Unsupervised learning of probabilistic models is a central yet challenging problem. Deep generative models have shown promising results in modeling complex distributions such as natural images (Radford et al.,2015), audio (Van Den Oord et al.,2016)and text (Bowman et al.,2015).

Lossless compression methods shorten the expected representation size of data without loss of information, using a statistical model. Flow-based models are attractive in this setting because they admit exact likelihood optimization, which is equivalent to minimizing the expected number of bits per message. However, conventional flows assume continuous data, which may lead to reconstruction errors when quantized for compression. For that reason, we introduce a flow-based generative model for ordinal discrete data called Integer Discrete Flow (IDF): a bijective integer map that can learn rich transformations on high-dimensional data. As building blocks for IDFs, we introduce a flexible transformation layer called integer discrete coupling. Our experiments show that IDFs are competitive with other flow-based generative models. Furthermore, we demonstrate that IDF based compression achieves state-of-the-art lossless compression rates on CIFAR10, ImageNet32, and ImageNet64. To the best of our knowledge, this is the first lossless compression method that uses invertible neural networks.

Our primary observation is that the velocity predictor's outputs in the

Unsupervised learning of probabilistic models is a central yet challenging problem. Deep generative models have shown promising results in modeling complex distributions such as natural images (Radford et al., 2015), audio (V an Den Oord et al., 2016) and text (Bowman et al., 2015).

Guiding pretrained flow-based generative models for conditional generation or to produce samples with desired target properties enables solving diverse tasks without retraining on paired data. We present ESS-Flow, a gradient-free method that leverages the typically Gaussian prior of the source distribution in flow-based models to perform Bayesian inference directly in the source space using Elliptical Slice Sampling. ESS-Flow only requires forward passes through the generative model and observation process, no gradient or Jacobian computations, and is applicable even when gradients are unreliable or unavailable, such as with simulation-based observations or quantization in the generation or observation process. We demonstrate its effectiveness on designing materials with desired target properties and predicting protein structures from sparse inter-residue distance measurements. In generative modeling, we are given data samples and aim to construct a sampler that approximates the data distribution. Diffusion models (Ho et al., 2020; Song et al., 2021) and continuous normalizing flows (Lipman et al., 2023; Liu et al., 2023; Albergo et al., 2023) achieve this by transporting samples from a simple source distribution to the data distribution.

Generative AI (GenAI) has revolutionized data-driven modeling by enabling the synthesis of high-dimensional data across various applications, including image generation, language modeling, biomedical signal processing, and anomaly detection. Flow-based generative models provide a powerful framework for capturing complex probability distributions, offering exact likelihood estimation, efficient sampling, and deterministic transformations between distributions. These models leverage invertible mappings governed by Ordinary Differential Equations (ODEs), enabling precise density estimation and likelihood evaluation. This tutorial presents an intuitive mathematical framework for flow-based generative models, formulating them as neural network-based representations of continuous probability densities. We explore key theoretical principles, including the Wasserstein metric, gradient flows, and density evolution governed by ODEs, to establish convergence guarantees and bridge empirical advancements with theoretical insights. By providing a rigorous yet accessible treatment, we aim to equip researchers and practitioners with the necessary tools to effectively apply flow-based generative models in signal processing and machine learning.

Flow-based generative models have recently shown impressive performance for conditional generation tasks, such as text-to-image generation. However, current methods transform a general unimodal noise distribution to a specific mode of the target data distribution. As such, every point in the initial source distribution can be mapped to every point in the target distribution, resulting in long average paths. To this end, in this work, we tap into a non-utilized property of conditional flow-based models: the ability to design a non-trivial prior distribution. Given an input condition, such as a text prompt, we first map it to a point lying in data space, representing an ``average" data point with the minimal average distance to all data points of the same conditional mode (e.g., class). We then utilize the flow matching formulation to map samples from a parametric distribution centered around this point to the conditional target distribution. Experimentally, our method significantly improves training times and generation efficiency (FID, KID and CLIP alignment scores) compared to baselines, producing high quality samples using fewer sampling steps.

Density estimation is a fundamental problem in statistics and machine learning. We consider a modern approach using flow-based generative models, and propose Local Flow Matching ($\texttt{LFM}$), a computational framework for density estimation based on such models, which learn a continuous and invertible flow to map noise samples to data samples. Unlike existing methods, $\texttt{LFM}$ employs a simulation-free scheme and incrementally learns a sequence of Flow Matching sub-models. Each sub-model matches a diffusion process over a small step size in the data-to-noise direction. This iterative process reduces the gap between the two distributions interpolated by the sub-models, enabling smaller models with faster training times. Theoretically, we prove a generation guarantee of the proposed flow model regarding the $\chi^2$-divergence between the generated and true data distributions. Experimentally, we demonstrate the improved training efficiency and competitive generative performance of $\texttt{LFM}$ compared to FM on the unconditional generation of tabular data and image datasets and its applicability to robotic manipulation policy learning.