Recent advances in quantized compressed sensing and high-dimensional estimation have shown that signal recovery is even feasible under strong non-linear distortions in the observation process. An important characteristic of associated guarantees is uniformity, i.e., recovery succeeds for an entire class of structured signals with a fixed measurement ensemble. However, despite significant results in various special cases, a general understanding of uniform recovery from non-linear observations is still missing. This paper develops a unified approach to this problem under the assumption of i.i.d. sub-Gaussian measurement vectors. Our main result shows that a simple least-squares estimator with any convex constraint can serve as a universal recovery strategy, which is outlier robust and does not require explicit knowledge of the underlying non-linearity. Based on empirical process theory, a key technical novelty is an approximative increment condition that can be implemented for all common types of non-linear models. This flexibility allows us to apply our approach to a variety of problems in non-linear compressed sensing and high-dimensional statistics, leading to several new and improved guarantees. Each of these applications is accompanied by a conceptually simple and systematic proof, which does not rely on any deeper properties of the observation model. On the other hand, known local stability properties can be incorporated into our framework in a plug-and-play manner, thereby implying near-optimal error bounds.
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This paper proposes a new technique based on a non-linear Minmax Detector Based (MDB) filter for image restoration. The aim of image enhancement is to reconstruct the true image from the corrupted image. The process of image acquisition frequently leads to degradation and the quality of the digitized image becomes inferior to the original image. Image degradation can be due to the addition of different types of noise in the original image. Image noise can be modelled of many types and impulse noise is one of them. Impulse noise generates pixels with gray value not consistent with their local neighbourhood. It appears as a sprinkle of both light and dark or only light spots in the image. Filtering is a technique for enhancing the image. Linear filter is the filtering in which the value of an output pixel is a linear combination of neighborhood values, which can produce blur in the image. Thus a variety of smoothing techniques have been developed that are non linear. Median filter is the one of the most popular non-linear filter. When considering a small neighborhood it is highly efficient but for large window and in case of high noise it gives rise to more blurring to image. The Centre Weighted Mean (CWM) filter has got a better average performance over the median filter. However the original pixel corrupted and noise reduction is substantial under high noise condition. Hence this technique has also blurring affect on the image. To illustrate the superiority of the proposed approach, the proposed new scheme has been simulated along with the standard ones and various restored performance measures have been compared.
Applications of Reinforcement Learning (RL), in which agents learn to make a sequence of decisions despite lacking complete information about the latent states of the controlled system, that is, they act under partial observability of the states, are ubiquitous. Partially observable RL can be notoriously difficult -- well-known information-theoretic results show that learning partially observable Markov decision processes (POMDPs) requires an exponential number of samples in the worst case. Yet, this does not rule out the existence of large subclasses of POMDPs over which learning is tractable. In this paper we identify such a subclass, which we call weakly revealing POMDPs. This family rules out the pathological instances of POMDPs where observations are uninformative to a degree that makes learning hard. We prove that for weakly revealing POMDPs, a simple algorithm combining optimism and Maximum Likelihood Estimation (MLE) is sufficient to guarantee polynomial sample complexity. To the best of our knowledge, this is the first provably sample-efficient result for learning from interactions in overcomplete POMDPs, where the number of latent states can be larger than the number of observations.
We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a "quasi-path" by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.
In the interdependent values (IDV) model introduced by Milgrom and Weber [1982], agents have private signals that capture their information about different social alternatives, and the valuation of every agent is a function of all agent signals. While interdependence has been mainly studied for auctions, it is extremely relevant for a large variety of social choice settings, including the canonical setting of public projects. The IDV model is very challenging relative to standard independent private values, and welfare guarantees have been achieved through two alternative conditions known as {\em single-crossing} and {\em submodularity over signals (SOS)}. In either case, the existing theory falls short of solving the public projects setting. Our contribution is twofold: (i) We give a workable characterization of truthfulness for IDV public projects for the largest class of valuations for which such a characterization exists, and term this class \emph{decomposable valuations}; (ii) We provide possibility and impossibility results for welfare approximation in public projects with SOS valuations. Our main impossibility result is that, in contrast to auctions, no universally truthful mechanism performs better for public projects with SOS valuations than choosing a project at random. Our main positive result applies to {\em excludable} public projects with SOS, for which we establish a constant factor approximation similar to auctions. Our results suggest that exclusion may be a key tool for achieving welfare guarantees in the IDV model.
We propose a stochastic conditional gradient method (CGM) for minimizing convex finite-sum objectives formed as a sum of smooth and non-smooth terms. Existing CGM variants for this template either suffer from slow convergence rates, or require carefully increasing the batch size over the course of the algorithm's execution, which leads to computing full gradients. In contrast, the proposed method, equipped with a stochastic average gradient (SAG) estimator, requires only one sample per iteration. Nevertheless, it guarantees fast convergence rates on par with more sophisticated variance reduction techniques. In applications we put special emphasis on problems with a large number of separable constraints. Such problems are prevalent among semidefinite programming (SDP) formulations arising in machine learning and theoretical computer science. We provide numerical experiments on matrix completion, unsupervised clustering, and sparsest-cut SDPs.
When researchers carry out a null hypothesis significance test, it is tempting to assume that a statistically significant result lowers Prob(H0), the probability of the null hypothesis being true. Technically, such a statement is meaningless for various reasons: e.g., the null hypothesis does not have a probability associated with it. However, it is possible to relax certain assumptions to compute the posterior probability Prob(H0) under repeated sampling. We show in a step-by-step guide that the intuitively appealing belief, that Prob(H0) is low when significant results have been obtained under repeated sampling, is in general incorrect and depends greatly on: (a) the prior probability of the null being true; (b) type-I error rate, (c) type-II error rate, and (d) replication of a result. Through step-by-step simulations using open-source code in the R System of Statistical Computing, we show that uncertainty about the null hypothesis being true often remains high despite a significant result. To help the reader develop intuitions about this common misconception, we provide a Shiny app (//danielschad.shinyapps.io/probnull/). We expect that this tutorial will help researchers better understand and judge results from null hypothesis significance tests.
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.
The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.
We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.
Proactive dialogue system is able to lead the conversation to a goal topic and has advantaged potential in bargain, persuasion and negotiation. Current corpus-based learning manner limits its practical application in real-world scenarios. To this end, we contribute to advance the study of the proactive dialogue policy to a more natural and challenging setting, i.e., interacting dynamically with users. Further, we call attention to the non-cooperative user behavior -- the user talks about off-path topics when he/she is not satisfied with the previous topics introduced by the agent. We argue that the targets of reaching the goal topic quickly and maintaining a high user satisfaction are not always converge, because the topics close to the goal and the topics user preferred may not be the same. Towards this issue, we propose a new solution named I-Pro that can learn Proactive policy in the Interactive setting. Specifically, we learn the trade-off via a learned goal weight, which consists of four factors (dialogue turn, goal completion difficulty, user satisfaction estimation, and cooperative degree). The experimental results demonstrate I-Pro significantly outperforms baselines in terms of effectiveness and interpretability.
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