See proof of Proposition 3 below for the form of the Jacobian. Theorem 4.7] and so is the product p Equation ( 50) is an element-wise division. The main preprocessing we did was to (i) remove the "label" attribute from each data set, and (ii) Descriptions for all data set are below. All data have been completely anonymized. The original task was to predict whether an applicant would be recommended for acceptance by hierarchical decision model, which has been removed during preprocessing.

In the past, normalizing generative flows have emerged as a promising class of generative models for natural images. This type of model has many modeling advantages: the ability to efficiently compute log-likelihood of the input data, fast generation and simple overall structure. Normalizing flows remained a topic of active research but later fell out of favor, as visual quality of the samples was not competitive with other model classes, such as GANs, VQ-VAE-based approaches or diffusion models. In this paper we revisit the design of the coupling-based normalizing flow models by carefully ablating prior design choices and using computational blocks based on the Vision Transformer architecture, not convolutional neural networks. As a result, we achieve state-of-the-art quantitative and qualitative performance with a much simpler architecture. While the overall visual quality is still behind the current state-of-the-art models, we argue that strong normalizing flow models can help advancing research frontier by serving as building components of more powerful generative models.

Modeling distributions that depend on external control parameters is a common scenario in diverse applications like molecular simulations, where system properties like temperature affect molecular configurations. Despite the relevance of these applications, existing solutions are unsatisfactory as they require severely restricted model architectures or rely on backward training, which is prone to unstable training. We introduce TRADE, which overcomes these limitations by formulating the learning process as a boundary value problem. By initially training the model for a specific condition using either i.i.d. samples or backward KL training, we establish a boundary distribution. We then propagate this information across other conditions using the gradient of the unnormalized density with respect to the external parameter. This formulation, akin to the principles of physics-informed neural networks, allows us to efficiently learn parameter-dependent distributions without restrictive assumptions. Experimentally, we demonstrate that TRADE achieves excellent results in a wide range of applications, ranging from Bayesian inference and molecular simulations to physical lattice models.

Image fusion typically employs non-invertible neural networks to merge multiple source images into a single fused image. However, for clinical experts, solely relying on fused images may be insufficient for making diagnostic decisions, as the fusion mechanism blends features from source images, thereby making it difficult to interpret the underlying tumor pathology. We introduce FusionINN, a novel decomposable image fusion framework, capable of efficiently generating fused images and also decomposing them back to the source images. FusionINN is designed to be bijective by including a latent image alongside the fused image, while ensuring minimal transfer of information from the source images to the latent representation. To the best of our knowledge, we are the first to investigate the decomposability of fused images, which is particularly crucial for life-sensitive applications such as medical image fusion compared to other tasks like multi-focus or multi-exposure image fusion. Our extensive experimentation validates FusionINN over existing discriminative and generative fusion methods, both subjectively and objectively. Moreover, compared to a recent denoising diffusion-based fusion model, our approach offers faster and qualitatively better fusion results.

We introduce an Invertible Symbolic Regression (ISR) method. It is a machine learning technique that generates analytical relationships between inputs and outputs of a given dataset via invertible maps (or architectures). The proposed ISR method naturally combines the principles of Invertible Neural Networks (INNs) and Equation Learner (EQL), a neural network-based symbolic architecture for function learning. In particular, we transform the affine coupling blocks of INNs into a symbolic framework, resulting in an end-to-end differentiable symbolic invertible architecture that allows for efficient gradient-based learning. The proposed ISR framework also relies on sparsity promoting regularization, allowing the discovery of concise and interpretable invertible expressions. We show that ISR can serve as a (symbolic) normalizing flow for density estimation tasks. Furthermore, we highlight its practical applicability in solving inverse problems, including a benchmark inverse kinematics problem, and notably, a geoacoustic inversion problem in oceanography aimed at inferring posterior distributions of underlying seabed parameters from acoustic signals.

We present a novel theoretical framework for understanding the expressive power of coupling-based normalizing flows such as RealNVP. Despite their prevalence in scientific applications, a comprehensive understanding of coupling flows remains elusive due to their restricted architectures. Existing theorems fall short as they require the use of arbitrarily ill-conditioned neural networks, limiting practical applicability. Additionally, we demonstrate that these constructions inherently lead to volume-preserving flows, a property which we show to be a fundamental constraint for expressivity. We propose a new distributional universality theorem for coupling-based normalizing flows, which overcomes several limitations of prior work. Our results support the general wisdom that the coupling architecture is expressive and provide a nuanced view for choosing the expressivity of coupling functions, bridging a gap between empirical results and theoretical understanding.