Uncertainty Quantification (UQ) is essential for creating trustworthy machine learning models. Recent years have seen a steep rise in UQ methods that can flag suspicious examples, however, it is often unclear what exactly these methods identify. In this work, we propose a framework for categorizing uncertain examples flagged by UQ methods in classification tasks. We introduce the confusion density matrix--a kernel-based approximation of the misclassification density--and use this to categorize suspicious examples identified by a given uncertainty method into three classes: out-of-distribution (OOD) examples, boundary (Bnd) examples, and examples in regions of high in-distribution misclassification (IDM). Through extensive experiments, we show that our framework provides a new and distinct perspective for assessing differences between uncertainty quantification methods, thereby forming a valuable assessment benchmark.

Word meaning may change over time as a reflection of changes in human society. Therefore, modeling time in word representation is necessary for some diachronic tasks. Most existing diachronic word representation approaches train the embeddings separately for each pre-grouped time-stamped corpus and align these embeddings, e.g., by orthogonal projections, vector initialization, temporal referencing, and compass. However, not only does word meaning change in a short time, word meaning may also be subject to evolution over long timespans, thus resulting in a unified continuous process. A recent approach called'DiffTime' models semantic evolution as functions parameterized by multiple-layer nonlinear neural networks over time. In this paper, we will carry on this line of work by learning explicit functions over time for each word. Our approach, called'Word2Fun', reduces the space complexity from O(TVD) to O(kVD) where kis a small constant (k T). In particular, a specific instance based on polynomial functions could provably approximate any function modeling word evolution with a given negligible error thanks to the Weierstrass Approximation Theorem. The effectiveness of the proposed approach is evaluated in diverse tasks including timeaware word clustering, temporal analogy, and semantic change detection.

In this supplementary material, we present additional details of the proposed Flare7K dataset and experimental settings and show more results. Figure 1: Illustration of a simplified lens system. In the lens and aperture plane, the light passes through the dirty aperture and lens system, leaving a scattering flare on the image plane. In this section, we use a simplified Fourier optics model to illustrate how different kinds of scattering flares occur. A basic lens system can be viewed as a combination of one convex lens, one aperture, and an image plane as shown in Figure 1. We set the optical center as the origin of a coordinate system. Then, the light source's position is (x0,y0, z0). It is a combination of aperture function eAλ(x,y) and a lens function eTL(x,y). Supposing the focus of the lens is f and the lens is ideal. After adjusting the origin of x1 and x2, Equation (11) can be viewed as a standard Fourier transformation. Thus, the point spread function (PSF) which is the square of the amplitude of the image plane's optical field can be written as: PSFλ = |F{eAλ(x,y)}|2. Since stains with depth may bring phase shift for the aperture function, the PSFλ may vary with the wavelength λof the light source.

Artificial lights commonly leave strong lens flare artifacts on images captured at night. Nighttime flare not only affects the visual quality but also degrades the performance of vision algorithms. Existing flare removal methods mainly focus on removing daytime flares and fail in nighttime. Nighttime flare removal is challenging because of the unique luminance and spectrum of artificial lights and the diverse patterns and image degradation of the flares captured at night. The scarcity of nighttime flare removal datasets limits the research on this crucial task.

This supplemental material introduces implementation details, additional comparison experiments, complete proofs and checklist of our proposed model. A.1 Implementation Details Network Architecture: Inspired by [33], we utilize a pre-trained ResNet-50 [20] as the feature extractor for object recognition tasks (i.e., Office-31 [22], Office-Caltech [18] and Office-Home [46]). The penultimate fully-connected layer is replaced with a bottleneck layer and a classifier with weight normalization. Batch normalization is employed to normalize the outputs of bottleneck layer. For digit recognition task (i.e., Digits-Five [41]), we utilize a variant of the LeNet [27] as the feature extractor and classifier.

In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization algorithm. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and which offers strong problem-independent convergence guarantees. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching, without substantially affecting its convergence properties. This approach, called Newton-LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings, not just up to constant factors but even down to lower order terms, for a large class of optimization tasks. In particular, this leads to a new state-of-the-art convergence result for an iterative least squares solver. Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which can be implemented efficiently and which perform well in numerical experiments.