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We examine a family of discrete probability distributions that describes the "spillage number" in the extended balls-in-bins model. The spillage number is defined as the number of balls that occupy their bins minus the total number of occupied bins. This probability distribution can be characterised as a normed version of the expansion of the noncentral Stirling numbers of the second kind in terms of the central Stirling numbers of the second kind. Alternatively it can be derived in a natural way from the extended balls-in-bins model. We derive the generating functions for this distribution and important moments of the distribution. We also derive an algorithm for recursive computation of the mass values for the distribution. Finally, we examine the asymptotic behaviour of the spillage distribution and the performance of an approximation to the distribution.

Astrometry -- the precise measurement of positions and motions of celestial objects -- has emerged as a promising avenue for characterizing the dark matter population in our Galaxy. By leveraging recent advances in simulation-based inference and neural network architectures, we introduce a novel method to search for global dark matter-induced gravitational lensing signatures in astrometric datasets. Our method based on neural likelihood-ratio estimation shows significantly enhanced sensitivity to a cold dark matter population and more favorable scaling with measurement noise compared to existing approaches based on two-point correlation statistics. We demonstrate the real-world viability of our method by showing it to be robust to non-trivial modeled as well as unmodeled noise features expected in astrometric measurements. This establishes machine learning as a powerful tool for characterizing dark matter using astrometric data.

Various approaches have been proposed for out-of-distribution (OOD) detection by augmenting models, input examples, training sets, and optimization objectives. Deviating from existing work, we have a simple hypothesis that standard off-the-shelf models may already contain sufficient information about the training set distribution which can be leveraged for reliable OOD detection. Our empirical study on validating this hypothesis, which measures the model activation's mean for OOD and in-distribution (ID) mini-batches, surprisingly finds that activation means of OOD mini-batches consistently deviate more from those of the training data. In addition, training data's activation means can be computed offline efficiently or retrieved from batch normalization layers as a `free lunch'. Based upon this observation, we propose a novel metric called Neural Mean Discrepancy (NMD), which compares neural means of the input examples and training data. Leveraging the simplicity of NMD, we propose an efficient OOD detector that computes neural means by a standard forward pass followed by a lightweight classifier. Extensive experiments show that NMD outperforms state-of-the-art OOD approaches across multiple datasets and model architectures in terms of both detection accuracy and computational cost.

In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from $n$ independent observations of a $p$-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that $n,p\to \infty$ with $p/n$ tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix ${\mathbf R}$ and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of ${\mathbf R}$ and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of ${\mathbf R}$ in a generalized spiked correlation model are analyzed.

Approaches based on deep neural networks have achieved striking performance when testing data and training data share similar distribution, but can significantly fail otherwise. Therefore, eliminating the impact of distribution shifts between training and testing data is crucial for building performance-promising deep models. Conventional methods assume either the known heterogeneity of training data (e.g. domain labels) or the approximately equal capacities of different domains. In this paper, we consider a more challenging case where neither of the above assumptions holds. We propose to address this problem by removing the dependencies between features via learning weights for training samples, which helps deep models get rid of spurious correlations and, in turn, concentrate more on the true connection between discriminative features and labels. Extensive experiments clearly demonstrate the effectiveness of our method on multiple distribution generalization benchmarks compared with state-of-the-art counterparts. Through extensive experiments on distribution generalization benchmarks including PACS, VLCS, MNIST-M, and NICO, we show the effectiveness of our method compared with state-of-the-art counterparts.

We present self-supervised geometric perception (SGP), the first general framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels (e.g., camera poses, rigid transformations). Our first contribution is to formulate geometric perception as an optimization problem that jointly optimizes the feature descriptor and the geometric models given a large corpus of visual measurements (e.g., images, point clouds). Under this optimization formulation, we show that two important streams of research in vision, namely robust model fitting and deep feature learning, correspond to optimizing one block of the unknown variables while fixing the other block. This analysis naturally leads to our second contribution -- the SGP algorithm that performs alternating minimization to solve the joint optimization. SGP iteratively executes two meta-algorithms: a teacher that performs robust model fitting given learned features to generate geometric pseudo-labels, and a student that performs deep feature learning under noisy supervision of the pseudo-labels. As a third contribution, we apply SGP to two perception problems on large-scale real datasets, namely relative camera pose estimation on MegaDepth and point cloud registration on 3DMatch. We demonstrate that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.

We propose an approach for unsupervised adaptation of object detectors from label-rich to label-poor domains which can significantly reduce annotation costs associated with detection. Recently, approaches that align distributions of source and target images using an adversarial loss have been proven effective for adapting object classifiers. However, for object detection, fully matching the entire distributions of source and target images to each other at the global image level may fail, as domains could have distinct scene layouts and different combinations of objects. On the other hand, strong matching of local features such as texture and color makes sense, as it does not change category level semantics. This motivates us to propose a novel approach for detector adaptation based on strong local alignment and weak global alignment. Our key contribution is the weak alignment model, which focuses the adversarial alignment loss on images that are globally similar and puts less emphasis on aligning images that are globally dissimilar. Additionally, we design the strong domain alignment model to only look at local receptive fields of the feature map. We empirically verify the effectiveness of our approach on several detection datasets comprising both large and small domain shifts.

With the development of deep learning, Deep Metric Learning (DML) has achieved great improvements in face recognition. Specifically, the widely used softmax loss in the training process often bring large intra-class variations, and feature normalization is only exploited in the testing process to compute the pair similarities. To bridge the gap, we impose the intra-class cosine similarity between the features and weight vectors in softmax loss larger than a margin in the training step, and extend it from four aspects. First, we explore the effect of a hard sample mining strategy. To alleviate the human labor of adjusting the margin hyper-parameter, a self-adaptive margin updating strategy is proposed. Then, a normalized version is given to take full advantage of the cosine similarity constraint. Furthermore, we enhance the former constraint to force the intra-class cosine similarity larger than the mean inter-class cosine similarity with a margin in the exponential feature projection space. Extensive experiments on Labeled Face in the Wild (LFW), Youtube Faces (YTF) and IARPA Janus Benchmark A (IJB-A) datasets demonstrate that the proposed methods outperform the mainstream DML methods and approach the state-of-the-art performance.

Clustering and classification critically rely on distance metrics that provide meaningful comparisons between data points. We present mixed-integer optimization approaches to find optimal distance metrics that generalize the Mahalanobis metric extensively studied in the literature. Additionally, we generalize and improve upon leading methods by removing reliance on pre-designated "target neighbors," "triplets," and "similarity pairs." Another salient feature of our method is its ability to enable active learning by recommending precise regions to sample after an optimal metric is computed to improve classification performance. This targeted acquisition can significantly reduce computational burden by ensuring training data completeness, representativeness, and economy. We demonstrate classification and computational performance of the algorithms through several simple and intuitive examples, followed by results on real image and medical datasets.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.