Published as a conference paper at ICLR 2025T OWARDSH IERARCHICAL R ECTIFIED F LOW Yichi Zhang 1, Yici Y an 1, Alex Schwing 1, Zhizhen Zhao 1 1 University of Illinois Urbana-Champaign A BSTRACT We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a known source distribution. Each ODE resembles the ODE that is solved in a classic rectified flow, but differs in its domain, i.e., location, velocity, acceleration, etc. Unlike the classic rectified flow formulation, which formulates a single ODE in the location domain and only captures the expected velocity field (sufficient to capture a multi-modal data distribution), the hierarchical rectified flow formulation models the multi-modal random velocity field, acceleration field, etc., in their entirety. This more faithful modeling of the random velocity field enables integration paths to intersect when the underlying ODE is solved during data generation. Intersecting paths in turn lead to integration trajectories that are more straight than those obtained in the classic rectified flow formulation, where integration paths cannot intersect. This leads to modeling of data distributions with fewer neural function evaluations. We empirically verify this on synthetic 1D and 2D data as well as MNIST, CIFAR-10, and ImageNet-32 data. Our code is available at: https://riccizz.github.io/HRF/ . 1 I NTRODUCTION Diffusion models (Ho et al., 2020; Song et al., 2021a;b) and particularly also flow matching (Liu et al., 2023; Lipman et al., 2023; Albergo & V anden-Eijnden, 2023; Albergo et al., 2023) have gained significant attention recently. This is partly due to impressive results that have been reported across domains from computer vision (Ho et al., 2020) and medical imaging (Song et al., 2022) to robotics (Kapelyukh et al., 2023) and computational biology (Guo et al., 2024). Beyond impressive results, flow matching was also reported to faithfully model multimodal data distributions. In addition, sampling is reasonably straightforward: it requires to solve an ordinary differential equation (ODE) via forward integration of a set of source distribution points along an estimated velocity field from time zero to time one. The source distribution points are sampled from a simple and known source distribution, e.g., a standard Gaussian. The velocity field is obtained by matching velocities from a constructed "ground-truth" integration path with a parametric deep net using a mean squared error (MSE) objective. See Figure 1(a) for the "ground-truth" integration paths of classic rectified flow. Studying the "ground-truth" velocity distribution at a distinct location and time for rectified flow reveals a multimodal distribution.

Backpropagation, which uses the chain rule, is the de-facto standard algorithm for optimizing neural networks nowadays. Recently, Hinton (2022) proposed the forward-forward algorithm, a promising alternative that optimizes neural nets layer-by-layer, without propagating gradients throughout the network. Although such an approach has several advantages over back-propagation and shows promising results, the fact that each layer is being trained independently limits the optimization process. Specifically, it prevents the network's layers from collaborating to learn complex and rich features. In this work, we study layer collaboration in the forward-forward algorithm. We show that the current version of the forward-forward algorithm is suboptimal when considering information flow in the network, resulting in a lack of collaboration between layers of the network. We propose an improved version that supports layer collaboration to better utilize the network structure, while not requiring any additional assumptions or computations. We empirically demonstrate the efficacy of the proposed version when considering both information flow and objective metrics. Additionally, we provide a theoretical motivation for the proposed method, inspired by functional entropy theory.

Diffusion probabilistic models have quickly become a major approach for generative modeling of images, 3D geometry, video and other domains. However, to adapt diffusion generative modeling to these domains the denoising network needs to be carefully designed for each domain independently, oftentimes under the assumption that data lives in a Euclidean grid. In this paper we introduce Diffusion Probabilistic Fields (DPF), a diffusion model that can learn distributions over continuous functions defined over metric spaces, commonly known as fields. We extend the formulation of diffusion probabilistic models to deal with this field parametrization in an explicit way, enabling us to define an end-to-end learning algorithm that side-steps the requirement of representing fields with latent vectors as in previous approaches (Dupont et al., 2022a; Du et al., 2021). We empirically show that, while using the same denoising network, DPF effectively deals with different modalities like 2D images and 3D geometry, in addition to modeling distributions over fields defined on non-Euclidean metric spaces.

Generative adversarial nets (GANs) have been remarkably successful at learning to sample from distributions specified by a given dataset, particularly if the given dataset is reasonably large compared to its dimensionality. However, given limited data, classical GANs have struggled, and strategies like output-regularization, data-augmentation, use of pre-trained models and pruning have been shown to lead to improvements. Notably, the applicability of these strategies is 1) often constrained to particular settings, e.g., availability of a pretrained GAN; or 2) increases training time, e.g., when using pruning. In contrast, we propose a Discriminator gradIent Gap regularized GAN (DigGAN) formulation which can be added to any existing GAN. DigGAN augments existing GANs by encouraging to narrow the gap between the norm of the gradient of a discriminator's prediction w.r.t.\ real images and w.r.t.\ the generated samples. We observe this formulation to avoid bad attractors within the GAN loss landscape, and we find DigGAN to significantly improve the results of GAN training when limited data is available. Code is available at \url{https://github.com/AilsaF/DigGAN}.

Understanding of GAN training is still very limited. One major challenge is its non-convex-non-concave min-max objective, which may lead to sub-optimal local minima. In this work, we perform a global landscape analysis of the empirical loss of GANs. We prove that a class of separable-GAN, including the original JS-GAN, has exponentially many bad basins which are perceived as mode-collapse. We also study the relativistic pairing GAN (RpGAN) loss which couples the generated samples and the true samples. We prove that RpGAN has no bad basins. Experiments on synthetic data show that the predicted bad basin can indeed appear in training. We also perform experiments to support our theory that RpGAN has a better landscape than separable-GAN. For instance, we empirically show that RpGAN performs better than separable-GAN with relatively narrow neural nets. The code is available at https://github.com/AilsaF/RS-GAN.

In this paper, we prove that every multivariate polynomial with even degree can be decomposed into a sum of convex and concave polynomials. Motivated by this property, we exploit the concave-convex procedure to perform inference on continuous Markov random fields with polynomial potentials. In particular, we show that the concave-convex decomposition of polynomials can be expressed as a sum-of-squares optimization, which can be efficiently solved via semidefinite programming. We demonstrate the effectiveness of our approach in the context of 3D reconstruction, shape from shading and image denoising, and show that our approach significantly outperforms existing approaches in terms of efficiency as well as the quality of the retrieved solution. Papers published at the Neural Information Processing Systems Conference.

Inference in Markov random fields subject to consistency structure is a fundamental problem that arises in many real-life applications. In order to enforce consistency, classical approaches utilize consistency potentials or encode constraints over feasible instances. Unfortunately this comes at the price of a serious computational bottleneck. In this paper we suggest to tackle consistency by incorporating constraints on beliefs. This permits derivation of a closed-form message-passing algorithm which we refer to as the Constraints Based Convex Belief Propagation (CBCBP). Experiments show that CBCBP outperforms the standard approach while being at least an order of magnitude faster.

In this paper we aim at facilitating generalization for deep networks while supporting interpretability of the learned representations. Towards this goal, we propose a clustering based regularization that encourages parsimonious representations. Our k-means style objective is easy to optimize and flexible supporting various forms of clustering, including sample and spatial clustering as well as co-clustering. We demonstrate the effectiveness of our approach on the tasks of unsupervised learning, classification, fine grained categorization and zero-shot learning.

Maximum a-posteriori (MAP) inference is an important task for many applications. Although the standard formulation gives rise to a hard combinatorial optimization problem, several effective approximations have been proposed and studied in recent years. We focus on linear programming (LP) relaxations, which have achieved state-of-the-art performance in many applications. However, optimization of the resulting program is in general challenging due to non-smoothness and complex non-separable constraints.Therefore, in this work we study the benefits of augmenting the objective function of the relaxation with strong convexity. Specifically, we introduce strong convexity by adding a quadratic term to the LP relaxation objective. We provide theoretical guarantees for the resulting programs, bounding the difference between their optimal value and the original optimum. Further, we propose suitable optimization algorithms and analyze their convergence.

In this paper, we prove that every multivariate polynomial with even degree can be decomposed into a sum of convex and concave polynomials. Motivated by this property, we exploit the concave-convex procedure to perform inference on continuous Markov random fields with polynomial potentials. In particular, we show that the concave-convex decomposition of polynomials can be expressed as a sum-of-squares optimization, which can be efficiently solved via semidefinite programming. We demonstrate the effectiveness of our approach in the context of 3D reconstruction, shape from shading and image denoising, and show that our approach significantly outperforms existing approaches in terms of efficiency as well as the quality of the retrieved solution.