NeurIPS 2025

Linearly Constrained Diffusion Implicit Models
Vivek Jayaram1
Ira Kemelmacher-Shlizerman1
Steven M. Seitz1
John Thickstun2
1University of Washington 2Cornell University
Abstract
We introduce Linearly Constrained Diffusion Implicit Models (CDIM), a fast and
accurate approach to solving noisy linear inverse problems using diffusion models.
Traditional diffusion-based inverse methods rely on numerous projection steps to
enforce measurement consistency in addition to unconditional denoising steps.
CDIM achieves a 10–50× reduction in projection steps by dynamically adjusting
the number and size of projection steps to align a residual measurement energy
with its theoretical distribution under the forward diffusion process. This adap-
tive alignment preserves measurement consistency while substantially accelerat-
ing constrained inference. For noise-free linear inverse problems, CDIM exactly
satisfies the measurement constraints with few projection steps, even when exist-
ing methods fail. We demonstrate CDIM’s effectiveness across a range of applica-
tions, including super-resolution, denoising, inpainting, deblurring, and 3D point
cloud reprojection. Code and an interactive demo can be found on our project
website. 1
1
Introduction
Recovering an unknown signal from noisy linear measurements is a fundamental challenge encom-
passing tasks like super-resolution, inpainting, and denoising. These are examples of linear inverse
problems, for which we observe a partial measurements y ∈Rd that was generated through a linear
operator A ∈Rd×n applied to a signal x ∈Rn. In the noisy case, measurements y are corrupted
with noise sampled i.i.d. from a gaussian distribution with variance σ2
y:
y = Ax + σ,
where σ ∼N(0, σ2
yI)
(1)
We aim to use a pretrained diffusion model [1] as a prior to solve these underdetermined inverse
problems. However, adapting diffusion models to solve inverse problems has traditionally presented
two key challenges. First, recovering the original signal from...