Partial differential equations (PDEs) are widely used for description of physical and engineering phenomena. Some key parameters involved in PDEs, which represents certain physical properties with important scientific interpretations, are difficult or even impossible to be measured directly. Estimation of these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations of numerical solution of PDE through algorithms such as finite element method, which can be time-consuming especially for nonlinear PDEs. In this paper, we propose a novel method for estimating unknown parameters in PDEs, called PDE-Informed Gaussian Process Inference (PIGPI). Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transfers the nonlinear PDE into an equivalent PDE system linear in all derivatives that our PIGPI can handle. PIGPI can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. The method completely bypasses the numerical solver for PDE, thus achieving drastic savings in computation time, especially for nonlinear PDEs. Moreover, the PIGPI method can give the uncertainty quantification for both the unknown parameters and the PDE solution. The proposed method is demonstrated by several application examples from different areas.

We present a way to apply topological data analysis for classifying encrypted bits into distinct classes. Persistent homology is applied to generate topological features of a point cloud obtained from sets of encryptions. We see that this machine learning pipeline is able to classify our data successfully where classical models of machine learning fail to perform the task. We also see that this pipeline works as a dimensionality reduction method making this approach to classify encrypted data a realistic method to classify the given encryptioned bits.

Gaussian process regression in its most simplified form assumes normal homoscedastic noise and utilizes analytically tractable mean and covariance functions of predictive posterior distribution using Gaussian conditioning. Its hyperparameters are estimated by maximizing the evidence, commonly known as type II maximum likelihood estimation. Unfortunately, Bayesian inference based on Gaussian likelihood is not robust to outliers, which are often present in the observational training data sets. To overcome this problem, we propose a robust process model in the Gaussian process framework with the likelihood of observed data expressed as the Huber probability distribution. The proposed model employs weights based on projection statistics to scale residuals and bound the influence of vertical outliers and bad leverage points on the latent functions estimates while exhibiting a high statistical efficiency at the Gaussian and thick tailed noise distributions. The proposed method is demonstrated by two real world problems and two numerical examples using datasets with additive errors following thick tailed distributions such as Students t, Laplace, and Cauchy distribution.

We investigate $k$-means clustering in the online no-substitution setting when the input arrives in \emph{arbitrary} order. In this setting, points arrive one after another, and the algorithm is required to instantly decide whether to take the current point as a center before observing the next point. Decisions are irrevocable. The goal is to minimize both the number of centers and the $k$-means cost. Previous works in this setting assume that the input's order is random, or that the input's aspect ratio is bounded. It is known that if the order is arbitrary and there is no assumption on the input, then any algorithm must take all points as centers. Moreover, assuming a bounded aspect ratio is too restrictive -- it does not include natural input generated from mixture models. We introduce a new complexity measure that quantifies the difficulty of clustering a dataset arriving in arbitrary order. We design a new random algorithm and prove that if applied on data with complexity $d$, the algorithm takes $O(d\log(n) k\log(k))$ centers and is an $O(k^3)$-approximation. We also prove that if the data is sampled from a ``natural" distribution, such as a mixture of $k$ Gaussians, then the new complexity measure is equal to $O(k^2\log(n))$. This implies that for data generated from those distributions, our new algorithm takes only $\text{poly}(k\log(n))$ centers and is a $\text{poly}(k)$-approximation. In terms of negative results, we prove that the number of centers needed to achieve an $\alpha$-approximation is at least $\Omega\left(\frac{d}{k\log(n\alpha)}\right)$.

Federated learning (FL) is a general framework for learning across heterogeneous clients while preserving data privacy, under the orchestration of a central server. FL methods often compute gradients of loss functions purely locally (ie. entirely at each client, or entirely at the server), typically using automatic differentiation (AD) techniques. We propose a federated automatic differentiation (FAD) framework that 1) enables computing derivatives of functions involving client and server computation as well as communication between them and 2) operates in a manner compatible with existing federated technology. In other words, FAD computes derivatives across communication boundaries. We show, in analogy with traditional AD, that FAD may be implemented using various accumulation modes, which introduce distinct computation-communication trade-offs and systems requirements. Further, we show that a broad class of federated computations is closed under these various modes of FAD, implying in particular that if the original computation can be implemented using privacy-preserving primitives, its derivative may be computed using only these same primitives. We then show how FAD can be used to create algorithms that dynamically learn components of the algorithm itself. In particular, we show that FedAvg-style algorithms can exhibit significantly improved performance by using FAD to adjust the server optimization step automatically, or by using FAD to learn weighting schemes for computing weighted averages across clients.

Quantifying cause and effect relationships is an important problem in many domains. The gold standard solution is to conduct a randomised controlled trial. However, in many situations such trials cannot be performed. In the absence of such trials, many methods have been devised to quantify the causal impact of an intervention from observational data given certain assumptions. One widely used method are synthetic control models. While identifiability of the causal estimand in such models has been obtained from a range of assumptions, it is widely and implicitly assumed that the underlying assumptions are satisfied for all time periods both pre- and post-intervention. This is a strong assumption, as synthetic control models can only be learned in pre-intervention period. In this paper we address this challenge, and prove identifiability can be obtained without the need for this assumption, by showing it follows from the principle of invariant causal mechanisms. Moreover, for the first time, we formulate and study synthetic control models in Pearl's structural causal model framework. Importantly, we provide a general framework for sensitivity analysis of synthetic control causal inference to violations of the assumptions underlying non-parametric identifiability. We end by providing an empirical demonstration of our sensitivity analysis framework on simulated and real data in the widely-used linear synthetic control framework.

Predictive process monitoring is a subfield of process mining that aims to estimate case or event features for running process instances. Such predictions are of significant interest to the process stakeholders. However, most of the state-of-the-art methods for predictive monitoring require the training of complex machine learning models, which is often inefficient. Moreover, most of these methods require a hyper-parameter optimization that requires several repetitions of the training process which is not feasible in many real-life applications. In this paper, we propose an instance selection procedure that allows sampling training process instances for prediction models. We show that our instance selection procedure allows for a significant increase of training speed for next activity and remaining time prediction methods while maintaining reliable levels of prediction accuracy.

Robots deployed in human-centric environments may need to manipulate a diverse range of articulated objects, such as doors, dishwashers, and cabinets. Articulated objects often come with unexpected articulation mechanisms that are inconsistent with categorical priors: for example, a drawer might rotate about a hinge joint instead of sliding open. We propose a category-independent framework for predicting the articulation models of unknown objects from sequences of RGB-D images. The prediction is performed by a two-step process: first, a visual perception module tracks object part poses from raw images, and second, a factor graph takes these poses and infers the articulation model including the current configuration between the parts as a 6D twist. We also propose a manipulation-oriented metric to evaluate predicted joint twists in terms of how well a compliant robot controller would be able to manipulate the articulated object given the predicted twist. We demonstrate that our visual perception and factor graph modules outperform baselines on simulated data and show the applicability of our factor graph on real world data.

Reinforcement learning (RL) has been widely used in text generation to alleviate the exposure bias issue or to utilize non-parallel datasets. The reward function plays an important role in making RL training successful. However, previous reward functions are typically task-specific and sparse, restricting the use of RL. In our work, we propose a task-agnostic approach that derives a step-wise reward function directly from a model trained with teacher forcing. We additionally propose a simple modification to stabilize the RL training on non-parallel datasets with our induced reward function. Empirical results show that our method outperforms self-training and reward regression methods on several text generation tasks, confirming the effectiveness of our reward function.

Data augmentation has recently seen increased interest in graph machine learning given its demonstrated ability to improve model performance and generalization by added training data. Despite this recent surge, the area is still relatively under-explored, due to the challenges brought by complex, non-Euclidean structure of graph data, which limits the direct analogizing of traditional augmentation operations on other types of image, video or text data. Our work aims to give a necessary and timely overview of existing graph data augmentation methods; notably, we present a comprehensive and systematic survey of graph data augmentation approaches, summarizing the literature in a structured manner. We first introduce three different taxonomies for categorizing graph data augmentation methods from the data, task, and learning perspectives, respectively. Next, we introduce recent advances in graph data augmentation, differentiated by their methodologies and applications. We conclude by outlining currently unsolved challenges and directions for future research. Overall, our work aims to clarify the landscape of existing literature in graph data augmentation and motivates additional work in this area, providing a helpful resource for researchers and practitioners in the broader graph machine learning domain. Additionally, we provide a continuously updated reading list at https://github.com/zhao-tong/graph-data-augmentation-papers.