We tackle the problem of text-driven 3D generation from a geometry alignment perspective. Given a set of text prompts, we aim to generate a collection of objects with semantically corresponding parts aligned across them. Recent methods based on Score Distillation have succeeded in distilling the knowledge from 2D diffusion models to high-quality representations of the 3D objects. These methods handle multiple text queries separately, and therefore the resulting objects have a high variability in object pose and structure. However, in some applications, such as 3D asset design, it may be desirable to obtain a set of objects aligned with each other. In order to achieve the alignment of the corresponding parts of the generated objects, we propose to embed these objects into a common latent space and optimize the continuous transitions between these objects. We enforce two kinds of properties of these transitions: smoothness of the transition and plausibility of the intermediate objects along the transition. We demonstrate that both of these properties are essential for good alignment. We provide several practical scenarios that benefit from alignment between the objects, including 3D editing and object hybridization, and experimentally demonstrate the effectiveness of our method.

Learning diffusion bridge models is easy; making them fast and practical is an art. Diffusion bridge models (DBMs) are a promising extension of diffusion models for applications in image-to-image translation. However, like many modern diffusion and flow models, DBMs suffer from the problem of slow inference. To address it, we propose a novel distillation technique based on the inverse bridge matching formulation and derive the tractable objective to solve it in practice. Unlike previously developed DBM distillation techniques, the proposed method can distill both conditional and unconditional types of DBMs, distill models in a one-step generator, and use only the corrupted images for training. We evaluate our approach for both conditional and unconditional types of bridge matching on a wide set of setups, including super-resolution, JPEG restoration, sketch-to-image, and other tasks, and show that our distillation technique allows us to accelerate the inference of DBMs from 4x to 100x and even provide better generation quality than used teacher model depending on particular setup. We provide the code at https://github.com/ngushchin/IBMD

The Schrödinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-time stochastic processes. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations. To address this limitation, we propose a novel Discrete-time IMF (D-IMF) procedure in which learning of stochastic processes is replaced by learning just a few transition probabilities in discrete time. Its great advantage is that in practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), an already well-established adversarial generative modeling technique. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds.

In many unpaired image domain translation problems, e.g., style transfer or super-resolution, it is important to keep the translated image similar to its respective input image. We propose the extremal transport (ET) which is a mathematical formalization of the theoretically best possible unpaired translation between a pair of domains w.r.t. the given similarity function. Inspired by the recent advances in neural optimal transport (OT), we propose a scalable algorithm to approximate ET maps as a limit of partial OT maps. We test our algorithm on toy examples and on the unpaired image-to-image translation task. The code is publicly available at https://github.com/milenagazdieva/ExtremalNeuralOptimalTransport

In this merged paper, we consider the problem of minimizing a convex function with Lipschitz-continuous $p$-th order derivatives. Given an oracle which when queried at a point returns the first $p$-derivatives of the function at that point we provide some methods which compute an $\e$ approximate minimizer in $O\left(\e^{-\frac{2}{3p+1}} \right)$ iterations. These methods match known lower bounds up to polylogarithmic factors for constant $p$.

We consider convex optimization problems with the objective function having Lipshitz-continuous $p$-th order derivative, where $p\geq 1$. We propose a new tensor method, which closes the gap between the lower $\Omega\left(\e^{-\frac{2}{3p+1}} \right)$ and upper $O\left(\e^{-\frac{1}{p+1}} \right)$ iteration complexity bounds for this class of optimization problems. We also consider uniformly convex functions, and show how the proposed method can be accelerated under this additional assumption. Moreover, we introduce a $p$-th order condition number which naturally arises in the complexity analysis of tensor methods under this assumption. Finally, we make a numerical study of the proposed optimal method and show that in practice it is faster than the best known accelerated tensor method. We also compare the performance of tensor methods for $p=2$ and $p=3$ and show that the 3rd-order method is superior to the 2nd-order method in practice.