A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. In this paper, we develop a geometric theory that improves and generalises this construction to any manifold. We thereby demonstrate that the completeness result is a direct consequence of the topology of the underlying manifold and the geometry induced by the target measure $P$; there is no need to introduce other structures such as a Riemannian metric, local coordinates, or a reference measure. Instead, our framework relies on the intrinsic geometry of $P$ and in particular its canonical derivative, the deRham rotationnel, which allows us to parametrise the Fokker--Planck currents of measure-preserving diffusions using potentials. The geometric formalism can easily incorporate constraints and symmetries, and deliver new important insights, for example, a new complete recipe of Langevin-like diffusions that are suited to the construction of samplers. We also analyse the reversibility and dissipative properties of the diffusions, the associated deterministic flow on the space of measures, and the geometry of Langevin processes. Our article connects ideas from various literature and frames the theory of measure-preserving diffusions in its appropriate mathematical context.

Gaussian Process Regression is a popular nonparametric regression method based on Bayesian principles that provides uncertainty estimates for its predictions. However, these estimates are of a Bayesian nature, whereas for some important applications, like learning-based control with safety guarantees, frequentist uncertainty bounds are required. Although such rigorous bounds are available for Gaussian Processes, they are too conservative to be useful in applications. This often leads practitioners to replacing these bounds by heuristics, thus breaking all theoretical guarantees. To address this problem, we introduce new uncertainty bounds that are rigorous, yet practically useful at the same time. In particular, the bounds can be explicitly evaluated and are much less conservative than state of the art results. Furthermore, we show that certain model misspecifications lead to only graceful degradation. We demonstrate these advantages and the usefulness of our results for learning-based control with numerical examples.

The potential for machine learning systems to amplify social inequities and unfairness is receiving increasing popular and academic attention. Much recent work has focused on developing algorithmic tools to assess and mitigate such unfairness. However, there is little work on enhancing fairness in graph algorithms. Here, we develop a simple, effective and general method, CrossWalk, that enhances fairness of various graph algorithms, including influence maximization, link prediction and node classification, applied to node embeddings. CrossWalk is applicable to any random walk based node representation learning algorithm, such as DeepWalk and Node2Vec. The key idea is to bias random walks to cross group boundaries, by upweighting edges which (1) are closer to the groups' peripheries or (2) connect different groups in the network. CrossWalk pulls nodes that are near groups' peripheries towards their neighbors from other groups in the embedding space, while preserving the necessary structural information from the graph. Extensive experiments show the effectiveness of our algorithm to enhance fairness in various graph algorithms, including influence maximization, link prediction and node classification in synthetic and real networks, with only a very small decrease in performance.

We propose a new ensemble framework for supervised learning, named machine collaboration (MaC), based on a collection of base machines for prediction tasks. Different from bagging/stacking (a parallel & independent framework) and boosting (a sequential & top-down framework), MaC is a type of circular & interactive learning framework. The circular & interactive feature helps the base machines to transfer information circularly and update their own structures and parameters accordingly. The theoretical result on the risk bound of the estimator based on MaC shows that circular & interactive feature can help MaC reduce the risk via a parsimonious ensemble. We conduct extensive experiments on simulated data and 119 benchmark real data sets. The results of the experiments show that in most cases, MaC performs much better than several state-of-the-art methods, including CART, neural network, stacking, and boosting.

The discovery of causal mechanisms from time series data is a key problem in fields working with complex systems. Most identifiability results and learning algorithms assume the underlying dynamics to be discrete in time. Comparatively few, in contrast, explicitly define causal associations in infinitesimal intervals of time, independently of the scale of observation and of the regularity of sampling. In this paper, we consider causal discovery in continuous-time for the study of dynamical systems. We prove that for vector fields parameterized in a large class of neural networks, adaptive regularization schemes consistently recover causal graphs in systems of ordinary differential equations (ODEs). Using this insight, we propose a causal discovery algorithm based on penalized Neural ODEs that we show to be applicable to the general setting of irregularly-sampled multivariate time series and to strongly outperform the state of the art.

Undirected graphical models have been widely used to model the conditional independence structure of high-dimensional random vector data for years. In many modern applications such as EEG and fMRI data, the observations are multivariate random functions rather than scalars. To model the conditional independence of this type of data, functional graphical models are proposed and have attracted an increasing attention in recent years. In this paper, we propose a neighborhood selection approach to estimate Gaussian functional graphical models. We first estimate the neighborhood of all nodes via function-on-function regression, and then we can recover the whole graph structure based on the neighborhood information. By estimating conditional structure directly, we can circumvent the need of a well-defined precision operator which generally does not exist. Besides, we can better explore the effect of the choice of function basis for dimension reduction. We give a criterion for choosing the best function basis and motivate two practically useful choices, which we justified by both theory and experiments and show that they are better than expanding each function onto its own FPCA basis as in previous literature. In addition, the neighborhood selection approach is computationally more efficient than fglasso as it is more easy to do parallel computing. The statistical consistency of our proposed methods in high-dimensional setting are supported by both theory and experiment.

Multiple data types naturally co-occur when describing real-world phenomena and learning from them is a long-standing goal in machine learning research. However, existing self-supervised generative models approximating an ELBO are not able to fulfill all desired requirements of multimodal models: their posterior approximation functions lead to a trade-off between the semantic coherence and the ability to learn the joint data distribution. We propose a new, generalized ELBO formulation for multimodal data that overcomes these limitations. The new objective encompasses two previous methods as special cases and combines their benefits without compromises. In extensive experiments, we demonstrate the advantage of the proposed method compared to state-of-the-art models in self-supervised, generative learning tasks.

Exploratory Data Analysis is one of the critical processes of performing initial investigations on data analysis. Basic idea is to discover the patterns, anomalies, test hypotheses, and check the assumptions with the help of summary statistics and graphical representations. The dataset we are using for exploratory data analysis is from gss_cat. The datasets contains 21,483 observations and 9 variables. Above function will provide basic idea about variables in the datasets.

Figure 4 shows different analyses of the experimental results. From the first two rows of the figure we can infer that there is a strong correlation between the error on synthetic data, ϵs, and the error on the test set, ϵr. We evaluate this correlation using Spearman's rank correlation coefficient, as it is appropriate for measuring rank preservation. From the Spearman correlation plot (second row) we learn that the ranking capability of the synthetic data is comparable to that of the real data validation set. This strengthens our premise that using synthetic data for model selection is appropriate.

\textit{Stochastic gradient descent} (SGD) is of fundamental importance in deep learning. Despite its simplicity, elucidating its efficacy remains challenging. Conventionally, the success of SGD is attributed to the \textit{stochastic gradient noise} (SGN) incurred in the training process. Based on this general consensus, SGD is frequently treated and analyzed as the Euler-Maruyama discretization of a \textit{stochastic differential equation} (SDE) driven by either Brownian or L\'evy stable motion. In this study, we argue that SGN is neither Gaussian nor stable. Instead, inspired by the long-time correlation emerging in SGN series, we propose that SGD can be viewed as a discretization of an SDE driven by \textit{fractional Brownian motion} (FBM). Accordingly, the different convergence behavior of SGD dynamics is well grounded. Moreover, the first passage time of an SDE driven by FBM is approximately derived. This indicates a lower escaping rate for a larger Hurst parameter, and thus SGD stays longer in flat minima. This happens to coincide with the well-known phenomenon that SGD favors flat minima that generalize well. Four groups of experiments are conducted to validate our conjecture, and it is demonstrated that long-range memory effects persist across various model architectures, datasets, and training strategies. Our study opens up a new perspective and may contribute to a better understanding of SGD.