Diffusion models have emerged as powerful generative approaches for missing-data imputation, yet most existing methods operate directly in data space and degrade when training data are heavily incomplete. We investigate whether shifting diffusion to a learned latent representation improves robustness under missing-completely-at-random (MCAR) corruption. To this end, we propose a two-stage framework: a robust VAE-based imputer first learns compact semantic features from incomplete observations, and a diffusion model is then trained in the resulting latent space. Across training missing rates, we perform a controlled comparison against pixel-space diffusion models under the same incomplete-data setting. The latent diffusion model maintains high sample quality and remains stable up to 50\% missingness, while pixel-space diffusion degrades progressively as missingness increases. For downstream imputation, latent diffusion also achieves consistently better performance than pixel-space diffusion. These findings indicate that latent-space modeling mitigates artifact amplification from zero-imputed inputs and provides a more robust generative prior for incomplete-data learning. Overall, our results support latent diffusion as a strong and practically useful alternative to pixel-space diffusion for missing-data problems.

The proof follows from the following equality and the fact that Zγ is independent of q(z). All experiments are run on Nvidia GPUs. The exact softwares can be found in the supplemental code. The'letter' split of the EMNIST dataset was used as the auxiliary dataset. The images are resized to are 32x32.

B.3 Derivations of Eq. (19) Similar to derivation above, we give the gradient with respect to weight vector w RM+, which is given by wDKL = w log Z(U,w) wEU,w (log pθ(X |z))T1N + wEU,w (log pθ(U |z))Tw . The learning rate of each stochastic gradient descent step is γt t 1, where t {1,,T}denotes the iteration for optimization. We already report the t-SNE visualization of ByPE-VAE and standard VAE in Figure. Here we give more t-SNE visualization results. First, we randomly sample from ByPE-VAEs trained on different datasets, namely, MNIST, Fashion MNIST, and Celeba, as shown in Fig.7.

The Gaussian homotopy (GH) method is a popular approach to finding better stationary points for non-convex optimization problems by gradually reducing a parameter value t, which changes the problem to be solved from an almost convex one to the original target one. Existing GH-based methods repeatedly call an iterative optimization solver to find a stationary point every time t is updated, which incurs high computational costs. We propose a novel single loop framework for GH methods (SLGH) that updates the parameter tand the optimization decision variables at the same. Computational complexity analysis is performed on the SLGH algorithm under various situations: either a gradient or gradient-free oracle of a GH function can be obtained for both deterministic and stochastic settings. The convergence rate of SLGH with a tuned hyperparameter becomes consistent with the convergence rate of gradient descent, even though the problem to be solved is gradually changed due to t. In numerical experiments, our SLGH algorithms show faster convergence than an existing double loop GH method while outperforming gradient descent-based methods in terms of finding a better solution.

Let, y be a point in that intersection. Since, by definition, ˆr(x0,) is the radius of the smallest ball with 1/2 + probability mass of f(x0 + P) over all possible centers in Rk and ˆRis the radius of the smallest such ball centered at ˆf(x), we must have ˆr(x0,) ˆR. Consider the smallest ball B(z0,ˆr(x, 1)) that encloses at least 1/2 + 1 probability mass of f(x+ P). Since, r is the radius of the minimum enclosing ball that contains at least half of the points in Z, we have r ˆr(x, 1). Now, using the definition of ˆRand following the same reasoning as theorem 2, we can say that, d( ˆf(x), ˆf(x0)) βˆr(x0,) + ˆR (1 + β) ˆR.