Deep neural operators (DNOs) have been utilized to approximate nonlinear mappings between function spaces. However, DNOs face the challenge of increased dimensionality and computational cost associated with unaligned observation data. In this study, we propose a hybrid Decoder-DeepONet operator regression framework to handle unaligned data effectively. Additionally, we introduce a Multi-Decoder-DeepONet, which utilizes an average field of training data as input augmentation. The consistencies of the frameworks with the operator approximation theory are provided, on the basis of the universal approximation theorem. Two numerical experiments, Darcy problem and flow-field around an airfoil, are conducted to validate the efficiency and accuracy of the proposed methods. Results illustrate the advantages of Decoder-DeepONet and Multi-Decoder-DeepONet in handling unaligned observation data and showcase their potentials in improving prediction accuracy.
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Discovering causal relationships from observational data is a fundamental yet challenging task. Invariant causal prediction (ICP, Peters et al., 2016) is a method for causal feature selection which requires data from heterogeneous settings and exploits that causal models are invariant. ICP has been extended to general additive noise models and to nonparametric settings using conditional independence tests. However, the latter often suffer from low power (or poor type I error control) and additive noise models are not suitable for applications in which the response is not measured on a continuous scale, but reflects categories or counts. Here, we develop transformation-model (TRAM) based ICP, allowing for continuous, categorical, count-type, and uninformatively censored responses (these model classes, generally, do not allow for identifiability when there is no exogenous heterogeneity). As an invariance test, we propose TRAM-GCM based on the expected conditional covariance between environments and score residuals with uniform asymptotic level guarantees. For the special case of linear shift TRAMs, we also consider TRAM-Wald, which tests invariance based on the Wald statistic. We provide an open-source R package 'tramicp' and evaluate our approach on simulated data and in a case study investigating causal features of survival in critically ill patients.
This paper addresses the problem of designing the {\it continuous-discrete} unscented Kalman filter (UKF) implementation methods. More precisely, the aim is to propose the MATLAB-based UKF algorithms for {\it accurate} and {\it robust} state estimation of stochastic dynamic systems. The accuracy of the {\it continuous-discrete} nonlinear filters heavily depends on how the implementation method manages the discretization error arisen at the filter prediction step. We suggest the elegant and accurate implementation framework for tracking the hidden states by utilizing the MATLAB built-in numerical integration schemes developed for solving ordinary differential equations (ODEs). The accuracy is boosted by the discretization error control involved in all MATLAB ODE solvers. This keeps the discretization error below the tolerance value provided by users, automatically. Meanwhile, the robustness of the UKF filtering methods is examined in terms of the stability to roundoff. In contrast to the pseudo-square-root UKF implementations established in engineering literature, which are based on the one-rank Cholesky updates, we derive the stable square-root methods by utilizing the $J$-orthogonal transformations for calculating the Cholesky square-root factors.
We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.
Analysis of higher-order organizations, usually small connected subgraphs called motifs, is a fundamental task on complex networks. This paper studies a new problem of testing higher-order clusterability: given query access to an undirected graph, can we judge whether this graph can be partitioned into a few clusters of highly-connected motifs? This problem is an extension of the former work proposed by Czumaj et al. (STOC' 15), who recognized cluster structure on graphs using the framework of property testing. In this paper, a good graph cluster on high dimensions is first defined for higher-order clustering. Then, query lower bound is given for testing whether this kind of good cluster exists. Finally, an optimal sublinear-time algorithm is developed for testing clusterability based on triangles.
Subspace clustering methods which embrace a self-expressive model that represents each data point as a linear combination of other data points in the dataset provide powerful unsupervised learning techniques. However, when dealing with large datasets, representation of each data point by referring to all data points via a dictionary suffers from high computational complexity. To alleviate this issue, we introduce a parallelizable multi-subset based self-expressive model (PMS) which represents each data point by combining multiple subsets, with each consisting of only a small proportion of the samples. The adoption of PMS in subspace clustering (PMSSC) leads to computational advantages because the optimization problems decomposed over each subset are small, and can be solved efficiently in parallel. Furthermore, PMSSC is able to combine multiple self-expressive coefficient vectors obtained from subsets, which contributes to an improvement in self-expressiveness. Extensive experiments on synthetic and real-world datasets show the efficiency and effectiveness of our approach in comparison to other methods.
As an optical processor, a Diffractive Deep Neural Network (D2NN) utilizes engineered diffractive surfaces designed through machine learning to perform all-optical information processing, completing its tasks at the speed of light propagation through thin optical layers. With sufficient degrees-of-freedom, D2NNs can perform arbitrary complex-valued linear transformations using spatially coherent light. Similarly, D2NNs can also perform arbitrary linear intensity transformations with spatially incoherent illumination; however, under spatially incoherent light, these transformations are non-negative, acting on diffraction-limited optical intensity patterns at the input field-of-view (FOV). Here, we expand the use of spatially incoherent D2NNs to complex-valued information processing for executing arbitrary complex-valued linear transformations using spatially incoherent light. Through simulations, we show that as the number of optimized diffractive features increases beyond a threshold dictated by the multiplication of the input and output space-bandwidth products, a spatially incoherent diffractive visual processor can approximate any complex-valued linear transformation and be used for all-optical image encryption using incoherent illumination. The findings are important for the all-optical processing of information under natural light using various forms of diffractive surface-based optical processors.
The joint modeling of longitudinal and time-to-event outcomes has become a popular tool in follow-up studies. However, fitting Bayesian joint models to large datasets, such as patient registries, can require extended computing times. To speed up sampling, we divided a patient registry dataset into subsamples, analyzed them in parallel, and combined the resulting Markov chain Monte Carlo draws into a consensus distribution. We used a simulation study to investigate how different consensus strategies perform with joint models. In particular, we compared grouping all draws together with using equal- and precision-weighted averages. We considered scenarios reflecting different sample sizes, numbers of data splits, and processor characteristics. Parallelization of the sampling process substantially decreased the time required to run the model. We found that the weighted-average consensus distributions for large sample sizes were nearly identical to the target posterior distribution. The proposed algorithm has been made available in an R package for joint models, JMbayes2. This work was motivated by the clinical interest in investigating the association between ppFEV1, a commonly measured marker of lung function, and the risk of lung transplant or death, using data from the US Cystic Fibrosis Foundation Patient Registry (35,153 individuals with 372,366 years of cumulative follow-up). Splitting the registry into five subsamples resulted in an 85\% decrease in computing time, from 9.22 to 1.39 hours. Splitting the data and finding a consensus distribution by precision-weighted averaging proved to be a computationally efficient and robust approach to handling large datasets under the joint modeling framework.
Gaussian processes (GP) regression has gained substantial popularity in machine learning applications. The behavior of a GP regression depends on the choice of covariance function. Stationary covariance functions are favorite in machine learning applications. However, (non-periodic) stationary covariance functions are always mean reverting and can therefore exhibit pathological behavior when applied to data that does not relax to a fixed global mean value. In this paper, we show that it is possible to use improper GP prior with infinite variance to define processes that are stationary but not mean reverting. To this aim, we introduce a large class of improper kernels that can only be defined in this improper regime. Specifically, we introduce the Smooth Walk kernel, which produces infinitely smooth samples, and a family of improper Mat\'ern kernels, which can be defined to be $j$-times differentiable for any integer $j$. The resulting posterior distributions can be computed analytically and it involves a simple correction of the usual formulas. By analyzing both synthetic and real data, we demonstrate that these improper kernels solve some known pathologies of mean reverting GP regression while retaining most of the favourable properties of ordinary smooth stationary kernels.
One central theme in machine learning is function estimation from sparse and noisy data. An example is supervised learning where the elements of the training set are couples, each containing an input location and an output response. In the last decades, a substantial amount of work has been devoted to design estimators for the unknown function and to study their convergence to the optimal predictor, also characterizing the learning rate. These results typically rely on stationary assumptions where input locations are drawn from a probability distribution that does not change in time. In this work, we consider kernel-based ridge regression and derive convergence conditions under non stationary distributions, addressing also cases where stochastic adaption may happen infinitely often. This includes the important exploration-exploitation problems where e.g. a set of agents/robots has to monitor an environment to reconstruct a sensorial field and their movements rules are continuously updated on the basis of the acquired knowledge on the field and/or the surrounding environment.
We present ResMLP, an architecture built entirely upon multi-layer perceptrons for image classification. It is a simple residual network that alternates (i) a linear layer in which image patches interact, independently and identically across channels, and (ii) a two-layer feed-forward network in which channels interact independently per patch. When trained with a modern training strategy using heavy data-augmentation and optionally distillation, it attains surprisingly good accuracy/complexity trade-offs on ImageNet. We will share our code based on the Timm library and pre-trained models.
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