmehr, Ali Mohades, and Johannes Lederer. 2026. "How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra." https://omanscience.com/en/articles/how-many-directions-must-a-truncated-diffusion-sampler-retain-matching-bounds-under-power-law-spectra.
Harvard
Karimian, R., Mohades, A. and Lederer, J. (2026) 'How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra', Available at: https://omanscience.com/en/articles/how-many-directions-must-a-truncated-diffusion-sampler-retain-matching-bounds-under-power-law-spectra.
Vancouver
Karimian R, Mohades A, Lederer J. How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra. https://omanscience.com/en/articles/how-many-directions-must-a-truncated-diffusion-sampler-retain-matching-bounds-under-power-law-spectra
IEEE
R. Karimian, A. Mohades, and J. Lederer, "How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra," https://omanscience.com/en/articles/how-many-directions-must-a-truncated-diffusion-sampler-retain-matching-bounds-under-power-law-spectra.