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Solving Inverse Problems by Joint Posterior Maximization with a VAE Prior2019-11-14   ${\displaystyle \cong }$
In this paper we address the problem of solving ill-posed inverse problems in imaging where the prior is a neural generative model. Specifically we consider the decoupled case where the prior is trained once and can be reused for many different log-concave degradation models without retraining. Whereas previous MAP-based approaches to this problem lead to highly non-convex optimization algorithms, our approach computes the joint (space-latent) MAP that naturally leads to alternate optimization algorithms and to the use of a stochastic encoder to accelerate computations. The resulting technique is called JPMAP because it performs Joint Posterior Maximization using an Autoencoding Prior. We show theoretical and experimental evidence that the proposed objective function is quite close to bi-convex. Indeed it satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima.
Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior2021-03-02   ${\displaystyle \cong }$
In this work we address the problem of solving ill-posed inverse problems in imaging where the prior is a variational autoencoder (VAE). Specifically we consider the decoupled case where the prior is trained once and can be reused for many different log-concave degradation models without retraining. Whereas previous MAP-based approaches to this problem lead to highly non-convex optimization algorithms, our approach computes the joint (space-latent) MAP that naturally leads to alternate optimization algorithms and to the use of a stochastic encoder to accelerate computations. The resulting technique (JPMAP) performs Joint Posterior Maximization using an Autoencoding Prior. We show theoretical and experimental evidence that the proposed objective function is quite close to bi-convex. Indeed it satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. We also highlight the importance of correctly training the VAE using a denoising criterion, in order to ensure that the encoder generalizes well to out-of-distribution images, without affecting the quality of the generative model. This simple modification is key to providing robustness to the whole procedure. Finally we show how our joint MAP methodology relates to more common MAP approaches, and we propose a continuation scheme that makes use of our JPMAP algorithm to provide more robust MAP estimates. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima.
Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors2020-03-27   ${\displaystyle \cong }$
Deep neural networks as image priors have been recently introduced for problems such as denoising, super-resolution and inpainting with promising performance gains over hand-crafted image priors such as sparsity and low-rank. Unlike learned generative priors they do not require any training over large datasets. However, few theoretical guarantees exist in the scope of using untrained neural network priors for inverse imaging problems. We explore new applications and theory for untrained neural network priors. Specifically, we consider the problem of solving linear inverse problems, such as compressive sensing, as well as non-linear problems, such as compressive phase retrieval. We model images to lie in the range of an untrained deep generative network with a fixed seed. We further present a projected gradient descent scheme that can be used for both compressive sensing and phase retrieval and provide rigorous theoretical guarantees for its convergence. We also show both theoretically as well as empirically that with deep network priors, one can achieve better compression rates for the same image quality compared to hand crafted priors.
Learning Convex Optimization Models2020-06-18   ${\displaystyle \cong }$
A convex optimization model predicts an output from an input by solving a convex optimization problem. The class of convex optimization models is large, and includes as special cases many well-known models like linear and logistic regression. We propose a heuristic for learning the parameters in a convex optimization model given a dataset of input-output pairs, using recently developed methods for differentiating the solution of a convex optimization problem with respect to its parameters. We describe three general classes of convex optimization models, maximum a posteriori (MAP) models, utility maximization models, and agent models, and present a numerical experiment for each.
Compressive Phase Retrieval: Optimal Sample Complexity with Deep Generative Priors2020-08-24   ${\displaystyle \cong }$
Advances in compressive sensing provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data-driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm in this challenging nonlinear inverse problem. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non-convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements.
Provably Convergent Algorithms for Solving Inverse Problems Using Generative Models2021-05-13   ${\displaystyle \cong }$
The traditional approach of hand-crafting priors (such as sparsity) for solving inverse problems is slowly being replaced by the use of richer learned priors (such as those modeled by deep generative networks). In this work, we study the algorithmic aspects of such a learning-based approach from a theoretical perspective. For certain generative network architectures, we establish a simple non-convex algorithmic approach that (a) theoretically enjoys linear convergence guarantees for certain linear and nonlinear inverse problems, and (b) empirically improves upon conventional techniques such as back-propagation. We support our claims with the experimental results for solving various inverse problems. We also propose an extension of our approach that can handle model mismatch (i.e., situations where the generative network prior is not exactly applicable). Together, our contributions serve as building blocks towards a principled use of generative models in inverse problems with more complete algorithmic understanding.
Solving Inverse Computational Imaging Problems using Deep Pixel-level Prior2018-04-23   ${\displaystyle \cong }$
Signal reconstruction is a challenging aspect of computational imaging as it often involves solving ill-posed inverse problems. Recently, deep feed-forward neural networks have led to state-of-the-art results in solving various inverse imaging problems. However, being task specific, these networks have to be learned for each inverse problem. On the other hand, a more flexible approach would be to learn a deep generative model once and then use it as a signal prior for solving various inverse problems. We show that among the various state of the art deep generative models, autoregressive models are especially suitable for our purpose for the following reasons. First, they explicitly model the pixel level dependencies and hence are capable of reconstructing low-level details such as texture patterns and edges better. Second, they provide an explicit expression for the image prior which can then be used for MAP based inference along with the forward model. Third, they can model long range dependencies in images which make them ideal for handling global multiplexing as encountered in various compressive imaging systems. We demonstrate the efficacy of our proposed approach in solving three computational imaging problems: Single Pixel Camera (SPC), LiSens and FlatCam. For both real and simulated cases, we obtain better reconstructions than the state-of-the-art methods in terms of perceptual and quantitative metrics.
Exact priors of finite neural networks2021-04-23   ${\displaystyle \cong }$
Bayesian neural networks are theoretically well-understood only in the infinite-width limit, where Gaussian priors over network weights yield Gaussian priors over network outputs. Recent work has suggested that finite Bayesian networks may outperform their infinite counterparts, but their non-Gaussian output priors have been characterized only though perturbative approaches. Here, we derive exact solutions for the output priors for individual input examples of a class of finite fully-connected feedforward Bayesian neural networks. For deep linear networks, the prior has a simple expression in terms of the Meijer $G$-function. The prior of a finite ReLU network is a mixture of the priors of linear networks of smaller widths, corresponding to different numbers of active units in each layer. Our results unify previous descriptions of finite network priors in terms of their tail decay and large-width behavior.
Bayesian Optimization with Automatic Prior Selection for Data-Efficient Direct Policy Search2018-03-13   ${\displaystyle \cong }$
One of the most interesting features of Bayesian optimization for direct policy search is that it can leverage priors (e.g., from simulation or from previous tasks) to accelerate learning on a robot. In this paper, we are interested in situations for which several priors exist but we do not know in advance which one fits best the current situation. We tackle this problem by introducing a novel acquisition function, called Most Likely Expected Improvement (MLEI), that combines the likelihood of the priors and the expected improvement. We evaluate this new acquisition function on a transfer learning task for a 5-DOF planar arm and on a possibly damaged, 6-legged robot that has to learn to walk on flat ground and on stairs, with priors corresponding to different stairs and different kinds of damages. Our results show that MLEI effectively identifies and exploits the priors, even when there is no obvious match between the current situations and the priors.
A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton2020-07-02   ${\displaystyle \cong }$
In recent years, a variety of gradient-based first-order methods have been developed to solve bi-level optimization problems for learning applications. However, theoretical guarantees of these existing approaches heavily rely on the simplification that for each fixed upper-level variable, the lower-level solution must be a singleton (a.k.a., Lower-Level Singleton, LLS). In this work, we first design a counter-example to illustrate the invalidation of such LLS condition. Then by formulating BLPs from the view point of optimistic bi-level and aggregating hierarchical objective information, we establish Bi-level Descent Aggregation (BDA), a flexible and modularized algorithmic framework for generic bi-level optimization. Theoretically, we derive a new methodology to prove the convergence of BDA without the LLS condition. Our investigations also demonstrate that BDA is indeed compatible to a verify of particular first-order computation modules. Additionally, as an interesting byproduct, we also improve these conventional first-order bi-level schemes (under the LLS simplification). Particularly, we establish their convergences with weaker assumptions. Extensive experiments justify our theoretical results and demonstrate the superiority of the proposed BDA for different tasks, including hyper-parameter optimization and meta learning.
Posterior Model Adaptation With Updated Priors2020-07-02   ${\displaystyle \cong }$
Classification approaches based on the direct estimation and analysis of posterior probabilities will degrade if the original class priors begin to change. We prove that a unique (up to scale) solution is possible to recover the data likelihoods for a test example from its original class posteriors and dataset priors. Given the recovered likelihoods and a set of new priors, the posteriors can be re-computed using Bayes' Rule to reflect the influence of the new priors. The method is simple to compute and allows a dynamic update of the original posteriors.
Image Restoration using Plug-and-Play CNN MAP Denoisers2019-12-20   ${\displaystyle \cong }$
Plug-and-play denoisers can be used to perform generic image restoration tasks independent of the degradation type. These methods build on the fact that the Maximum a Posteriori (MAP) optimization can be solved using smaller sub-problems, including a MAP denoising optimization. We present the first end-to-end approach to MAP estimation for image denoising using deep neural networks. We show that our method is guaranteed to minimize the MAP denoising objective, which is then used in an optimization algorithm for generic image restoration. We provide theoretical analysis of our approach and show the quantitative performance of our method in several experiments. Our experimental results show that the proposed method can achieve 70x faster performance compared to the state-of-the-art, while maintaining the theoretical perspective of MAP.
Accelerating Optimization Algorithms With Dynamic Parameter Selections Using Convolutional Neural Networks For Inverse Problems In Image Processing2019-11-18   ${\displaystyle \cong }$
Recent advances using deep neural networks (DNNs) for solving inverse problems in image processing have significantly outperformed conventional optimization algorithm based methods. Most works train DNNs to learn 1) forward models and image priors implicitly for direct mappings from given measurements to solutions, 2) data-driven priors as proximal operators in conventional iterative algorithms, or 3) forward models, priors and/or static stepsizes in unfolded structures of optimization iterations. Here we investigate another way of utilizing convolutional neural network (CNN) for empirically accelerating conventional optimization for solving inverse problems in image processing. We propose a CNN to yield parameters in optimization algorithms that have been chosen heuristically, but have shown to be crucial for good empirical performance. Our CNN-incorporated scaled gradient projection methods, without compromising theoretical properties, significantly improve empirical convergence rate over conventional optimization based methods in large-scale inverse problems such as image inpainting, compressive image recovery with partial Fourier samples, deblurring and sparse view CT. During testing, our proposed methods dynamically select parameters every iterations to speed up convergence robustly for different degradation levels, noise, or regularization parameters as compared to direct mapping approach.
All You Need is a Good Functional Prior for Bayesian Deep Learning2020-11-25   ${\displaystyle \cong }$
The Bayesian treatment of neural networks dictates that a prior distribution is specified over their weight and bias parameters. This poses a challenge because modern neural networks are characterized by a large number of parameters, and the choice of these priors has an uncontrolled effect on the induced functional prior, which is the distribution of the functions obtained by sampling the parameters from their prior distribution. We argue that this is a hugely limiting aspect of Bayesian deep learning, and this work tackles this limitation in a practical and effective way. Our proposal is to reason in terms of functional priors, which are easier to elicit, and to "tune" the priors of neural network parameters in a way that they reflect such functional priors. Gaussian processes offer a rigorous framework to define prior distributions over functions, and we propose a novel and robust framework to match their prior with the functional prior of neural networks based on the minimization of their Wasserstein distance. We provide vast experimental evidence that coupling these priors with scalable Markov chain Monte Carlo sampling offers systematically large performance improvements over alternative choices of priors and state-of-the-art approximate Bayesian deep learning approaches. We consider this work a considerable step in the direction of making the long-standing challenge of carrying out a fully Bayesian treatment of neural networks, including convolutional neural networks, a concrete possibility.
Solving Electrical Impedance Tomography with Deep Learning2019-11-11   ${\displaystyle \cong }$
This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map. Both the forward map from the electrical conductivity to the DtN map and the inverse map are high-dimensional and nonlinear. Motivated by the linear perturbative analysis of the forward map and based on a numerically low-rank property, we propose compact neural network architectures for the forward and inverse maps for both 2D and 3D problems. Numerical results demonstrate the efficiency of the proposed neural networks.
Variational Algorithms for Marginal MAP2013-07-17   ${\displaystyle \cong }$
The marginal maximum a posteriori probability (MAP) estimation problem, which calculates the mode of the marginal posterior distribution of a subset of variables with the remaining variables marginalized, is an important inference problem in many models, such as those with hidden variables or uncertain parameters. Unfortunately, marginal MAP can be NP-hard even on trees, and has attracted less attention in the literature compared to the joint MAP (maximization) and marginalization problems. We derive a general dual representation for marginal MAP that naturally integrates the marginalization and maximization operations into a joint variational optimization problem, making it possible to easily extend most or all variational-based algorithms to marginal MAP. In particular, we derive a set of "mixed-product" message passing algorithms for marginal MAP, whose form is a hybrid of max-product, sum-product and a novel "argmax-product" message updates. We also derive a class of convergent algorithms based on proximal point methods, including one that transforms the marginal MAP problem into a sequence of standard marginalization problems. Theoretically, we provide guarantees under which our algorithms give globally or locally optimal solutions, and provide novel upper bounds on the optimal objectives. Empirically, we demonstrate that our algorithms significantly outperform the existing approaches, including a state-of-the-art algorithm based on local search methods.
Robust priors for regularized regression2020-10-06   ${\displaystyle \cong }$
Induction benefits from useful priors. Penalized regression approaches, like ridge regression, shrink weights toward zero but zero association is usually not a sensible prior. Inspired by simple and robust decision heuristics humans use, we constructed non-zero priors for penalized regression models that provide robust and interpretable solutions across several tasks. Our approach enables estimates from a constrained model to serve as a prior for a more general model, yielding a principled way to interpolate between models of differing complexity. We successfully applied this approach to a number of decision and classification problems, as well as analyzing simulated brain imaging data. Models with robust priors had excellent worst-case performance. Solutions followed from the form of the heuristic that was used to derive the prior. These new algorithms can serve applications in data analysis and machine learning, as well as help in understanding how people transition from novice to expert performance.
Prior-guided Bayesian Optimization2020-06-25   ${\displaystyle \cong }$
While Bayesian Optimization (BO) is a very popular method for optimizing expensive black-box functions, it fails to leverage the experience of domain experts. This causes BO to waste function evaluations on commonly known bad regions of design choices, e.g., hyperparameters of a machine learning algorithm. To address this issue, we introduce Prior-guided Bayesian Optimization (PrBO). PrBO allows users to inject their knowledge into the optimization process in the form of priors about which parts of the input space will yield the best performance, rather than BO's standard priors over functions which are much less intuitive for users. PrBO then combines these priors with BO's standard probabilistic model to yield a posterior. We show that PrBO is more sample efficient than state-of-the-art methods without user priors and 10,000$\times$ faster than random search, on a common suite of benchmarks and a real-world hardware design application. We also show that PrBO converges faster even if the user priors are not entirely accurate and that it robustly recovers from misleading priors.
Optimization of Smooth Functions with Noisy Observations: Local Minimax Rates2018-03-22   ${\displaystyle \cong }$
We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy evaluations of function values at the queried points (i.e. the algorithm has access to zeroth-order information). Optimization performance is evaluated by the expected difference of function values at the estimated optimum and the true optimum. In contrast to the classical optimization setup, first-order information like gradients are not directly accessible to the optimization algorithm. We show that the classical minimax framework of analysis, which roughly characterizes the worst-case query complexity of an optimization algorithm in this setting, leads to excessively pessimistic results. We propose a local minimax framework to study the fundamental difficulty of optimizing smooth functions with adaptive function evaluations, which provides a refined picture of the intrinsic difficulty of zeroth-order optimization. We show that for functions with fast level set growth around the global minimum, carefully designed optimization algorithms can identify a near global minimizer with many fewer queries. For the special case of strongly convex and smooth functions, our implied convergence rates match the ones developed for zeroth-order convex optimization problems. At the other end of the spectrum, for worst-case smooth functions no algorithm can converge faster than the minimax rate of estimating the entire unknown function in the $\ell_\infty$-norm. We provide an intuitive and efficient algorithm that attains the derived upper error bounds.
A Novel Meta-Heuristic Optimization Algorithm Inspired by the Spread of Viruses2020-06-11   ${\displaystyle \cong }$
According to the no-free-lunch theorem, there is no single meta-heuristic algorithm that can optimally solve all optimization problems. This motivates many researchers to continuously develop new optimization algorithms. In this paper, a novel nature-inspired meta-heuristic optimization algorithm called virus spread optimization (VSO) is proposed. VSO loosely mimics the spread of viruses among hosts, and can be effectively applied to solving many challenging and continuous optimization problems. We devise a new representation scheme and viral operations that are radically different from previously proposed virus-based optimization algorithms. First, the viral RNA of each host in VSO denotes a potential solution for which different viral operations will help to diversify the searching strategies in order to largely enhance the solution quality. In addition, an imported infection mechanism, inheriting the searched optima from another colony, is introduced to possibly avoid the prematuration of any potential solution in solving complex problems. VSO has an excellent capability to conduct adaptive neighborhood searches around the discovered optima for achieving better solutions. Furthermore, with a flexible infection mechanism, VSO can quickly escape from local optima. To clearly demonstrate both its effectiveness and efficiency, VSO is critically evaluated on a series of well-known benchmark functions. Moreover, VSO is validated on its applicability through two real-world examples including the financial portfolio optimization and optimization of hyper-parameters of support vector machines for classification problems. The results show that VSO has attained superior performance in terms of solution fitness, convergence rate, scalability, reliability, and flexibility when compared to those results of the conventional as well as state-of-the-art meta-heuristic optimization algorithms.