Cascaded models are multi-scale generative models with a marked capacity for producing perceptually impressive samples at high resolutions. In this work, we show that they can also be excellent likelihood models, so long as we overcome a fundamental difficulty with probabilistic multi-scale models: the intractability of the likelihood function. Chiefly, in cascaded models each intermediary scale introduces extraneous variables that cannot be tractably marginalized out for likelihood evaluation. This issue vanishes by modeling the diffusion process on latent spaces induced by a class of transformations we call hierarchical volume-preserving maps, which decompose spatially structured data in a hierarchical fashion without introducing local distortions in the latent space. We demonstrate that two such maps are well-known in the literature for multiscale modeling: Laplacian pyramids and wavelet transforms. Not only do such reparameterizations allow the likelihood function to be directly expressed as a joint likelihood over the scales, we show that the Laplacian pyramid and wavelet transform also produces significant improvements to the state-of-the-art on a selection of benchmarks in likelihood modeling, including density estimation, lossless compression, and out-of-distribution detection. Investigating the theoretical basis of our empirical gains we uncover deep connections to score matching under the Earth Mover's Distance (EMD), which is a well-known surrogate for perceptual similarity. Code can be found at \href{https://github.com/lihenryhfl/pcdm}{this https url}.

Diffusion models have gained tremendous success in text-to-image generation, yet still lag behind with visual understanding tasks, an area dominated by autoregressive vision-language models. We propose a large-scale and fully end-to-end diffusion model for multi-modal understanding and generation that significantly improves on existing diffusion-based multimodal models, and is the first of its kind to support the full suite of vision-language modeling capabilities. Inspired by the multimodal diffusion transformer (MM-DiT) and recent advances in discrete diffusion language modeling, we leverage a cross-modal maximum likelihood estimation framework that simultaneously trains the conditional likelihoods of both images and text jointly under a single loss function, which is back-propagated through both branches of the diffusion transformer. The resulting model is highly flexible and capable of a wide range of tasks including image generation, captioning, and visual question answering. Our model attained competitive performance compared to recent unified image understanding and generation models, demonstrating the potential of multimodal diffusion modeling as a promising alternative to autoregressive next-token prediction models.

Existing approaches to diffusion-based inverse problem solvers frame the signal recovery task as a probabilistic sampling episode, where the solution is drawn from the desired posterior distribution. This framework suffers from several critical drawbacks, including the intractability of the conditional likelihood function, strict dependence on the score network approximation, and poor $\mathbf{x}_0$ prediction quality. We demonstrate that these limitations can be sidestepped by reframing the generative process as a discrete optimal control episode. We derive a diffusion-based optimal controller inspired by the iterative Linear Quadratic Regulator (iLQR) algorithm. This framework is fully general and able to handle any differentiable forward measurement operator, including super-resolution, inpainting, Gaussian deblurring, nonlinear deblurring, and even highly nonlinear neural classifiers. Furthermore, we show that the idealized posterior sampling equation can be recovered as a special case of our algorithm. We then evaluate our method against a selection of neural inverse problem solvers, and establish a new baseline in image reconstruction with inverse problems.

Diffusion models generate samples by incrementally reversing a process that turns data into noise. We show that when the step size goes to zero, the reversed process is invariant to the distribution of these increments. This reveals a previously unconsidered parameter in the design of diffusion models: the distribution of the diffusion step $\Delta x_k := x_{k} - x_{k + 1}$. This parameter is implicitly set by default to be normally distributed in most diffusion models. By lifting this assumption, we generalize the framework for designing diffusion models and establish an expanded class of diffusion processes with greater flexibility in the choice of loss function used during training. We demonstrate the effectiveness of these models on density estimation and generative modeling tasks on standard image datasets, and show that different choices of the distribution of $\Delta x_k$ result in qualitatively different generated samples.

Large-scale generative models have shown impressive image-generation capabilities, propelled by massive data. However, this often inadvertently leads to the generation of harmful or inappropriate content and raises copyright concerns. Driven by these concerns, machine unlearning has become crucial to effectively purge undesirable knowledge from models. While existing literature has studied various unlearning techniques, these often suffer from either poor unlearning quality or degradation in text-image alignment after unlearning, due to the competitive nature of these objectives. To address these challenges, we propose a framework that seeks an optimal model update at each unlearning iteration, ensuring monotonic improvement on both objectives. We further derive the characterization of such an update. In addition, we design procedures to strategically diversify the unlearning and remaining datasets to boost performance improvement. Our evaluation demonstrates that our method effectively removes target classes from recent diffusion-based generative models and concepts from stable diffusion models while maintaining close alignment with the models' original trained states, thus outperforming state-of-the-art baselines. Our code will be made available at \url{https://github.com/reds-lab/Restricted_gradient_diversity_unlearning.git}.

The exponential moving average (EMA) is a commonly used statistic for providing stable estimates of stochastic quantities in deep learning optimization. Recently, EMA has seen considerable use in generative models, where it is computed with respect to the model weights, and significantly improves the stability of the inference model during and after training. While the practice of weight averaging at the end of training is well-studied and known to improve estimates of local optima, the benefits of EMA over the course of training is less understood. In this paper, we derive an explicit connection between EMA and a damped harmonic system between two particles, where one particle (the EMA weights) is drawn to the other (the model weights) via an idealized zero-length spring. We then leverage this physical analogy to analyze the effectiveness of EMA, and propose an improved training algorithm, which we call BELAY. Finally, we demonstrate theoretically and empirically several advantages enjoyed by BELAY over standard EMA.

Density estimation based anomaly detection schemes typically model anomalies as examples that reside in low-density regions. We propose a modified density estimation problem and demonstrate its effectiveness for anomaly detection. Specifically, we assume the density function of normal samples is uniform in some compact domain. This assumption implies the density function is more stable (with lower variance) around normal samples than anomalies. We first corroborate this assumption empirically using a wide range of real-world data. Then, we design a variance stabilized density estimation problem for maximizing the likelihood of the observed samples while minimizing the variance of the density around normal samples. We introduce an ensemble of autoregressive models to learn the variance stabilized distribution. Finally, we perform an extensive benchmark with 52 datasets demonstrating that our method leads to state-of-the-art results while alleviating the need for data-specific hyperparameter tuning.

We analyze the problem of simultaneous support recovery and estimation of the coefficient vector ($\beta^*$) in a linear model with independent and identically distributed Normal errors. We apply the penalized least square estimator based on non-linear penalties of stochastic gates (STG) [YLNK20] to estimate the coefficients. Considering Gaussian design matrices we show that under reasonable conditions on dimension and sparsity of $\beta^*$ the STG based estimator converges to the true data generating coefficient vector and also detects its support set with high probability. We propose a new projection based algorithm for linear models setup to improve upon the existing STG estimator that was originally designed for general non-linear models. Our new procedure outperforms many classical estimators for support recovery in synthetic data analysis.

Phase retrieval is the inverse problem of recovering a signal from magnitude-only Fourier measurements, and underlies numerous imaging modalities, such as Coherent Diffraction Imaging (CDI). A variant of this setup, known as holography, includes a reference object that is placed adjacent to the specimen of interest before measurements are collected. The resulting inverse problem, known as holographic phase retrieval, is well-known to have improved problem conditioning relative to the original. This innovation, i.e. Holographic CDI, becomes crucial at the nanoscale, where imaging specimens such as viruses, proteins, and crystals require low-photon measurements. This data is highly corrupted by Poisson shot noise, and often lacks low-frequency content as well. In this work, we introduce a dataset-free deep learning framework for holographic phase retrieval adapted to these challenges. The key ingredients of our approach are the explicit and flexible incorporation of the physical forward model into an automatic differentiation procedure, the Poisson log-likelihood objective function, and an optional untrained deep image prior. We perform extensive evaluation under realistic conditions. Compared to competing classical methods, our method recovers signal from higher noise levels and is more resilient to suboptimal reference design, as well as to large missing regions of low frequencies in the observations. To the best of our knowledge, this is the first work to consider a dataset-free machine learning approach for holographic phase retrieval.

Variational autoencoders (VAEs) have become one of the most popular deep learning approaches to unsupervised learning and data generation. However, traditional VAEs suffer from the constraint that the latent space must distributionally match a simple prior (e.g. normal, uniform), independent of the initial data distribution. This leads to a number of issues around modeling manifold data, as there is no function with a bounded Jacobian that maps a normal distribution to certain manifolds (e.g. a hypersphere). Similarly, there are not many theoretical guarantees on the encoder and decoder created by the VAE. In this work, we propose a variational autoencoder that maps manifold valued data to its diffusion map coordinates in the latent space, resamples in a neighborhood around a given point in the latent space, and learns a decoder that maps the newly resampled points back to the manifold. The framework is built off of SpectralNet [Shaham et al., 2018a] and is capable of learning this data dependent latent space without computing the eigenfunction of the Laplacian explicitly. We prove that our method is capable of learning a locally bi-Lipschitz map between the manifold and the latent space, and that our resampling method around a point in the latent space $\psi(x)$ maps points back to the manifold around the point $x$, specifically into a neighborbood on the tangent space at the point $x$ on the manifold. We also provide empirical evidence of the benefits of using a diffusion map latent space on manifold data.