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For solving linear inverse problems, particularly of the type that appear in tomographic imaging and compressive sensing, this paper develops two new approaches. The first approach is an iterative algorithm that minimizers a regularized least squares objective function where the regularization is based on a compound Gaussian prior distribution. The Compound Gaussian prior subsumes many of the commonly used priors in image reconstruction, including those of sparsity-based approaches. The developed iterative algorithm gives rise to the paper's second new approach, which is a deep neural network that corresponds to an "unrolling" or "unfolding" of the iterative algorithm. Unrolled deep neural networks have interpretable layers and outperform standard deep learning methods. This paper includes a detailed computational theory that provides insight into the construction and performance of both algorithms. The conclusion is that both algorithms outperform other state-of-the-art approaches to tomographic image formation and compressive sensing, especially in the difficult regime of low training.

Langevin Dynamics has been extensively employed in global non-convex optimization due to the concentration of its stationary distribution around the global minimum of the potential function at low temperatures. In this paper, we propose to utilize a more comprehensive class of stochastic processes, known as reversible diffusion, and apply the Euler-Maruyama discretization for global non-convex optimization. We design the diffusion coefficient to be larger when distant from the optimum and smaller when near, thus enabling accelerated convergence while regulating discretization error, a strategy inspired by landscape modifications. Our proposed method can also be seen as a time change of Langevin Dynamics, and we prove convergence with respect to KL divergence, investigating the trade-off between convergence speed and discretization error. The efficacy of our proposed method is demonstrated through numerical experiments.

In this work, we propose a novel framework for estimating the dimension of the data manifold using a trained diffusion model. A diffusion model approximates the score function i.e. the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that, if the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximal likelihood increase. Therefore, for small levels of corruption, the diffusion model provides us with access to an approximation of the normal bundle of the data manifold. This allows us to estimate the dimension of the tangent space, thus, the intrinsic dimension of the data manifold. To the best of our knowledge, our method is the first estimator of the data manifold dimension based on diffusion models and it outperforms well established statistical estimators in controlled experiments on both Euclidean and image data.

Transformer-based models have delivered impressive results on many tasks, particularly vision and language tasks. In many model training situations, conventional configurations are typically adopted. For example, we often set the base model with hidden dimensions (i.e. model width) to be 768 and the number of transformer layers (i.e. model depth) to be 12. In this paper, we revisit these conventional configurations. Through theoretical analysis and experimental evaluation, we show that the masked autoencoder is effective in alleviating the over-smoothing issue in deep transformer training. Based on this finding, we propose Bamboo, an idea of using deeper and narrower transformer configurations, for masked autoencoder training. On ImageNet, with such a simple change in configuration, re-designed model achieves 87.1% top-1 accuracy and outperforms SoTA models like MAE and BEiT. On language tasks, re-designed model outperforms BERT with default setting by 1.1 points on average, on GLUE datasets.

We study the voting game with information uncertainty where agents can coordinate in groups. We show that strategic voting behaviors have a positive impact on leading to the ``correct'' decision, outperforming the common non-strategic behavior of informative voting and sincere voting. Our results give merit to strategic voting for making good decisions. We reveal the surprising equivalence between a strategy profile being a strong equilibrium and leading to the decision favored by the majority (the {\em informed majority decision}): as the size of the vote goes to infinity, every $\varepsilon$-strong Bayes Nash Equilibrium with $\varepsilon$ converging to $0$ formed by strategic agents leads to the informed majority decision with probability converging to $1$. On the other hand, we show that informative voting leads to the informed majority decision only under unbiased instances, and sincere voting leads to the informed majority decision only when it also forms an equilibrium. In our model, voters' preferences between two alternatives depend on a discrete state variable that is not directly observable. Each voter receives a private signal that is correlated with the state variable.

This paper focuses on a challenging class of inverse problems that is often encountered in applications. The forward model is a complex non-linear black-box, potentially non-injective, whose outputs cover multiple decades in amplitude. Observations are supposed to be simultaneously damaged by additive and multiplicative noises and censorship. As needed in many applications, the aim of this work is to provide uncertainty quantification on top of parameter estimates. The resulting log-likelihood is intractable and potentially non-log-concave. An adapted Bayesian approach is proposed to provide credibility intervals along with point estimates. An MCMC algorithm is proposed to deal with the multimodal posterior distribution, even in a situation where there is no global Lipschitz constant (or it is very large). It combines two kernels, namely an improved version of (Preconditioned Metropolis Adjusted Langevin) PMALA and a Multiple Try Metropolis (MTM) kernel. Whenever smooth, its gradient admits a Lipschitz constant too large to be exploited in the inference process. This sampler addresses all the challenges induced by the complex form of the likelihood. The proposed method is illustrated on classical test multimodal distributions as well as on a challenging and realistic inverse problem in astronomy.

We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the $n$ vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold. We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations. The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.

This work presents an effective depth-consistency self-prompt Transformer for image dehazing. It is motivated by an observation that the estimated depths of an image with haze residuals and its clear counterpart vary. Enforcing the depth consistency of dehazed images with clear ones, therefore, is essential for dehazing. For this purpose, we develop a prompt based on the features of depth differences between the hazy input images and corresponding clear counterparts that can guide dehazing models for better restoration. Specifically, we first apply deep features extracted from the input images to the depth difference features for generating the prompt that contains the haze residual information in the input. Then we propose a prompt embedding module that is designed to perceive the haze residuals, by linearly adding the prompt to the deep features. Further, we develop an effective prompt attention module to pay more attention to haze residuals for better removal. By incorporating the prompt, prompt embedding, and prompt attention into an encoder-decoder network based on VQGAN, we can achieve better perception quality. As the depths of clear images are not available at inference, and the dehazed images with one-time feed-forward execution may still contain a portion of haze residuals, we propose a new continuous self-prompt inference that can iteratively correct the dehazing model towards better haze-free image generation. Extensive experiments show that our method performs favorably against the state-of-the-art approaches on both synthetic and real-world datasets in terms of perception metrics including NIQE, PI, and PIQE.

Diffusion models have shown incredible capabilities as generative models; indeed, they power the current state-of-the-art models on text-conditioned image generation such as Imagen and DALL-E 2. In this work we review, demystify, and unify the understanding of diffusion models across both variational and score-based perspectives. We first derive Variational Diffusion Models (VDM) as a special case of a Markovian Hierarchical Variational Autoencoder, where three key assumptions enable tractable computation and scalable optimization of the ELBO. We then prove that optimizing a VDM boils down to learning a neural network to predict one of three potential objectives: the original source input from any arbitrary noisification of it, the original source noise from any arbitrarily noisified input, or the score function of a noisified input at any arbitrary noise level. We then dive deeper into what it means to learn the score function, and connect the variational perspective of a diffusion model explicitly with the Score-based Generative Modeling perspective through Tweedie's Formula. Lastly, we cover how to learn a conditional distribution using diffusion models via guidance.

In this paper, we focus on three problems in deep learning based medical image segmentation. Firstly, U-net, as a popular model for medical image segmentation, is difficult to train when convolutional layers increase even though a deeper network usually has a better generalization ability because of more learnable parameters. Secondly, the exponential ReLU (ELU), as an alternative of ReLU, is not much different from ReLU when the network of interest gets deep. Thirdly, the Dice loss, as one of the pervasive loss functions for medical image segmentation, is not effective when the prediction is close to ground truth and will cause oscillation during training. To address the aforementioned three problems, we propose and validate a deeper network that can fit medical image datasets that are usually small in the sample size. Meanwhile, we propose a new loss function to accelerate the learning process and a combination of different activation functions to improve the network performance. Our experimental results suggest that our network is comparable or superior to state-of-the-art methods.