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Recent works have explored the fundamental role of depth estimation in multi-view stereo (MVS) and semantic scene completion (SSC). They generally construct 3D cost volumes to explore geometric correspondence in depth, and estimate such volumes in a single step relying directly on the ground truth approximation. However, such problem cannot be thoroughly handled in one step due to complex empirical distributions, especially in challenging regions like occlusions, reflections, etc. In this paper, we formulate the depth estimation task as a multi-step distribution approximation process, and introduce a new paradigm of modeling the Volumetric Probability Distribution progressively (step-by-step) following a Markov chain with Diffusion models (VPDD). Specifically, to constrain the multi-step generation of volume in VPDD, we construct a meta volume guidance and a confidence-aware contextual guidance as conditional geometry priors to facilitate the distribution approximation. For the sampling process, we further investigate an online filtering strategy to maintain consistency in volume representations for stable training. Experiments demonstrate that our plug-and-play VPDD outperforms the state-of-the-arts for tasks of MVS and SSC, and can also be easily extended to different baselines to get improvement. It is worth mentioning that we are the first camera-based work that surpasses LiDAR-based methods on the SemanticKITTI dataset.

Many machine learning tasks can be formulated as a stochastic compositional optimization (SCO) problem such as reinforcement learning, AUC maximization, and meta-learning, where the objective function involves a nested composition associated with an expectation. While a significant amount of studies has been devoted to studying the convergence behavior of SCO algorithms, there is little work on understanding their generalization, i.e., how these learning algorithms built from training examples would behave on future test examples. In this paper, we provide the stability and generalization analysis of stochastic compositional gradient descent algorithms through the lens of algorithmic stability in the framework of statistical learning theory. Firstly, we introduce a stability concept called compositional uniform stability and establish its quantitative relation with generalization for SCO problems. Then, we establish the compositional uniform stability results for two popular stochastic compositional gradient descent algorithms, namely SCGD and SCSC. Finally, we derive dimension-independent excess risk bounds for SCGD and SCSC by trade-offing their stability results and optimization errors. To the best of our knowledge, these are the first-ever-known results on stability and generalization analysis of stochastic compositional gradient descent algorithms.

Shared Mobility Services (SMS), e.g., Demand-Responsive Transit (DRT) or ride-sharing, can improve mobility in low-density areas, often poorly served by conventional Public Transport (PT). Such improvement is mostly quantified via basic performance indicators, like wait or travel time. However, accessibility indicators, measuring the ease of reaching surrounding opportunities (e.g., jobs, schools, shops, ...), would be a more comprehensive indicator. To date, no method exists to quantify the accessibility of SMS based on empirical measurements. Indeed, accessibility is generally computed on graph representations of PT networks, but SMS are dynamic and do not follow a predefined network. We propose a spatial-temporal statistical method that takes as input observed trips of a SMS acting as a feeder for PT and summarized such trips in a graph. On such a graph, we compute classic accessibility indicators. We apply our method to a MATSim simulation study concerning DRT in Paris-Saclay.

Time-series datasets are central in numerous fields of science and engineering, such as biomedicine, Earth observation, and network analysis. Extensive research exists on state-space models (SSMs), which are powerful mathematical tools that allow for probabilistic and interpretable learning on time series. Estimating the model parameters in SSMs is arguably one of the most complicated tasks, and the inclusion of prior knowledge is known to both ease the interpretation but also to complicate the inferential tasks. Very recent works have attempted to incorporate a graphical perspective on some of those model parameters, but they present notable limitations that this work addresses. More generally, existing graphical modeling tools are designed to incorporate either static information, focusing on statistical dependencies among independent random variables (e.g., graphical Lasso approach), or dynamic information, emphasizing causal relationships among time series samples (e.g., graphical Granger approaches). However, there are no joint approaches combining static and dynamic graphical modeling within the context of SSMs. This work proposes a novel approach to fill this gap by introducing a joint graphical modeling framework that bridges the static graphical Lasso model and a causal-based graphical approach for the linear-Gaussian SSM. We present DGLASSO (Dynamic Graphical Lasso), a new inference method within this framework that implements an efficient block alternating majorization-minimization algorithm. The algorithm's convergence is established by departing from modern tools from nonlinear analysis. Experimental validation on synthetic and real weather variability data showcases the effectiveness of the proposed model and inference algorithm.

We establish a framework of random inverse problems with real-time observations over graphs, and present a decentralized online learning algorithm based on online data streams, which unifies the distributed parameter estimation in Hilbert space and the least mean square problem in reproducing kernel Hilbert space (RKHS-LMS). We transform the algorithm convergence into the asymptotic stability of randomly time-varying difference equations in Hilbert space with L2-bounded martingale difference terms and develop the L2 -asymptotic stability theory. It is shown that if the network graph is connected and the sequence of forward operators satisfies the infinitedimensional spatio-temporal persistence of excitation condition, then the estimates of all nodes are mean square and almost surely strongly consistent. By equivalently transferring the distributed learning problem in RKHS to the random inverse problem over graphs, we propose a decentralized online learning algorithm in RKHS based on non-stationary and non-independent online data streams, and prove that the algorithm is mean square and almost surely strongly consistent if the operators induced by the random input data satisfy the infinite-dimensional spatio-temporal persistence of excitation condition.

Mutual coherence is a measure of similarity between two opinions. Although the notion comes from philosophy, it is essential for a wide range of technologies, e.g., the Wahl-O-Mat system. In Germany, this system helps voters to find candidates that are the closest to their political preferences. The exact computation of mutual coherence is highly time-consuming due to the iteration over all subsets of an opinion. Moreover, for every subset, an instance of the SAT model counting problem has to be solved which is known to be a hard problem in computer science. This work is the first study to accelerate this computation. We model the distribution of the so-called confirmation values as a mixture of three Gaussians and present efficient heuristics to estimate its model parameters. The mutual coherence is then approximated with the expected value of the distribution. Some of the presented algorithms are fully polynomial-time, others only require solving a small number of instances of the SAT model counting problem. The average squared error of our best algorithm lies below 0.0035 which is insignificant if the efficiency is taken into account. Furthermore, the accuracy is precise enough to be used in Wahl-O-Mat-like systems.

Statistical data by their very nature are indeterminate in the sense that if one repeated the process of collecting the data the new data set would be somewhat different from the original. Therefore, a statistical method, a map $\Phi$ taking a data set $x$ to a point in some space F, should be stable at $x$: Small perturbations in $x$ should result in a small change in $\Phi(x)$. Otherwise, $\Phi$ is useless at $x$ or -- and this is important -- near $x$. So one doesn't want $\Phi$ to have "singularities," data sets $x$ such that the the limit of $\Phi(y)$ as $y$ approaches $x$ doesn't exist. (Yes, the same issue arises elsewhere in applied math.) However, broad classes of statistical methods have topological obstructions of continuity: They must have singularities. We show why and give lower bounds on the Hausdorff dimension, even Hausdorff measure, of the set of singularities of such data maps. There seem to be numerous examples. We apply mainly topological methods to study the (topological) singularities of functions defined (on dense subsets of) "data spaces" and taking values in spaces with nontrivial homology. At least in this book, data spaces are usually compact manifolds. The purpose is to gain insight into the numerical conditioning of statistical description, data summarization, and inference and learning methods. We prove general results that can often be used to bound below the dimension of the singular set. We apply our topological results to develop lower bounds on Hausdorff measure of the singular set. We apply these methods to the study of plane fitting and measuring location of data on spheres. This is not a "final" version, merely another attempt.

Foundation models pretrained on diverse data at scale have demonstrated extraordinary capabilities in a wide range of vision and language tasks. When such models are deployed in real world environments, they inevitably interface with other entities and agents. For example, language models are often used to interact with human beings through dialogue, and visual perception models are used to autonomously navigate neighborhood streets. In response to these developments, new paradigms are emerging for training foundation models to interact with other agents and perform long-term reasoning. These paradigms leverage the existence of ever-larger datasets curated for multimodal, multitask, and generalist interaction. Research at the intersection of foundation models and decision making holds tremendous promise for creating powerful new systems that can interact effectively across a diverse range of applications such as dialogue, autonomous driving, healthcare, education, and robotics. In this manuscript, we examine the scope of foundation models for decision making, and provide conceptual tools and technical background for understanding the problem space and exploring new research directions. We review recent approaches that ground foundation models in practical decision making applications through a variety of methods such as prompting, conditional generative modeling, planning, optimal control, and reinforcement learning, and discuss common challenges and open problems in the field.

Imitation learning aims to extract knowledge from human experts' demonstrations or artificially created agents in order to replicate their behaviors. Its success has been demonstrated in areas such as video games, autonomous driving, robotic simulations and object manipulation. However, this replicating process could be problematic, such as the performance is highly dependent on the demonstration quality, and most trained agents are limited to perform well in task-specific environments. In this survey, we provide a systematic review on imitation learning. We first introduce the background knowledge from development history and preliminaries, followed by presenting different taxonomies within Imitation Learning and key milestones of the field. We then detail challenges in learning strategies and present research opportunities with learning policy from suboptimal demonstration, voice instructions and other associated optimization schemes.

It is a common paradigm in object detection frameworks to treat all samples equally and target at maximizing the performance on average. In this work, we revisit this paradigm through a careful study on how different samples contribute to the overall performance measured in terms of mAP. Our study suggests that the samples in each mini-batch are neither independent nor equally important, and therefore a better classifier on average does not necessarily mean higher mAP. Motivated by this study, we propose the notion of Prime Samples, those that play a key role in driving the detection performance. We further develop a simple yet effective sampling and learning strategy called PrIme Sample Attention (PISA) that directs the focus of the training process towards such samples. Our experiments demonstrate that it is often more effective to focus on prime samples than hard samples when training a detector. Particularly, On the MSCOCO dataset, PISA outperforms the random sampling baseline and hard mining schemes, e.g. OHEM and Focal Loss, consistently by more than 1% on both single-stage and two-stage detectors, with a strong backbone ResNeXt-101.