In practice, the use of rounding is ubiquitous. Although researchers have looked at the implications of rounding continuous random variables, rounding may be applied to functions of discrete random variables as well. For example, to infer on suicide excess deaths after a national emergency, authorities may provide a rounded average of deaths before and after the emergency started. Suicide rates tend to be relatively low around the world and such rounding may seriously affect inference on the change of suicide rate. In this paper, we study the scenario when a rounded to nearest integer average is used as a proxy for a non-negative discrete random variable. Specifically, our interest is in drawing inference on a parameter from the pmf of Y , when we get U = n[Y /n] as a proxy for Y . The probability generating function of U , E(U ), and Var(U ) capture the effect of the coarsening of the support of Y . Also, moments and estimators of distribution parameters are explored for some special cases. We show that under certain conditions, there is little impact from rounding. However, we also find scenarios where rounding can significantly affect statistical inference as demonstrated in three examples. The simple methods we propose are able to partially counter rounding error effects. While for some probability distributions it may be difficult to derive maximum likelihood estimators as a function of U , we provide a framework to obtain an estimator numerically.
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Purpose of Review: To effectively synthesise and analyse multi-robot behaviour, we require formal task-level models which accurately capture multi-robot execution. In this paper, we review modelling formalisms for multi-robot systems under uncertainty, and discuss how they can be used for planning, reinforcement learning, model checking, and simulation. Recent Findings: Recent work has investigated models which more accurately capture multi-robot execution by considering different forms of uncertainty, such as temporal uncertainty and partial observability, and modelling the effects of robot interactions on action execution. Other strands of work have presented approaches for reducing the size of multi-robot models to admit more efficient solution methods. This can be achieved by decoupling the robots under independence assumptions, or reasoning over higher level macro actions. Summary: Existing multi-robot models demonstrate a trade off between accurately capturing robot dependencies and uncertainty, and being small enough to tractably solve real world problems. Therefore, future research should exploit realistic assumptions over multi-robot behaviour to develop smaller models which retain accurate representations of uncertainty and robot interactions; and exploit the structure of multi-robot problems, such as factored state spaces, to develop scalable solution methods.
Since the control of the Lipschitz constant has a great impact on the training stability, generalization, and robustness of neural networks, the estimation of this value is nowadays a real scientific challenge. In this paper we introduce a precise, fast, and differentiable upper bound for the spectral norm of convolutional layers using circulant matrix theory and a new alternative to the Power iteration. Called the Gram iteration, our approach exhibits a superlinear convergence. First, we show through a comprehensive set of experiments that our approach outperforms other state-of-the-art methods in terms of precision, computational cost, and scalability. Then, it proves highly effective for the Lipschitz regularization of convolutional neural networks, with competitive results against concurrent approaches. Code is available at //github.com/blaisedelattre/lip4conv.
Accurate maps are a prerequisite for virtually all autonomous vehicle tasks. Most state-of-the-art maps assume a static world, and therefore dynamic objects are filtered out of the measurements. However, this division ignores movable but non-moving, i.e. semi-static, objects, which are usually recorded in the map and treated as static objects, violating the static world assumption, causing error in the localization. In this paper, we present a method for modeling moving and movable objects for matching the map and the measurements consistently. This reduces the error resulting from inconsistent categorization and treatment of non-static measurements. A semantic segmentation network is used to categorize the measurements into static and semi-static classes, and a background subtraction-based filtering method is used to remove dynamic measurements. Experimental comparison against a state-of-the-art baseline solution using real-world data from Oxford Radar RobotCar data set shows that consistent assumptions over dynamics increase localization accuracy.
When machine learning models are trained continually on a sequence of tasks, they are liable to forget what they learned on previous tasks -- a phenomenon known as catastrophic forgetting. Proposed solutions to catastrophic forgetting tend to involve storing information about past tasks, meaning that memory usage is a chief consideration in determining their practicality. This paper proposes a memory-efficient solution to catastrophic forgetting, improving upon an established algorithm known as orthogonal gradient descent (OGD). OGD utilizes prior model gradients to find weight updates that preserve performance on prior datapoints. However, since the memory cost of storing prior model gradients grows with the runtime of the algorithm, OGD is ill-suited to continual learning over arbitrarily long time horizons. To address this problem, this paper proposes SketchOGD. SketchOGD employs an online sketching algorithm to compress model gradients as they are encountered into a matrix of a fixed, user-determined size. In contrast to existing memory-efficient variants of OGD, SketchOGD runs online without the need for advance knowledge of the total number of tasks, is simple to implement, and is more amenable to analysis. We provide theoretical guarantees on the approximation error of the relevant sketches under a novel metric suited to the downstream task of OGD. Experimentally, we find that SketchOGD tends to outperform current state-of-the-art variants of OGD given a fixed memory budget.
A core principle in statistical learning is that smoothness of target functions allows to break the curse of dimensionality. However, learning a smooth function through Taylor expansions requires enough samples close to one another to get meaningful estimate of high-order derivatives, which seems hard in machine learning problems where the ratio between number of data and input dimension is relatively small. Should we really hope to break the curse of dimensionality based on Taylor expansion estimation? What happens if Taylor expansions are replaced by Fourier or wavelet expansions? By deriving a new lower bound on the generalization error, this paper investigates the role of constants and transitory regimes which are usually not depicted beyond classical learning theory statements while that play a dominant role in practice.
Diffusion models, as a kind of powerful generative model, have given impressive results on image super-resolution (SR) tasks. However, due to the randomness introduced in the reverse process of diffusion models, the performances of diffusion-based SR models are fluctuating at every time of sampling, especially for samplers with few resampled steps. This inherent randomness of diffusion models results in ineffectiveness and instability, making it challenging for users to guarantee the quality of SR results. However, our work takes this randomness as an opportunity: fully analyzing and leveraging it leads to the construction of an effective plug-and-play sampling method that owns the potential to benefit a series of diffusion-based SR methods. More in detail, we propose to steadily sample high-quality SR images from pretrained diffusion-based SR models by solving diffusion ordinary differential equations (diffusion ODEs) with optimal boundary conditions (BCs) and analyze the characteristics between the choices of BCs and their corresponding SR results. Our analysis shows the route to obtain an approximately optimal BC via an efficient exploration in the whole space. The quality of SR results sampled by the proposed method with fewer steps outperforms the quality of results sampled by current methods with randomness from the same pretrained diffusion-based SR model, which means that our sampling method ``boosts'' current diffusion-based SR models without any additional training.
An important issue in medical image processing is to be able to estimate not only the performances of algorithms but also the precision of the estimation of these performances. Reporting precision typically amounts to reporting standard-error of the mean (SEM) or equivalently confidence intervals. However, this is rarely done in medical image segmentation studies. In this paper, we aim to estimate what is the typical confidence that can be expected in such studies. To that end, we first perform experiments for Dice metric estimation using a standard deep learning model (U-net) and a classical task from the Medical Segmentation Decathlon. We extensively study precision estimation using both Gaussian assumption and bootstrapping (which does not require any assumption on the distribution). We then perform simulations for other test set sizes and performance spreads. Overall, our work shows that small test sets lead to wide confidence intervals (e.g. $\sim$8 points of Dice for 20 samples with $\sigma \simeq 10$).
Distance covariance is a widely used statistical methodology for testing the dependency between two groups of variables. Despite the appealing properties of consistency and superior testing power, the testing results of distance covariance are often hard to be interpreted. This paper presents an elementary interpretation of the mechanism of distance covariance through an additive decomposition of correlations formula. Based on this formula, a visualization method is developed to provide practitioners with a more intuitive explanation of the distance covariance score.
Linear discriminant analysis (LDA) has been a useful tool in pattern recognition and data analysis research and practice. While linearity of class boundaries cannot always be expected, nonlinear projections through pre-trained deep neural networks have served to map complex data onto feature spaces in which linear discrimination has served well. The solution to binary LDA is obtained by eigenvalue analysis of within-class and between-class scatter matrices. It is well known that the multiclass LDA is solved by an extension to the binary LDA, a generalised eigenvalue problem, from which the largest subspace that can be extracted is of dimension one lower than the number of classes in the given problem. In this paper, we show that, apart from the first of the discriminant directions, the generalised eigenanalysis solution to multiclass LDA does neither yield orthogonal discriminant directions nor maximise discrimination of projected data along them. Surprisingly, to the best of our knowledge, this has not been noted in decades of literature on LDA. To overcome this drawback, we present a derivation with a strict theoretical support for sequentially obtaining discriminant directions that are orthogonal to previously computed ones and maximise in each step the Fisher criterion. We show distributions of projections along these axes and demonstrate that discrimination of data projected onto these discriminant directions has optimal separation, which is much higher than those from the generalised eigenvectors of the multiclass LDA. Using a wide range of benchmark tasks, we present a comprehensive empirical demonstration that on a number of pattern recognition and classification problems, the optimal discriminant subspaces obtained by the proposed method, referred to as GO-LDA (Generalised Optimal LDA), can offer superior accuracy.
Scientific machine learning has become an increasingly important tool in materials science and engineering. It is particularly well suited to tackle material problems involving many variables or to allow rapid construction of surrogates of material models, to name just a few. Mathematically, many problems in materials science and engineering can be cast as variational problems. However, handling of uncertainty, ever present in materials, in the context of variational formulations remains challenging for scientific machine learning. In this article, we propose a deep-learning-based numerical method for solving variational problems under uncertainty. Our approach seamlessly combines deep-learning approximation with Monte-Carlo sampling. The resulting numerical method is powerful yet remarkably simple. We assess its performance and accuracy on a number of variational problems.
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