Fitting a linear subspace to a dataset corrupted by outliers is a fundamental problem in machine learning and statistics, primarily known as(Robust) Principal Component Analysis (PCA)[10,2].

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In this appendix, we include all the material missing from the main paper. Moreover, we restate a key result which connects random sampling and submodular maximization. The original version of the theorem was due to Feige et al. In fact, in what follows we exclusively use S and O for their final versions. Before stating the next lemma, let us introduce some notation for the sake of readability.

The target network update frequency (TUF) is a central stabilization mechanism in (deep) Q-learning. However, their selection remains poorly understood and is often treated merely as another tunable hyperparameter rather than as a principled design decision. This work provides a theoretical analysis of target fixing in tabular Q-learning through the lens of approximate dynamic programming. We formulate periodic target updates as a nested optimization scheme in which each outer iteration applies an inexact Bellman optimality operator, approximated by a generic inner loop optimizer. Rigorous theory yields a finite-time convergence analysis for the asynchronous sampling setting, specializing to stochastic gradient descent in the inner loop. Our results deliver an explicit characterization of the bias-variance trade-off induced by the target update period, showing how to optimally set this critical hyperparameter. We prove that constant target update schedules are suboptimal, incurring a logarithmic overhead in sample complexity that is entirely avoidable with adaptive schedules. Our analysis shows that the optimal target update frequency increases geometrically over the course of the learning process.

Self-supervised multi-view stereopsis (MVS) attracts increasing attention for learning dense surface predictions from only a set of images without onerous ground-truth 3D training data for supervision. However, existing methods highly rely on the local photometric consistency, which fails to identify accurately dense correspondence in broad textureless and reflectance areas.In this paper, we show that geometric proximity such as surface connectedness and occlusion boundaries implicitly inferred from images could serve as reliable guidance for pixel-wise multi-view correspondences. With this insight, we present a novel elastic part representation which encodes physically-connected part segmentations with elastically-varying scales, shapes and boundaries. Meanwhile, a self-supervised MVS framework namely ElasticMVS is proposed to learn the representation and estimate per-view depth following a part-aware propagation and evaluation scheme. Specifically, the pixel-wise part representation is trained by a contrastive learning-based strategy, which increases the representation compactness in geometrically concentrated areas and contrasts otherwise. ElasticMVS iteratively optimizes a part-level consistency loss and a surface smoothness loss, based on a set of depth hypotheses propagated from the geometrically concentrated parts. Extensive evaluations convey the superiority of ElasticMVS in the reconstruction completeness and accuracy, as well as the efficiency and scalability. Particularly, for the challenging large-scale reconstruction benchmark, ElasticMVS demonstrates significant performance gain over both the supervised and self-supervised approaches.

We present a new method to reconstruct 3D human body pose and shape by fusing visual features from multiview images captured by uncalibrated cameras. Existing multiview approaches often use spatial camera calibration (intrinsic and extrinsic parameters) to geometrically align and fuse visual features. Despite remarkable performances, the requirement of camera calibration restricted their applicability to real-world scenarios, e.g., reconstruction from social videos with wide-baseline cameras. We address this challenge by leveraging the commonly observed human body as a semantic calibration target, which eliminates the requirement of camera calibration. Specifically, we map per-pixel image features to a canonical body surface coordinate system agnostic to views and poses using dense keypoints (correspondences). This feature mapping allows us to semantically, instead of geometrically, align and fuse visual features from multiview images. We learn a self-attention mechanism to reason about the confidence of visual features across and within views. With fused visual features, a regressor is learned to predict the parameters of a body model. We demonstrate that our calibration-free multiview fusion method reliably reconstructs 3D body pose and shape, outperforming state-of-the-art single view methods with post-hoc multiview fusion, particularly in the presence of non-trivial occlusion, and showing comparable accuracy to multiview methods that require calibration.

Monte Carlo sampling in high-dimensional, low-sample settings is important in many machine learning tasks. We improve current methods for sampling in Euclidean spaces by avoiding independence, and instead consider ways to couple samples. We show fundamental connections to optimal transport theory, leading to novel sampling algorithms, and providing new theoretical grounding for existing strategies. We compare our new strategies against prior methods for improving sample efficiency, including QMC, by studying discrepancy. We explore our findings empirically, and observe benefits of our sampling schemes for reinforcement learning and generative modelling.

The classical formulation of PCA, dating back to Carl F. Gauss, is based on minimizing the sum of

In this appendix, we include all the material missing from the main paper. Moreover, we restate a key result which connects random sampling and submodular maximization. The original version of the theorem was due to Feige et al. In fact, in what follows we exclusively use S and O for their final versions. Before stating the next lemma, let us introduce some notation for the sake of readability.