This is the appendix of the paper "Detecting Abrupt Changes in Sequential Pairwise Comparison It contains two parts: 1. Appendix A for some supplements to numerical results in Sections 4 and 5. 2. A.1 Wild binary segmentation based on likelihood It is based on the so-called CUSUM statistics. It is known that Binary Segmentation is consistent but not optimal (V enkatraman (1992)). Algorithm 3 shows the general framework of WBS algorithm. To get a rough sense of the number and locations of change points, we check the paths of the logarithm of generalized likelihood ratio statistics, which are shown in Figure 2. It should be noted that although In this subsection, we apply the potential competitor, the likelihood-based WBS method (i.e. Appendix A.1, WBS has another tuning parameter Here, we use samples with odd time indices as training data and even time indices as test data.

Conformal prediction provides a powerful framework for constructing distribution-free prediction regions with finite-sample coverage guarantees. While extensively studied in univariate settings, its extension to multi-output problems presents additional challenges, including complex output dependencies and high computational costs, and remains relatively underexplored. In this work, we present a unified comparative study of nine conformal methods with different multivariate base models for constructing multivariate prediction regions within the same framework. This study highlights their key properties while also exploring the connections between them. Additionally, we introduce two novel classes of conformity scores for multi-output regression that generalize their univariate counterparts. These scores ensure asymptotic conditional coverage while maintaining exact finite-sample marginal coverage. One class is compatible with any generative model, offering broad applicability, while the other is computationally efficient, leveraging the properties of invertible generative models. Finally, we conduct a comprehensive empirical evaluation across 13 tabular datasets, comparing all the multi-output conformal methods explored in this work. To ensure a fair and consistent comparison, all methods are implemented within a unified code base.

We develop a new method for creating prediction sets that combines the flexibility of conformal methods with an estimate of the conditional distribution $P_{Y \mid X}$. Most existing methods, such as conformalized quantile regression and probabilistic conformal prediction, only offer marginal coverage guarantees. Our approach extends these methods to achieve conditional coverage, which is essential for many practical applications. While exact conditional guarantees are impossible without assumptions on the data distribution, we provide non-asymptotic bounds that explicitly depend on the quality of the available estimate of the conditional distribution. Our confidence sets are highly adaptive to the local structure of the data, making them particularly useful in high heteroskedasticity situations. We demonstrate the effectiveness of our approach through extensive simulations, showing that it outperforms existing methods in terms of conditional coverage and improves the reliability of statistical inference in a wide range of applications.

Some of the most successful knowledge graph embedding (KGE) models for link prediction -- CP, RESCAL, TuckER, ComplEx -- can be interpreted as energy-based models. Under this perspective they are not amenable for exact maximum-likelihood estimation (MLE), sampling and struggle to integrate logical constraints. This work re-interprets the score functions of these KGEs as circuits -- constrained computational graphs allowing efficient marginalisation. Then, we design two recipes to obtain efficient generative circuit models by either restricting their activations to be non-negative or squaring their outputs. Our interpretation comes with little or no loss of performance for link prediction, while the circuits framework unlocks exact learning by MLE, efficient sampling of new triples, and guarantee that logical constraints are satisfied by design. Furthermore, our models scale more gracefully than the original KGEs on graphs with millions of entities.

Recent advances in self-supervised contrastive learning yield good image-level representation, which favors classification tasks but usually neglects pixel-level detailed information, leading to unsatisfactory transfer performance to dense prediction tasks such as semantic segmentation. In this work, we propose a pixel-wise contrastive learning method called CP2 (Copy-Paste Contrastive Pretraining), which facilitates both image- and pixel-level representation learning and therefore is more suitable for downstream dense prediction tasks. In detail, we copy-paste a random crop from an image (the foreground) onto different background images and pretrain a semantic segmentation model with the objective of 1) distinguishing the foreground pixels from the background pixels, and 2) identifying the composed images that share the same foreground.Experiments show the strong performance of CP2 in downstream semantic segmentation: By finetuning CP2 pretrained models on PASCAL VOC 2012, we obtain 78.6% mIoU with a ResNet-50 and 79.5% with a ViT-S.