Causal masking is a fundamental component in Generative Pre-Trained Transformer (GPT) models, playing a crucial role during training.Although GPTs can process the entire user prompt at once, the causal masking is applied to all input tokens step-by-step, mimicking the generation process.This imposes an unnecessary constraint during the initial "prefill" phase when the model processes the input prompt and generates the internal representations before producing any output tokens.In this work, attention is masked based on the known block structure at the prefill phase, followed by the conventional token-by-token autoregressive process after that.For example, in a typical chat prompt, the system prompt is treated as one block, and the user prompt as the next one.Each of these is treated as a unit for the purpose of masking, such that the first tokens in each block can access the subsequent tokens in a non-causal manner.Then, the model answer is generated in the conventional causal manner.The Segment-by-Segment scheme entails no additional computational overhead.When integrated using a lightweight fine-tuning into already trained models such as Llama and Qwen, MAS quickly increases models' performances.Our code will be available at: https://github.com/shacharKZ/ MAS-Segment-

We develop a framework to quantify predictive uncertainty in survival analysis, providing a reliable lower predictive bound (LPB) for the true, unknown patient survival time. Recently, conformal prediction has been used to construct such valid LPBs for *type-I right-censored data*, with the guarantee that the bound holds with high probability. Crucially, under the type-I setting, the censoring time is observed for all data points. As such, informative LPBs can be constructed by framing the calibration as an estimation task with covariate shift, relying on the conditionally independent censoring assumption. This paper expands the conformal toolbox for survival analysis, with the goal of handling the ubiquitous *general right-censored setting*, in which either the censoring or survival time is observed, but not both. The key challenge here is that the calibration cannot be directly formulated as a covariate shift problem anymore. Yet, we show how to construct LPBs with distribution-free finite-sample guarantees, under the same assumptions as conformal approaches for type-I censored data. Experiments demonstrate the informativeness and validity of our methods in simulated settings and showcase their practical utility using several real-world datasets.

Conformal prediction is a flexible framework for calibrating machine learning predictions, providing distribution-free statistical guarantees. In outlier detection, this calibration relies on a reference set of labeled inlier data to control the type-I error rate. However, obtaining a perfectly labeled inlier reference set is often unrealistic, and a more practical scenario involves access to a contaminated reference set containing a small fraction of outliers. This paper analyzes the impact of such contamination on the validity of conformal methods. We prove that under realistic, non-adversarial settings, calibration on contaminated data yields conservative type-I error control, shedding light on the inherent robustness of conformal methods. This conservativeness, however, typically results in a loss of power. To alleviate this limitation, we propose a novel, active data-cleaning framework that leverages a limited labeling budget and an outlier detection model to selectively annotate data points in the contaminated reference set that are suspected as outliers. By removing only the annotated outliers in this ``suspicious'' subset, we can effectively enhance power while mitigating the risk of inflating the type-I error rate, as supported by our theoretical analysis. Experiments on real datasets validate the conservative behavior of conformal methods under contamination and show that the proposed data-cleaning strategy improves power without sacrificing validity.

Conformal prediction is a framework for predictive inference with a distribution-free, finite-sample guarantee. However, it tends to provide uninformative prediction sets when calibration data are scarce. This paper introduces Synthetic-powered predictive inference (SPI), a novel framework that incorporates synthetic data---e.g., from a generative model---to improve sample efficiency. At the core of our method is a score transporter: an empirical quantile mapping that aligns nonconformity scores from trusted, real data with those from synthetic data. By carefully integrating the score transporter into the calibration process, SPI provably achieves finite-sample coverage guarantees without making any assumptions about the real and synthetic data distributions. When the score distributions are well aligned, SPI yields substantially tighter and more informative prediction sets than standard conformal prediction. Experiments on image classification---augmenting data with synthetic diffusion-model generated images---and on tabular regression demonstrate notable improvements in predictive efficiency in data-scarce settings.

Prediction-Powered Inference (PPI) is a recently proposed statistical inference technique for parameter estimation that leverages pseudo-labels on both labeled and unlabeled data to construct an unbiased, low-variance estimator. In this work, we extend its core idea to semi-supervised learning (SSL) for model training, introducing a novel unbiased gradient estimator. This extension addresses a key challenge in SSL: while unlabeled data can improve model performance, its benefit heavily depends on the quality of pseudo-labels. Inaccurate pseudo-labels can introduce bias, leading to suboptimal models. To balance the contributions of labeled and pseudo-labeled data, we utilize an interpolation parameter and tune it on the fly, alongside the model parameters, using a one-dimensional online learning algorithm. We verify the practical advantage of our approach through experiments on both synthetic and real datasets, demonstrating improved performance over classic SSL baselines and PPI methods that tune the interpolation parameter offline.

The risk-controlling prediction sets (RCPS) framework is a general tool for transforming the output of any machine learning model to design a predictive rule with rigorous error rate control. The key idea behind this framework is to use labeled hold-out calibration data to tune a hyper-parameter that affects the error rate of the resulting prediction rule. However, the limitation of such a calibration scheme is that with limited hold-out data, the tuned hyper-parameter becomes noisy and leads to a prediction rule with an error rate that is often unnecessarily conservative. To overcome this sample-size barrier, we introduce a semi-supervised calibration procedure that leverages unlabeled data to rigorously tune the hyper-parameter without compromising statistical validity. Our procedure builds upon the prediction-powered inference framework, carefully tailoring it to risk-controlling tasks. We demonstrate the benefits and validity of our proposal through two real-data experiments: few-shot image classification and early time series classification.

Conformal prediction is a powerful tool to generate uncertainty sets with guaranteed coverage using any predictive model, under the assumption that the training and test data are i.i.d.. Recently, it has been shown that adversarial examples are able to manipulate conformal methods to construct prediction sets with invalid coverage rates, as the i.i.d. assumption is violated. To address this issue, a recent work, Randomized Smoothed Conformal Prediction (RSCP), was first proposed to certify the robustness of conformal prediction methods to adversarial noise. However, RSCP has two major limitations: (i) its robustness guarantee is flawed when used in practice and (ii) it tends to produce large uncertainty sets. To address these limitations, we first propose a novel framework called RSCP+ to provide provable robustness guarantee in evaluation, which fixes the issues in the original RSCP method. Next, we propose two novel methods, Post-Training Transformation (PTT) and Robust Conformal Training (RCT), to effectively reduce prediction set size with little computation overhead. Experimental results in CIFAR10, CIFAR100, and ImageNet suggest the baseline method only yields trivial predictions including full label set, while our methods could boost the efficiency by up to $4.36\times$, $5.46\times$, and $16.9\times$ respectively and provide practical robustness guarantee.

Early time classification algorithms aim to label a stream of features without processing the full input stream, while maintaining accuracy comparable to that achieved by applying the classifier to the entire input. In this paper, we introduce a statistical framework that can be applied to any sequential classifier, formulating a calibrated stopping rule. This data-driven rule attains finite-sample, distribution-free control of the accuracy gap between full and early-time classification. We start by presenting a novel method that builds on the Learn-then-Test calibration framework to control this gap marginally, on average over i.i.d. instances. As this algorithm tends to yield an excessively high accuracy gap for early halt times, our main contribution is the proposal of a framework that controls a stronger notion of error, where the accuracy gap is controlled conditionally on the accumulated halt times. Numerical experiments demonstrate the effectiveness, applicability, and usefulness of our method. We show that our proposed early stopping mechanism reduces up to 94% of timesteps used for classification while achieving rigorous accuracy gap control.

We present a novel approach for test-time adaptation via online self-training, consisting of two components. First, we introduce a statistical framework that detects distribution shifts in the classifier's entropy values obtained on a stream of unlabeled samples. Second, we devise an online adaptation mechanism that utilizes the evidence of distribution shifts captured by the detection tool to dynamically update the classifier's parameters. The resulting adaptation process drives the distribution of test entropy values obtained from the self-trained classifier to match those of the source domain, building invariance to distribution shifts. This approach departs from the conventional self-training method, which focuses on minimizing the classifier's entropy. Our approach combines concepts in betting martingales and online learning to form a detection tool capable of quickly reacting to distribution shifts. We then reveal a tight relation between our adaptation scheme and optimal transport, which forms the basis of our novel self-supervised loss. Experimental results demonstrate that our approach improves test-time accuracy under distribution shifts while maintaining accuracy and calibration in their absence, outperforming leading entropy minimization methods across various scenarios.

We develop a method to generate prediction sets with a guaranteed coverage rate that is robust to corruptions in the training data, such as missing or noisy variables. Our approach builds on conformal prediction, a powerful framework to construct prediction sets that are valid under the i.i.d assumption. Importantly, naively applying conformal prediction does not provide reliable predictions in this setting, due to the distribution shift induced by the corruptions. To account for the distribution shift, we assume access to privileged information (PI). The PI is formulated as additional features that explain the distribution shift, however, they are only available during training and absent at test time. We approach this problem by introducing a novel generalization of weighted conformal prediction and support our method with theoretical coverage guarantees. Empirical experiments on both real and synthetic datasets indicate that our approach achieves a valid coverage rate and constructs more informative predictions compared to existing methods, which are not supported by theoretical guarantees.

We study the robustness of conformal prediction, a powerful tool for uncertainty quantification, to label noise. Our analysis tackles both regression and classification problems, characterizing when and how it is possible to construct uncertainty sets that correctly cover the unobserved noiseless ground truth labels. We further extend our theory and formulate the requirements for correctly controlling a general loss function, such as the false negative proportion, with noisy labels. Our theory and experiments suggest that conformal prediction and risk-controlling techniques with noisy labels attain conservative risk over the clean ground truth labels whenever the noise is dispersive and increases variability. In other adversarial cases, we can also correct for noise of bounded size in the conformal prediction algorithm in order to ensure achieving the correct risk of the ground truth labels without score or data regularity.

Uncertainty quantification for inverse problems in imaging has drawn much attention lately. Existing approaches towards this task define uncertainty regions based on probable values per pixel, while ignoring spatial correlations within the image, resulting in an exaggerated volume of uncertainty. In this paper, we propose PUQ (Principal Uncertainty Quantification) - a novel definition and corresponding analysis of uncertainty regions that takes into account spatial relationships within the image, thus providing reduced volume regions. Using recent advancements in generative models, we derive uncertainty intervals around principal components of the empirical posterior distribution, forming an ambiguity region that guarantees the inclusion of true unseen values with a user-defined confidence probability. To improve computational efficiency and interpretability, we also guarantee the recovery of true unseen values using only a few principal directions, resulting in more informative uncertainty regions. Our approach is verified through experiments on image colorization, super-resolution, and inpainting; its effectiveness is shown through comparison to baseline methods, demonstrating significantly tighter uncertainty regions.

This paper develops a model-free sequential test for conditional independence. The proposed test allows researchers to analyze an incoming i.i.d. data stream with any arbitrary dependency structure, and safely conclude whether a feature is conditionally associated with the response under study. We allow the processing of data points online, as soon as they arrive, and stop data acquisition once significant results are detected, rigorously controlling the type-I error rate. Our test can work with any sophisticated machine learning algorithm to enhance data efficiency to the extent possible. The developed method is inspired by two statistical frameworks. The first is the model-X conditional randomization test, a test for conditional independence that is valid in offline settings where the sample size is fixed in advance. The second is testing by betting, a “game-theoretic” approach for sequential hypothesis testing. We conduct synthetic experiments to demonstrate the advantage of our test over out-of-the-box sequential tests that account for the multiplicity of tests in the time horizon, and demonstrate the practicality of our proposal by applying it to real-world tasks.

Conformal prediction is a theoretically grounded framework for constructing predictive intervals. We study conformal prediction with missing values in the covariates – a setting that brings new challenges to uncertainty quantification. We first show that the marginal coverage guarantee of conformal prediction holds on imputed data for any missingness distribution and almost all imputation functions. However, we emphasize that the average coverage varies depending on the pattern of missing values: conformal methods tend to construct prediction intervals that under-cover the response conditionally to some missing patterns. This motivates our novel generalized conformalized quantile regression framework, missing data augmentation, which yields prediction intervals that are valid conditionally to the patterns of missing values, despite their exponential number. We then show that a universally consistent quantile regression algorithm trained on the imputed data is Bayes optimal for the pinball risk, thus achieving valid coverage conditionally to any given data point. Moreover, we examine the case of a linear model, which demonstrates the importance of our proposal in overcoming the heteroskedasticity induced by missing values. Using synthetic and data from critical care, we corroborate our theory and report improved performance of our methods.

Conformal inference provides a general distribution-free method to rigorously calibrate the output of any machine learning algorithm for novelty detection. While this approach has many strengths, it has the limitation of being randomized, in the sense that it may lead to different results when analyzing twice the same data and this can hinder the interpretation of any findings. We propose to make conformal inferences more stable by leveraging suitable conformal e-values instead of p-values to quantify statistical significance. This solution allows the evidence gathered from multiple analyses of the same data to be aggregated effectively while provably controlling the false discovery rate. Further, we show that the proposed method can reduce randomness without much loss of power compared to standard conformal inference, partly thanks to an innovative way of weighting conformal e-values based on additional side information carefully extracted from the same data. Simulations with synthetic and real data confirm this solution can be effective at eliminating random noise in the inferences obtained with state-of-the-art alternative techniques, sometimes also leading to higher power.

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

Image-to-image regression is an important learning task, used frequently in biological imaging. Current algorithms, however, do not generally offer statistical guarantees that protect against a model’s mistakes and hallucinations. To address this, we develop uncertainty quantification techniques with rigorous statistical guarantees for image-to-image regression problems. In particular, we show how to derive uncertainty intervals around each pixel that are guaranteed to contain the true value with a user-specified confidence probability. Our methods work in conjunction with any base machine learning model, such as a neural network, and endow it with formal mathematical guarantees{—}regardless of the true unknown data distribution or choice of model. Furthermore, they are simple to implement and computationally inexpensive. We evaluate our procedure on three image-to-image regression tasks: quantitative phase microscopy, accelerated magnetic resonance imaging, and super-resolution transmission electron microscopy of a Drosophila melanogaster brain.

Double machine learning is a statistical method for leveraging complex black-box models to construct approximately unbiased treatment effect estimates given observational data with high-dimensional covariates, under the assumption of a partially linear model. The idea is to first fit on a subset of the samples two non-linear predictive models, one for the continuous outcome of interest and one for the observed treatment, and then to estimate a linear coefficient for the treatment using the remaining samples through a simple orthogonalized regression. While this methodology is flexible and can accommodate arbitrary predictive models, typically trained independently of one another, this paper argues that a carefully coordinated learning algorithm for deep neural networks may reduce the estimation bias. The improved empirical performance of the proposed method is demonstrated through numerical experiments on both simulated and real data.

We propose a new conditional dependence measure and a statistical test for conditional independence. The measure is based on the difference between analytic kernel embeddings of two well-suited distributions evaluated at a finite set of locations. We obtain its asymptotic distribution under the null hypothesis of conditional independence and design a consistent statistical test from it. We conduct a series of experiments showing that our new test outperforms state-of-the-art methods both in terms of type-I and type-II errors even in the high dimensional setting.

Deep neural networks are powerful tools to detect hidden patterns in data and leverage them to make predictions, but they are not designed to understand uncertainty and estimate reliable probabilities. In particular, they tend to be overconfident. We begin to address this problem in the context of multi-class classification by developing a novel training algorithm producing models with more dependable uncertainty estimates, without sacrificing predictive power. The idea is to mitigate overconfidence by minimizing a loss function, inspired by advances in conformal inference, that quantifies model uncertainty by carefully leveraging hold-out data. Experiments with synthetic and real data demonstrate this method can lead to smaller conformal prediction sets with higher conditional coverage, after exact calibration with hold-out data, compared to state-of-the-art alternatives.

Meaningful uncertainty quantification in computer vision requires reasoning about semantic information---say, the hair color of the person in a photo or the location of a car on the street. To this end, recent breakthroughs in generative modeling allow us to represent semantic information in disentangled latent spaces, but providing uncertainties on the semantic latent variables has remained challenging. In this work, we provide principled uncertainty intervals that are guaranteed to contain the true semantic factors for any underlying generative model. The method does the following: (1) it uses quantile regression to output a heuristic uncertainty interval for each element in the latent space (2) calibrates these uncertainties such that they contain the true value of the latent for a new, unseen input. The endpoints of these calibrated intervals can then be propagated through the generator to produce interpretable uncertainty visualizations for each semantic factor. This technique reliably communicates semantically meaningful, principled, and instance-adaptive uncertainty in inverse problems like image super-resolution and image completion. Project page: https://swamiviv.github.io/semanticuncertaintyintervals/

We develop a method to generate prediction intervals that have a user-specified coverage level across all regions of feature-space, a property called conditional coverage. A typical approach to this task is to estimate the conditional quantiles with quantile regression---it is well-known that this leads to correct coverage in the large-sample limit, although it may not be accurate in finite samples. We find in experiments that traditional quantile regression can have poor conditional coverage. To remedy this, we modify the loss function to promote independence between the size of the intervals and the indicator of a miscoverage event. For the true conditional quantiles, these two quantities are independent (orthogonal), so the modified loss function continues to be valid. Moreover, we empirically show that the modified loss function leads to improved conditional coverage, as evaluated by several metrics. We also introduce two new metrics that check conditional coverage by looking at the strength of the dependence between the interval size and the indicator of miscoverage.

This paper develops a conformal method to compute prediction intervals for non-parametric regression that can automatically adapt to skewed data. Leveraging black-box machine learning algorithms to estimate the conditional distribution of the outcome using histograms, it translates their output into the shortest prediction intervals with approximate conditional coverage. The resulting prediction intervals provably have marginal coverage in finite samples, while asymptotically achieving conditional coverage and optimal length if the black-box model is consistent. Numerical experiments with simulated and real data demonstrate improved performance compared to state-of-the-art alternatives, including conformalized quantile regression and other distributional conformal prediction approaches.

We present a flexible framework for learning predictive models that approximately satisfy the equalized odds notion of fairness. This is achieved by introducing a general discrepancy functional that rigorously quantifies violations of this criterion. This differentiable functional is used as a penalty driving the model parameters towards equalized odds. To rigorously evaluate fitted models, we develop a formal hypothesis test to detect whether a prediction rule violates this property, the first such test in the literature. Both the model fitting and hypothesis testing leverage a resampled version of the sensitive attribute obeying equalized odds, by construction. We demonstrate the applicability and validity of the proposed framework both in regression and multi-class classification problems, reporting improved performance over state-of-the-art methods. Lastly, we show how to incorporate techniques for equitable uncertainty quantification---unbiased for each group under study---to communicate the results of the data analysis in exact terms.

Conformal inference, cross-validation+, and the jackknife+ are hold-out methods that can be combined with virtually any machine learning algorithm to construct prediction sets with guaranteed marginal coverage. In this paper, we develop specialized versions of these techniques for categorical and unordered response labels that, in addition to providing marginal coverage, are also fully adaptive to complex data distributions, in the sense that they perform favorably in terms of approximate conditional coverage compared to alternative methods. The heart of our contribution is a novel conformity score, which we explicitly demonstrate to be powerful and intuitive for classification problems, but whose underlying principle is potentially far more general. Experiments on synthetic and real data demonstrate the practical value of our theoretical guarantees, as well as the statistical advantages of the proposed methods over the existing alternatives.

Conformal prediction is a technique for constructing prediction intervals that attain valid coverage in finite samples, without making distributional assumptions. Despite this appeal, existing conformal methods can be unnecessarily conservative because they form intervals of constant or weakly varying length across the input space. In this paper we propose a new method that is fully adaptive to heteroscedasticity. It combines conformal prediction with classical quantile regression, inheriting the advantages of both. We establish a theoretical guarantee of valid coverage, supplemented by extensive experiments on popular regression datasets. We compare the efficiency of conformalized quantile regression to other conformal methods, showing that our method tends to produce shorter intervals.

Single image super-resolution (SISR) aims to recover a high-resolution image from a given low-resolution version of it. Video super-resolution (VSR) targets a series of given images, aiming to fuse them to create a higher resolution outcome. Although SISR and VSR seem to have a lot in common, most SISR algorithms do not have a simple and direct extension to VSR. VSR is considered a more challenging inverse problem, mainly due to its reliance on a sub-pixel accurate motion-estimation, which has no parallel in SISR. Another complication is the dynamics of the video, often addressed by simply generating a single frame instead of a complete output sequence. In this paper, we suggest a simple and robust super-resolution framework that can be applied to single images and easily extended to video. Our work relies on the observation that denoising of images and videos is well-managed and very effectively treated by a variety of methods. We exploit the plug-and-play-prior framework and the regularization-by-denoising (RED) approach that extends it, and show how to use such denoisers in order to handle the SISR and the VSR problems using a unified formulation and framework. This way, we benefit from the effectiveness and efficiency of existing image/video denoising algorithms, while solving much more challenging problems. More specifically, harnessing the VBM3D video denoiser, we obtain a strongly competitive motion-estimation free VSR algorithm, showing tendency to a high-quality output and fast processing.

Image denoising is the most fundamental image enhancement task, and many algorithms have been proposed over the years for its solution. Interestingly, such an image denoising “engine” can be used to solve general inverse problems. Indeed, in our recent work we have presented the Regularization by Denoising (RED) framework: using a denoising engine in defining the regularization of any inverse problem. We have shown how this scheme leads to well-founded iterative algorithms in which the denoiser is applied in each iteration. In this work we describe how a learned version of RED defines a novel convolutional neural network architecture, where the commonly used point-wise nonlinearities are replaced by a denoising engine. We show how this network can be optimized end- to-end using a back - propagation that relies on guided denoising algorithms. As a case-study, we concentrate on the image deblurring problem and show the superiority of the trainable variant of RED over its analytic form.

The recently proposed Multi-Layer Convolutional Sparse Coding (ML-CSC) model, consisting of a cascade of convolutional sparse layers, provides a new interpretation of Convolutional Neural Networks (CNNs). Under this framework, the forward pass in a CNN is equivalent to an algorithm that estimates nested sparse representation vectors from a given input signal. Despite having served as a pivotal connection between CNNs and sparse modeling, it is still unclear how to develop pursuit algorithms that serve this model exactly. In this work, we propose a new pursuit formulation by adopting a projection approach. We provide new and improved bounds on the stability of the resulting convolutional sparse representations, and we propose a multi-layer projection algorithm to retrieve them. We demonstrate this algorithm numerically, showing that it is superior to the Layered Basis Pursuit alternative in retrieving the representations of signals belonging to the ML-CSC model.

Image and texture synthesis is a challenging task that has long been drawing attention in the fields of image processing, graphics, and machine learning. This problem consists of modeling the desired type of images, either through training examples or via a parametric modeling, and then generating images that belong to the same statistical origin. This paper addresses the image synthesis task, focusing on two specific families of images-handwritten digits and face images. This paper offers two main contributions. First, we suggest a simple and intuitive algorithm capable of generating such images in a unified way. The proposed approach taken is pyramidal, consisting of upscaling and refining the estimated image several times. For each upscaling stage, the algorithm randomly draws small patches from a patch database and merges these to form a coherent and novel image with high visual quality. The second contribution is a general framework for the evaluation of the generation performance, which combines three aspects: the likelihood, the originality, and the spread of the synthesized images. We assess the proposed synthesis scheme and show that the results are similar in nature, and yet different from the ones found in the training set, suggesting that true synthesis effect has been obtained.

Convolutional sparse coding is an increasingly popular model in the signal and image processing communities, tackling some of the limitations of traditional patch-based sparse representations. Although several works have addressed the dictionary learning problem under this model, these relied on an ADMM formulation in the Fourier domain, losing the sense of locality and the relation to the traditional patch-based sparse pursuit. A recent work suggested a novel theoretical analysis of this global model, providing guarantees that rely on a localized sparsity measure. Herein, we extend this local-global relation by showing how one can efficiently solve the convolutional sparse pursuit problem and train the filters involved, while operating locally on image patches. Our approach provides an intuitive algorithm that can leverage standard techniques from the sparse representations field. The proposed method is fast to train, simple to implement, and flexible enough that it can be easily deployed in a variety of applications. We demonstrate the proposed training scheme for image inpainting and image separation, achieving state-of-the-art results.

Convolutional neural networks (CNN) have led to many state-of- the-art results spanning through various fields. However, a clear and profound theoretical understanding of the forward pass, the core algorithm of CNN, is still lacking. In parallel, within the wide field of sparse approximation, Convolutional Sparse Coding (CSC) has gained increasing attention in recent years. A theoretical study of this model was recently conducted, establishing it as a reliable and stable alternative to the commonly practiced patch-based processing. Herein, we propose a novel multi-layer model, ML-CSC, in which signals are assumed to emerge from a cascade of CSC layers. This is shown to be tightly connected to CNN, so much so that the forward pass of the CNN is in fact the thresholding pursuit serving the ML-CSC model. This connection brings a fresh view to CNN, as we are able to attribute to this architecture theoretical claims such as uniqueness of the representations throughout the network, and their stable estimation, all guaranteed under simple local sparsity conditions. Lastly, identifying the weaknesses in the above pursuit scheme, we propose an alternative to the forward pass, which is connected to deconvolutional and recurrent networks, and also has better theoretical guarantees.

Removal of noise from an image is an extensively studied problem in image processing. Indeed, the recent advent of sophisticated and highly effective denoising algorithms has led some to believe that existing methods are touching the ceiling in terms of noise removal performance. Can we leverage this impressive achievement to treat other tasks in image processing? Recent work has answered this question positively, in the form of the Plug-and-Play Prior ($P^3$) method, showing that any inverse problem can be handled by sequentially applying image denoising steps. This relies heavily on the ADMM optimization technique in order to obtain this chained denoising interpretation. Is this the only way in which tasks in image processing can exploit the image denoising engine? In this paper we provide an alternative, more powerful, and more flexible framework for achieving the same goal. As opposed to the $P^3$ method, we offer Regularization by Denoising (RED): using the denoising engine in defining the regularization of the inverse problem. We propose an explicit image-adaptive Laplacian-based regularization functional, making the overall objective functional clearer and better defined. With a complete flexibility to choose the iterative optimization procedure for minimizing the above functional, RED is capable of incorporating any image denoising algorithm, can treat general inverse problems very effectively, and is guaranteed to converge to the globally optimal result. We test this approach and demonstrate state-of-the-art results in the image deblurring and super-resolution problems.

Denoising and Super-Resolution are two inverse problems that have been extensively studied. Over the years, these two tasks were treated as two distinct problems that deserve a different algorithmic solution. In this paper we wish to exploit the recently introduced Plug-and-Play Prior (PPP) approach to connect between the two. Using the PPP, we turn leading denoisers into super-resolution solvers. As a case-study we demonstrate this on the NCSR algorithm, which has two variants: one for denoising and one for superresolution. We show that by using the NCSR denoiser, one can get equal or even better results when compared with the NCSR super-resolution.

Measuring the similarity between the patches in images is a fundamental building block in various tasks. Naturally, the patch size has a major impact on the matching quality and on the consequent application performance. Under the assumption that our patch database is sufficiently sampled, using large patches (e.g., 21 × 21 ) should be preferred over small ones (e.g., 7 × 7 ). However, this dense-sampling assumption is rarely true; in most cases, large patches cannot find relevant nearby examples. This phenomenon is a consequence of the curse of dimensionality, stating that the database size should grow exponentially with the patch size to ensure proper matches. This explains the favored choice of small patch size in most applications. Is there a way to keep the simplicity and work with small patches while getting some of the benefits that large patches provide? In this paper, we offer such an approach. We propose to concatenate the regular content of a conventional (small) patch with a compact representation of its (large) surroundings-its context. Therefore, with a minor increase of the dimensions (e.g., with additional ten values to the patch representation), we implicitly/softly describe the information of a large patch. The additional descriptors are computed based on a self-similarity behavior of the patch surrounding. We show that this approach achieves better matches, compared with the use of conventional-size patches, without the need to increase the database-size. Also, the effectiveness of the proposed method is tested on three distinct problems: 1) external natural image denoising; 2) depth image super-resolution; and 3) motion-compensated frame-rate up conversion.

In this paper we propose a way to improve the K-SVD image denoising algorithm. The suggested method aims to reduce the gap that exists between the local processing (sparse-coding of overlapping patches) and the global image recovery (obtained by averaging the overlapping patches). Inspired by game-theory ideas, we define a disagreement-patch as the difference between the intermediate locally denoised patch and its corresponding part in the final outcome. Our algorithm iterates the denoising process several times, applied on modified patches. Those are obtained by subtracting the disagreement-patches from their corresponding input noisy ones, thus pushing the overlapping patches towards an agreement. Experimental results demonstrate the improvement this algorithm leads to.

In this paper we propose a generic recursive algorithm for improving image denoising methods. Given the initial denoised image, we suggest repeating the following “SOS” procedure: (i) Strengthen the signal by adding the previous denoised image to the degraded input image, (ii) Operate the denoising method on the strengthened image, and (iii) Subtract the previous denoised image from the restored signal-strengthened outcome. The convergence of this process is studied for the K-SVD image denoising and related algorithms. Still in the context of K-SVD image denoising, we introduce an interesting interpretation of the SOS algorithm as a technique for closing the gap between the local patch-modeling and the global restoration task, thereby leading to improved performance. In a quest for the theoretical origin of the SOS algorithm, we provide a graph-based interpretation of our method, where the SOS recursive update effectively minimizes a penalty function that aims to denoise the image, while being regularized by the graph Laplacian. We demonstrate the SOS boosting algorithm for several leading denoising methods (K-SVD, NLM, BM3D, and EPLL), showing its tendency to further improve denoising performance.

Single image interpolation is a central and extensively studied problem in image processing. A common approach toward the treatment of this problem in recent years is to divide the given image into overlapping patches and process each of them based on a model for natural image patches. Adaptive sparse representation modeling is one such promising image prior, which has been shown to be powerful in filling-in missing pixels in an image. Another force that such algorithms may use is the self-similarity that exists within natural images. Processing groups of related patches together exploits their correspondence, leading often times to improved results. In this paper, we propose a novel image interpolation method, which combines these two forces-nonlocal self-similarities and sparse representation modeling. The proposed method is contrasted with competitive and related algorithms, and demonstrated to achieve state-of-the-art results.

Various patch-based image denoising algorithms have been shown to be very effective. Nevertheless, in most cases the difference between the noisy image and its denoised version (called “method-noise”) still contains traces of the original image content. In this paper we propose a novel technique for improving the K-SVD denoising results. Our scheme starts by applying the K-SVD on the given noisy image. Then, for each patch, we recover the “stolen” image content information from the method-noise by performing iterations of de-noising using the same atoms that represent the first-stage de-noised patch. Experimental results demonstrate the efficiency of this technique.