We present ElastoGen, a knowledge-driven AI model that generates physically accurate 4D elastodynamics. Unlike deep models that learn from video- or image-based observations, ElastoGen leverages the principles of physics and learns from established mathematical and optimization procedures. The core idea of ElastoGen is converting the differential equation, corresponding to the nonlinear force equilibrium, into a series of iterative local convolution-like operations, which naturally fit deep architectures. We carefully build our network module following this overarching design philosophy. ElastoGen is much more lightweight in terms of both training requirements and network scale than deep generative models. Because of its alignment with actual physical procedures, ElastoGen efficiently generates accurate dynamics for a wide range of hyperelastic materials and can be easily integrated with upstream and downstream deep modules to enable end-to-end 4D generation.

Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible Fourier Neural Operator (iFNO) for jointly tackling the forward and inverse problems. We developed a series of invertible Fourier blocks in the latent channel space to share the model parameters, exchange the information, and mutually regularize the learning for the bi-directional tasks. We integrated a variational auto-encoder to capture the intrinsic structures within the input space and to enable posterior inference so as to mitigate challenges of illposedness, data shortage, noises that are common in inverse problems. We proposed a three-step process to combine the invertible blocks and the VAE component for effective training. The evaluations on seven benchmark forward and inverse tasks have demonstrated the advantages of our approach. The code is available at \url{https://github.com/BayesianAIGroup/iFNO.}

A long-standing belief holds that Bayesian Optimization (BO) with standard Gaussian processes (GP) --- referred to as standard BO --- underperforms in high-dimensional optimization problems. While this belief seems plausible, it lacks both robust empirical evidence and theoretical justification. To address this gap, we present a systematic investigation. First, through a comprehensive evaluation across twelve benchmarks, we found that while the popular Square Exponential (SE) kernel often leads to poor performance, using Mat\'ern kernels enables standard BO to consistently achieve top-tier results, frequently surpassing methods specifically designed for high-dimensional optimization. Second, our theoretical analysis reveals that the SE kernel’s failure primarily stems from improper initialization of the length-scale parameters, which are commonly used in practice but can cause gradient vanishing in training. We provide a probabilistic bound to characterize this issue, showing that Mat\'ern kernels are less susceptible and can robustly handle much higher dimensions. Third, we propose a simple robust initialization strategy that dramatically improves the performance of the SE kernel, bringing it close to state-of-the-art methods, without requiring additional priors or regularization. We prove another probabilistic bound that demonstrates how the gradient vanishing issue can be effectively mitigated with our method. Our findings advocate for a re-evaluation of standard BO’s potential in high-dimensional settings.

Modern physics simulation often involves multiple functions of interests, and traditional numerical approaches are known to be complex and computationally costly. While machine learning-based surrogate models can offer significant cost reductions, most focus on a single task, such as forward prediction, and typically lack uncertainty quantification --- an essential component in many applications. To overcome these limitations, we propose Arbitrarily-Conditioned Multi-Functional Diffusion (ACM-FD), a versatile probabilistic surrogate model for multi-physics emulation. ACM-FD can perform a wide range of tasks within a single framework, including forward prediction, various inverse problems, and simulating data for entire systems or subsets of quantities conditioned on others. Specifically, we extend the standard Denoising Diffusion Probabilistic Model (DDPM) for multi-functional generation by modeling noise as Gaussian processes (GP). We propose a random-mask based, zero-regularized denoising loss to achieve flexible and robust conditional generation. We induce a Kronecker product structure in the GP covariance matrix, substantially reducing the computational cost and enabling efficient training and sampling. We demonstrate the effectiveness of ACM-FD across several fundamental multi-physics systems.

We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing challenges with a large number of collocation points, our approach eliminates these operators from the kernel. We model the solution using a standard kernel interpolation form and differentiate the interpolant to compute the derivatives. Our framework obviates the need for complex Gram matrix construction between solutions and their derivatives, allowing for a straightforward implementation and scalable computation. As an instance, we allocate the collocation points on a grid and adopt a product kernel, which yields a Kronecker product structure in the interpolation. This structure enables us to avoid computing the full Gram matrix, reducing costs and scaling efficiently to a large number of collocation points. We provide a proof of the convergence and rate analysis of our method under appropriate regularity assumptions. In numerical experiments, we demonstrate the advantages of our method in solving several benchmark PDEs.

Energy-saving methods like Discontinuous Reception (DRX) and Power Save Mode (PSM) are commonly used in Internet of Thing (IoT) applications, allowing for sleep and awake cycle adjustments to save energy. However, understanding and configuring these parameters on devices, especially actuator-type devices, is challenging for IoT service providers. Unlike sensor types, these devices must complete their sleep cycle before responding to infrequent downlink commands, making efficient parameter selection and traffic prediction vital for energy efficiency and command reception. To address this, we present ADDER, a network-side solution leveraging a context-aware traffic predictor. This predictor forecasts downlink arrival probabilities, guiding a deep deterministic policy gradient (DDPG) policymaker that selects energy-saving parameters based on thresholds defined by the IoT service providers. ADDER, leverages contextual information like day of week, hour, weather, holidays, and events, shifting the focus from individual device histories (often erratic) to analyzing broader service traffic patterns. This data-driven strategy enables ADDER to adjust energy-saving settings for the best balance between energy efficiency and latency, customizing to the unique requirements of each service and removing the burden of configuring complex network settings. We observed that ADDER meets latency needs while achieving a $\mathbf{5 . 9 \%}$ reduction in energy consumption for services requiring rapid responses. For applications prioritizing energy conservation, such as irrigation systems and city lighting, ADDER achieves a significant $32.7 \%$ reduction in energy consumption with a slight increase $(\mathbf{9 \%}$) in messages might not meet the strictest latency requirements. To evaluate the consequences of prediction inaccuracies from our predictor, we utilized a real-world shared mobility dataset provided by Austin’s Transportation Department for a case study.

Fourier Neural Operator (FNO) is a popular operator learning framework. It not only achieves the state-of-the-art performance in many tasks, but also is efficient in training and prediction. However, collecting training data for the FNO can be a costly bottleneck in practice, because it often demands expensive physical simulations. To overcome this problem, we propose Multi-Resolution Active Learning of FNO (MRA-FNO), which can dynamically select the input functions and resolutions to lower the data cost as much as possible while optimizing the learning efficiency. Specifically, we propose a probabilistic multi-resolution FNO and use ensemble Monte-Carlo to develop an effective posterior inference algorithm. To conduct active learning, we maximize a utility-cost ratio as the acquisition function to acquire new examples and resolutions at each step. We use moment matching and the matrix determinant lemma to enable tractable, efficient utility computation. Furthermore, we develop a cost annealing framework to avoid over-penalizing high-resolution queries at the early stage. The over-penalization is severe when the cost difference is significant between the resolutions, which renders active learning often stuck at low-resolution queries and inferior performance. Our method overcomes this problem and applies to general multi-fidelity active learning and optimization problems. We have shown the advantage of our method in several benchmark operator learning tasks. The code is available at https://github.com/shib0li/MRA-FNO.

Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity and noise issues, both of which are ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty quantification and/or are costly in training. To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We combine it with a Bayesian spike-and-slab prior — an ideal Bayesian sparse distribution — for effective operator selection and uncertainty quantification. We develop an expectation-propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation. To overcome the computational challenge of kernel regression, we place the function values on a mesh and induce a Kronecker product construction, and we use tensor algebra to enable efficient computation and optimization. We show the advantages of KBASS on a list of benchmark ODE and PDE discovery tasks. The code is available at \url{https://github.com/long-da/KBASS}.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at {https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition}.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student $t$ mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at {https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE}.

In real-world scenarios such as traffic and energy management, we frequently encounter large volumes of time-series data characterized by missing values, noise, and irregular sampling patterns. While numerous imputation methods have been proposed, the majority tend to operate within a local horizon, which involves dividing long sequences into batches of fixed-length segments for model training. This local horizon often leads to the overlooking of global trends and periodic patterns. More importantly, most methods assume the observations are sampled at regular timestamps, and fail to handle complex irregular sampled time series in various applications. Additionally, most existing methods are learned in an offline manner. Thus, it is not suitable for applications with rapidly arriving streaming data. To address these challenges, we propose BayOTIDE: Bayesian Online Multivariate Time series Imputation with functional decomposition. Our method conceptualizes multivariate time series as the weighted combination of groups of low-rank temporal factors with different patterns. We employ a suite of Gaussian Processes (GPs),each with a unique kernel, as functional priors to model these factors. For computational efficiency, we further convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE), and developing a scalable algorithm for online inference. The proposed method can not only handle imputation over arbitrary timestamps, but also offer uncertainty quantification and interpretability for the downstream application. We evaluate our method on both synthetic and real-world datasets. We release the code at https://github.com/xuangu-fang/BayOTIDE.

Model Agnostic Meta-Learning (MAML) is widely used to find a good initialization for a family of tasks. Despite its success, a critical challenge in MAML is to calculate the gradient w.r.t. the initialization of a long training trajectory for the sampled tasks, because the computation graph can rapidly explode and the computational cost is very expensive. To address this problem, we propose Adjoint MAML (A-MAML). We view gradient descent in the inner optimization as the evolution of an Ordinary Differential Equation (ODE). To efficiently compute the gradient of the validation loss w.r.t. the initialization, we use the adjoint method to construct a companion, backward ODE. To obtain the gradient w.r.t. the initialization, we only need to run the standard ODE solver twice — one is forward in time that evolves a long trajectory of gradient flow for the sampled task; the other is backward and solves the adjoint ODE. We need not create or expand any intermediate computational graphs, adopt aggressive approximations, or impose proximal regularizers in the training loss. Our approach is cheap, accurate, and adaptable to different trajectory lengths. We demonstrate the advantage of our approach in both synthetic and real-world meta-learning tasks. The code is available at https://github.com/shib0li/Adjoint-MAML.

The Schrödinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized optimal transport problem, which conducts projections onto every other marginal alternatingly. However, in practice, only approximated projections are accessible and their convergence is not well understood. To fill this gap, we present a first convergence analysis of the Schrödinger bridge algorithm based on approximated projections. As for its practical applications, we apply SBP to probabilistic time series imputation by generating missing values conditioned on observed data. We show that optimizing the transport cost improves the performance and the proposed algorithm achieves the state-of-the-art result in healthcare and environmental data while exhibiting the advantage of exploring both temporal and feature patterns in probabilistic time series imputation.

Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, apply different PINNs to solve the problem in each subdomain, and stitch the subdomains at the interface. Thereby, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of multi-domain PINNs is sensitive to the choice of the interface conditions. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine appropriate interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit (MAB) models. The first one applies to the entire training course, and online updates a Gaussian process (GP) reward that given the PDE parameters and interface conditions predicts the performance. We prove a sub-linear regret bound for both UCB and Thompson sampling, which in theory guarantees the effectiveness of our MAB. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP reward for each phase to enable different condition selections at the two stages to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.

Practical tensor data is often along with time information. Most existing temporal decomposition approaches estimate a set of fixed factors for the objects in each tensor mode, and hence cannot capture the temporal evolution of the objects' representation. More important, we lack an effective approach to capture such evolution from streaming data, which is common in real-world applications. To address these issues, we propose Streaming Factor Trajectory Learning (SFTL) for temporal tensor decomposition. We use Gaussian processes (GPs) to model the trajectory of factors so as to flexibly estimate their temporal evolution. To address the computational challenges in handling streaming data, we convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE). We develop an efficient online filtering algorithm to estimate a decoupled running posterior of the involved factor states upon receiving new data. The decoupled estimation enables us to conduct standard Rauch-Tung-Striebel smoothing to compute the full posterior of all the trajectories in parallel, without the need for revisiting any previous data. We have shown the advantage of SFTL in both synthetic tasks and real-world applications.

Tensor decomposition is an important tool for multiway data analysis. In practice, the data is often sparse yet associated with rich temporal information. Existing methods, however, often under-use the time information and ignore the structural knowledge within the sparsely observed tensor entries. To overcome these limitations and to better capture the underlying temporal structure, we propose Dynamic EMbedIngs fOr dynamic Tensor dEcomposition (DEMOTE). We develop a neural diffusion-reaction process to estimate dynamic embeddings for the entities in each tensor mode. Specifically, based on the observed tensor entries, we build a multi-partite graph to encode the correlation between the entities. We construct a graph diffusion process to co-evolve the embedding trajectories of the correlated entities and use a neural network to construct a reaction process for each individual entity. In this way, our model can capture both the commonalities and personalities during the evolution of the embeddings for different entities. We then use a neural network to model the entry value as a nonlinear function of the embedding trajectories. For model estimation, we combine ODE solvers to develop a stochastic mini-batch learning algorithm. We propose a stratified sampling method to balance the cost of processing each mini-batch so as to improve the overall efficiency. We show the advantage of our approach in both simulation studies and real-world applications. The code is available at https://github.com/wzhut/Dynamic-Tensor-Decomposition-via-Neural-Diffusion-Reaction-Processes.

Many applications, such as in physical simulation and engineering design, demand we estimate functions with high-dimensional outputs. To reduce the expensive cost of generating training examples, we usually choose several fidelities to enable a cost/quality trade-off. In this paper, we consider the active learning task to automatically identify the fidelities and training inputs to query new examples so as to achieve the best learning benefit-cost ratio. To this end, we propose DMFAL, a Deep Multi-Fidelity Active Learning approach. We first develop a deep neural network-based multi-fidelity model for high-dimensional outputs, which can flexibly capture strong complex correlations across the outputs and fidelities to enhance the learning of the target function. We then propose a mutual information based acquisition function that extends the predictive entropy principle. To overcome the computational challenges caused by large output dimensions, we use the multi-variate delta method and moment-matching to estimate the output posterior, and Weinstein-Aronszajn identity to calculate and optimize the acquisition function. We show the advantage of our method in several applications of computational physics and engineering design. The code is available at https://github.com/shib0li/DMFAL.

Deep kernel learning is a promising combination of deep neural networks and nonparametric function estimation. However, as a data driven approach, the performance of deep kernel learning can still be restricted by scarce or insufficient data, especially in extrapolation tasks. To address these limitations, we propose Physics Informed Deep Kernel Learning (PI-DKL) that exploits physics knowledge represented by differential equations with latent sources. Specifically, we use the posterior function sample of the Gaussian process as the surrogate for the solution of the differential equation, and construct a generative component to integrate the equation in a principled Bayesian hybrid framework. For efficient and effective inference, we marginalize out the latent variables in the joint probability and derive a collapsed model evidence lower bound (ELBO), based on which we develop a stochastic model estimation algorithm. Our ELBO can be viewed as a nice, interpretable posterior regularization objective. On synthetic datasets and real-world applications, we show the advantage of our approach in both prediction accuracy and uncertainty quantification. The code is available at https://github.com/GregDobby/PIDKL.

Neural networks tend to achieve better accuracy with training if they are larger {—} even if the resulting models are overparameterized. Nevertheless, carefully removing such excess of parameters before, during, or after training may also produce models with similar or even improved accuracy. In many cases, that can be curiously achieved by heuristics as simple as removing a percentage of the weights with the smallest absolute value {—} even though absolute value is not a perfect proxy for weight relevance. With the premise that obtaining significantly better performance from pruning depends on accounting for the combined effect of removing multiple weights, we revisit one of the classic approaches for impact-based pruning: the Optimal Brain Surgeon (OBS). We propose a tractable heuristic for solving the combinatorial extension of OBS, in which we select weights for simultaneous removal, and we combine it with a single-pass systematic update of unpruned weights. Our selection method outperforms other methods for high sparsity, and the single-pass weight update is also advantageous if applied after those methods.

Tensor decomposition is a dominant framework for multiway data analysis and prediction. Although practical data often contains timestamps for the observed entries, existing tensor decomposition approaches overlook or under-use this valuable time information. They either drop the timestamps or bin them into crude steps and hence ignore the temporal dynamics within each step or use simple parametric time coefficients. To overcome these limitations, we propose Bayesian Continuous-Time Tucker Decomposition. We model the tensor-core of the classical Tucker decomposition as a time-varying function, and place a Gaussian process prior to flexibly estimate all kinds of temporal dynamics. In this way, our model maintains the interpretability while is flexible enough to capture various complex temporal relationships between the tensor nodes. For efficient and high-quality posterior inference, we use the stochastic differential equation (SDE) representation of temporal GPs to build an equivalent state-space prior, which avoids huge kernel matrix computation and sparse/low-rank approximations. We then use Kalman filtering, RTS smoothing, and conditional moment matching to develop a scalable message passing inference algorithm. We show the advantage of our method in simulation and several real-world applications.

High-order interactions between multiple objects are common in real-world applications. Although tensor decomposition is a popular framework for high-order interaction analysis and prediction, most methods cannot well exploit the valuable timestamp information in data. The existent methods either discard the timestamps or convert them into discrete steps or use over-simplistic decomposition models. As a result, these methods might not be capable enough of capturing complex, fine-grained temporal dynamics or making accurate predictions for long-term interaction results. To overcome these limitations, we propose a novel Temporal High-order Interaction decompoSition model based on Ordinary Differential Equations (THIS-ODE). We model the time-varying interaction result with a latent ODE. To capture the complex temporal dynamics, we use a neural network (NN) to learn the time derivative of the ODE state. We use the representation of the interaction objects to model the initial value of the ODE and to constitute a part of the NN input to compute the state. In this way, the temporal relationships of the participant objects can be estimated and encoded into their representations. For tractable and scalable inference, we use forward sensitivity analysis to efficiently compute the gradient of ODE state, based on which we use integral transform to develop a stochastic mini-batch learning algorithm. We demonstrate the advantage of our approach in simulation and four real-world applications.

Physical modeling is critical for many modern science and engineering applications. From a data science or machine learning perspective, where more domain-agnostic, data-driven models are pervasive, physical knowledge {—} often expressed as differential equations {—} is valuable in that it is complementary to data, and it can potentially help overcome issues such as data sparsity, noise, and inaccuracy. In this work, we propose a simple, yet powerful and general framework {—} AutoIP, for Automatically Incorporating Physics {—} that can integrate all kinds of differential equations into Gaussian Processes (GPs) to enhance prediction accuracy and uncertainty quantification. These equations can be linear or nonlinear, spatial, temporal, or spatio-temporal, complete or incomplete with unknown source terms, and so on. Based on kernel differentiation, we construct a GP prior to sample the values of the target function, equation related derivatives, and latent source functions, which are all jointly from a multivariate Gaussian distribution. The sampled values are fed to two likelihoods: one to fit the observations, and the other to conform to the equation. We use the whitening method to evade the strong dependency between the sampled function values and kernel parameters, and we develop a stochastic variational learning algorithm. AutoIP shows improvement upon vanilla GPs in both simulation and several real-world applications, even using rough, incomplete equations.

We propose a nonparametric factorization approach for sparsely observed tensors. The sparsity does not mean zero-valued entries are massive or dominated. Rather, it implies the observed entries are very few, and even fewer with the growth of the tensor; this is ubiquitous in practice. Compared with the existent works, our model not only leverages the structural information underlying the observed entry indices, but also provides extra interpretability and flexibility {—} it can simultaneously estimate a set of location factors about the intrinsic properties of the tensor nodes, and another set of sociability factors reflecting their extrovert activity in interacting with others; users are free to choose a trade-off between the two types of factors. Specifically, we use hierarchical Gamma processes and Poisson random measures to construct a tensor-valued process, which can freely sample the two types of factors to generate tensors and always guarantees an asymptotic sparsity. We then normalize the tensor process to obtain hierarchical Dirichlet processes to sample each observed entry index, and use a Gaussian process to sample the entry value as a nonlinear function of the factors, so as to capture both the sparse structure properties and complex node relationships. For efficient inference, we use Dirichlet process properties over finite sample partitions, density transformations, and random features to develop a stochastic variational estimation algorithm. We demonstrate the advantage of our method in several benchmark datasets.

High-order interaction events are common in real-world applications. Learning embeddings that encode the complex relationships of the participants from these events is of great importance in knowledge mining and predictive tasks. Despite the success of existing approaches, e.g. Poisson tensor factorization, they ignore the sparse structure underlying the data, namely the occurred interactions are far less than the possible interactions among all the participants. In this paper, we propose Nonparametric Embeddings of Sparse High-order interaction events (NESH). We hybridize a sparse hypergraph (tensor) process and a matrix Gaussian process to capture both the asymptotic structural sparsity within the interactions and nonlinear temporal relationships between the participants. We prove strong asymptotic bounds (including both a lower and an upper bound ) of the sparse ratio, which reveals the asymptotic properties of the sampled structure. We use batch-normalization, stick-breaking construction and sparse variational GP approximations to develop an efficient, scalable model inference algorithm. We demonstrate the advantage of our approach in several real-world applications.

Tensor decomposition is a fundamental framework to analyze data that can be represented by multi-dimensional arrays. In practice, tensor data are often accompanied with temporal information, namely the time points when the entry values were generated. This information implies abundant, complex temporal variation patterns. However, current methods always assume the factor representations of the entities in each tensor mode are static, and never consider their temporal evolution. To fill this gap, we propose NONparametric FActor Trajectory learning for dynamic tensor decomposition (NONFAT). We place Gaussian process (GP) priors in the frequency domain and conduct inverse Fourier transform via Gauss-Laguerre quadrature to sample the trajectory functions. In this way, we can overcome data sparsity and obtain robust trajectory estimates across long time horizons. Given the trajectory values at specific time points, we use a second-level GP to sample the entry values and to capture the temporal relationship between the entities. For efficient and scalable inference, we leverage the matrix Gaussian structure in the model, introduce a matrix Gaussian posterior, and develop a nested sparse variational learning algorithm. We have shown the advantage of our method in several real-world applications.

Pruning techniques have been successfully used in neural networks to trade accuracy for sparsity. However, the impact of network pruning is not uniform: prior work has shown that the recall for underrepresented classes in a dataset may be more negatively affected. In this work, we study such relative distortions in recall by hypothesizing an intensification effect that is inherent to the model. Namely, that pruning makes recall relatively worse for a class with recall below accuracy and, conversely, that it makes recall relatively better for a class with recall above accuracy. In addition, we propose a new pruning algorithm aimed at attenuating such effect. Through statistical analysis, we have observed that intensification is less severe with our algorithm but nevertheless more pronounced with relatively more difficult tasks, less complex models, and higher pruning ratios. More surprisingly, we conversely observe a de-intensification effect with lower pruning ratios.

Multi-fidelity modeling and learning is important in physical simulation related applications. It can leverage both low-fidelity and high-fidelity examples for training so as to reduce the cost of data generation yet still achieving good performance. While existing approaches only model finite, discrete fidelities, in practice, the feasible fidelity choice is often infinite, which can correspond to a continuous mesh spacing or finite element length. In this paper, we propose Infinite Fidelity Coregionalization (IFC). Given the data, our method can extract and exploit rich information within infinite, continuous fidelities to bolster the prediction accuracy. Our model can interpolate and/or extrapolate the predictions to novel fidelities that are not covered by the training data. Specifically, we introduce a low-dimensional latent output as a continuous function of the fidelity and input, and multiple it with a basis matrix to predict high-dimensional solution outputs. We model the latent output as a neural Ordinary Differential Equation (ODE) to capture the complex relationships within and integrate information throughout the continuous fidelities. We then use Gaussian processes or another ODE to estimate the fidelity-varying bases. For efficient inference, we reorganize the bases as a tensor, and use a tensor-Gaussian variational posterior approximation to develop a scalable inference algorithm for massive outputs. We show the advantage of our method in several benchmark tasks in computational physics.

Learning functions with high-dimensional outputs is critical in many applications, such as physical simulation and engineering design. However, collecting training examples for these applications is often costly, e.g., by running numerical solvers. The recent work (Li et al., 2022) proposes the first multi-fidelity active learning approach for high-dimensional outputs, which can acquire examples at different fidelities to reduce the cost while improving the learning performance. However, this method only queries at one pair of fidelity and input at a time, and hence has a risk of bringing in strongly correlated examples to reduce the learning efficiency. In this paper, we propose Batch Multi-Fidelity Active Learning with Budget Constraints (BMFAL-BC), which can promote the diversity of training examples to improve the benefit-cost ratio, while respecting a given budget constraint for batch queries. Hence, our method can be more practically useful. Specifically, we propose a novel batch acquisition function that measures the mutual information between a batch of multi-fidelity queries and the target function, so as to penalize highly correlated queries and encourages diversity. The optimization of the batch acquisition function is challenging in that it involves a combinatorial search over many fidelities while subject to the budget constraint. To address this challenge, we develop a weighted greedy algorithm that can sequentially identify each (fidelity, input) pair, while achieving a near $(1 - 1/e)$-approximation of the optimum. We show the advantage of our method in several computational physics and engineering applications.

The key task of physical simulation is to solve partial differential equations (PDEs) on discretized domains, which is known to be costly. In particular, high-fidelity solutions are much more expensive than low-fidelity ones. To reduce the cost, we consider novel Gaussian process (GP) models that leverage simulation examples of different fidelities to predict high-dimensional PDE solution outputs. Existing GP methods are either not scalable to high-dimensional outputs or lack effective strategies to integrate multi-fidelity examples. To address these issues, we propose Multi-Fidelity High-Order Gaussian Process (MFHoGP) that can capture complex correlations both between the outputs and between the fidelities to enhance solution estimation, and scale to large numbers of outputs. Based on a novel nonlinear coregionalization model, MFHoGP propagates bases throughout fidelities to fuse information, and places a deep matrix GP prior over the basis weights to capture the (nonlinear) relationships across the fidelities. To improve inference efficiency and quality, we use bases decomposition to largely reduce the model parameters, and layer-wise matrix Gaussian posteriors to capture the posterior dependency and to simplify the computation. Our stochastic variational learning algorithm successfully handles millions of outputs without extra sparse approximations. We show the advantages of our method in several typical applications.

Despite the success of existing tensor factorization methods, most of them conduct a multilinear decomposition, and rarely exploit powerful modeling frameworks, like deep neural networks, to capture a variety of complicated interactions in data. More important, for highly expressive, deep factorization, we lack an effective approach to handle streaming data, which are ubiquitous in real-world applications. To address these issues, we propose SBTD, a Streaming Bayesian Deep Tensor factorization method. We first use Bayesian neural networks (NNs) to build a deep tensor factorization model. We assign a spike-and-slab prior over each NN weight to encourage sparsity and to prevent overfitting. We then use multivariate Delta’s method and moment matching to approximate the posterior of the NN output and calculate the running model evidence, based on which we develop an efficient streaming posterior inference algorithm in the assumed-density-filtering and expectation propagation framework. Our algorithm provides responsive incremental updates for the posterior of the latent factors and NN weights upon receiving newly observed tensor entries, and meanwhile identify and inhibit redundant/useless weights. We show the advantages of our approach in four real-world applications.

Tensor decomposition is a powerful framework for multiway data analysis. Despite the success of existing approaches, they ignore the sparse nature of the tensor data in many real-world applications, explicitly or implicitly assuming dense tensors. To address this model misspecification and to exploit the sparse tensor structures, we propose Nonparametric dEcomposition of Sparse Tensors (\ours), which can capture both the sparse structure properties and complex relationships between the tensor nodes to enhance the embedding estimation. Specifically, we first use completely random measures to construct tensor-valued random processes. We prove that the entry growth is much slower than that of the corresponding tensor size, which implies sparsity. Given finite observations (\ie projections), we then propose two nonparametric decomposition models that couple Dirichlet processes and Gaussian processes to jointly sample the sparse entry indices and the entry values (the latter as a nonlinear mapping of the embeddings), so as to encode both the structure properties and nonlinear relationships of the tensor nodes into the embeddings. Finally, we use the stick-breaking construction and random Fourier features to develop a scalable, stochastic variational learning algorithm. We show the advantage of our approach in sparse tensor generation, and entry index and value prediction in several real-world applications.

Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned. Importantly, we show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize. We then describe two promising solutions to address these failure modes. The first approach is to use curriculum regularization, where the PINN's loss term starts from a simple PDE regularization, and becomes progressively more complex as the NN gets trained. The second approach is to pose the problem as a sequence-to-sequence learning task, rather than learning to predict the entire space-time at once. Extensive testing shows that we can achieve up to 1-2 orders of magnitude lower error with these methods as compared to regular PINN training.

Bayesian optimization (BO) is a powerful approach for optimizing black-box, expensive-to-evaluate functions. To enable a flexible trade-off between the cost and accuracy, many applications allow the function to be evaluated at different fidelities. In order to reduce the optimization cost while maximizing the benefit-cost ratio, in this paper we propose Batch Multi-fidelity Bayesian Optimization with Deep Auto-Regressive Networks (BMBO-DARN). We use a set of Bayesian neural networks to construct a fully auto-regressive model, which is expressive enough to capture strong yet complex relationships across all the fidelities, so as to improve the surrogate learning and optimization performance. Furthermore, to enhance the quality and diversity of queries, we develop a simple yet efficient batch querying method, without any combinatorial search over the fidelities. We propose a batch acquisition function based on Max-value Entropy Search (MES) principle, which penalizes highly correlated queries and encourages diversity. We use posterior samples and moment matching to fulfill efficient computation of the acquisition function, and conduct alternating optimization over every fidelity-input pair, which guarantees an improvement at each step. We demonstrate the advantage of our approach on four real-world hyperparameter optimization applications.

Many applications involve multi-type event data. Understanding the complex influences of the events on each other is critical to discover useful knowledge and to predict future events and their types. Existing methods either ignore or partially account for these influences. Recent works use recurrent neural networks to model the event rate. While being highly expressive, they couple all the temporal dependencies in a black-box and can hardly extract meaningful knowledge. More important, most methods assume an exponential time decay of the influence strength, which is over-simplified and can miss many important strength varying patterns. To overcome these limitations, we propose SPRITE, a $\underline{S}$elf-adaptable $\underline{P}$oint p$\underline{R}$ocess w$\underline{I}$th nonparametric $\underline{T}$ime d$\underline{E}$cays, which can decouple the influences between every pair of the events and capture various time decays of the influence strengths. Specifically, we use an embedding to represent each event type and model the event influence as an unknown function of the embeddings and time span. We derive a general construction that can cover all possible time decaying functions. By placing Gaussian process (GP) priors over the latent functions and using Gauss-Legendre quadrature to obtain the integral in the construction, we can flexibly estimate all kinds of time-decaying influences, without restricting to any specific form or imposing derivative constraints that bring learning difficulties. We then use weight space augmentation of GPs to develop an efficient stochastic variational learning algorithm. We show the advantages of our approach in both the ablation study and real-world applications.

Tucker decomposition is a classical tensor factorization model. Compared with the most widely used CP decomposition, the Tucker model is much more flexible and interpretable in that it accounts for every possible (multiplicative) interaction between the factors in different modes. However, this also brings in the risk of overfitting and computational challenges, especially in the case of fast streaming data. To address these issues, we develop BASS-Tucker, a BAyesian Streaming Sparse Tucker decomposition method. We place a spike-and-slab prior over the core tensor elements to automatically select meaningful factor interactions so as to prevent overfitting and to further enhance the interpretability. To enable efficient streaming factorization, we use conditional moment matching and Delta’s method to develop one-shot incremental update of the latent factors and core tensor upon receiving each streaming batch. Thereby, we avoid processing the data points one by one as in the standard assumed density filtering, which needs to update the core tensor for each point and is quite inefficient. We explicitly introduce and update a sparse prior approximation in the running posterior to fulfill effective sparse estimation in the streaming inference. We show the advantage of BASS-Tucker in several real-world applications.

Learning compact representations for shapes (binary images) is important for many applications. Although neural network models are very powerful, they usually involve many parameters, require substantial tuning efforts and easily overfit small datasets, which are common in shape-related applications. The state-of-the-art approach, ShapeOdds, as a latent Gaussian model, can effectively prevent overfitting and is more robust. Nonetheless, it relies on a linear projection assumption and is incapable of capturing intrinsic nonlinear shape variations, hence may leading to inferior representations and structure discovery. To address these issues, we propose Infinite ShapeOdds (InfShapeOdds), a Bayesian nonparametric shape model, which is flexible enough to capture complex shape variations and discover hidden cluster structures, while still avoiding overfitting. Specifically, we use matrix Gaussian priors, nonlinear feature mappings and the kernel trick to generalize ShapeOdds to a shape-variate Gaussian process model, which can grasp various nonlinear correlations among the pixels within and across (different) shapes. To further discover the hidden structures in data, we place a Dirichlet process mixture (DPM) prior over the representations to jointly infer the cluster number and memberships. Finally, we exploit the Kronecker-product structure in our model to develop an efficient, truncated variational expectation-maximization algorithm for model estimation. On synthetic and real-world data, we show the advantage of our method in both representation learning and latent structure discovery.

Interaction events among multiple entities are ubiquitous in real-world applications. Although these interactions can be naturally represented by tensors and analyzed by tensor decomposition, most existing approaches are limited to multilinear decomposition forms, and cannot estimate complex, nonlinear relationships in data. More importantly, the existing approaches severely underexploit the time stamps information. They either drop/discretize the time stamps or set a local window to ignore the long-term dependency between the events. To address these issues, we propose a Bayesian nonparametric factorization model for high-order interaction events, which can flexibly estimate/embed the static, nonlinear relationships and capture various long-term and short-term excitations effects, encoding these effects and their decaying patterns into the latent factors. Specifically, we use the latent factors to construct a set of mutually excited Hawkes processes, where we place a Gaussian process prior over the background rates to estimate the static, nonlinear relationships of the entities and propose novel triggering kernels to embed the excitation strengths and their time decaying rates among the interactions. For scalable inference, we derive a fully-decomposed model evidence lower bound to dispose of the huge covariance matrix and expensive log summation terms. Then we develop an efficient stochastic optimization algorithm. We show the advantage of our approach in four real-world applications.

In many applications, a parsimonious model is often preferred for better interpretability and predictive performance. Online algorithms have been studied extensively for building such models in big data and fast evolving environments, with a prominent example, FTRL-proximal [1]. However, existing methods typically do not provide confidence levels, and with the usage of L1 regularization, the model estimation can be undermined by the uniform shrinkage on both relevant and irrelevant features. To address these issues, we developed OLSS, a Bayesian online sparse learning algorithm based on the spike-and-slab prior. OLSS achieves the same scalability as FTRL-proximal, but realizes appealing selective shrinkage and produces rich uncertainty information, such as posterior inclusion probabilities and feature weight variances. On the tasks of text classification and click-through-rate (CTR) prediction for Yahoo!'s display and search advertisement platforms, OLSS often demonstrates superior predictive performance to the state-of-the-art methods in industry, including Vowpal Wabbit [2] and FTRL-proximal.

Tensor decomposition is a fundamental framework to model and analyze multiway data, which are ubiquitous in realworld applications. A critical challenge of tensor decomposition is to capture a variety of complex relationships/interactions while avoiding overfitting the data that are usually very sparse. Although numerous tensor decomposition methods have been proposed, they are mostly based on a multilinear form and hence are incapable of estimating more complex, nonlinear relationships. To address the challenge, we propose POND, PrObabilistic Neural-kernel tensor Decomposition that unifies the self-adaptation of Bayes nonparametric function learning and the expressive power of neural networks. POND uses Gaussian processes (GPs) to model the hidden relationships and can automatically detect their complexity in tensors, preventing both underfitting and overfitting. POND then incorporates convolutional neural networks to construct the GP kernel to greatly promote the capability of estimating highly nonlinear relationships. To scale POND to large data, we use the sparse variational GP framework and reparameterization trick to develop an efficient stochastic variational learning algorithm. On both synthetic and real-world benchmark datasets, POND often exhibits better predictive performance than the state-of-the-art nonlinear tensor decomposition methods. In addition, as a Bayesian approach, POND provides the posterior distribution of the latent factors, and hence can conveniently quantify their uncertainty and the confidence levels for predictions.

Tensor factorization is a fundamental framework to analyze high-order interactions in data. Despite the success of the existing methods, the valuable temporal information are severely underused. The timestamps of the interactions are either ignored or discretized into crude steps. The recent work although formulates event-tensors to keep the timestamps in factorization and can capture mutual excitation effects among the interaction events, it overlooks another important type of temporal influence, inhibition. In addition, it uses a local window to exclude all the long-term dependencies. To overcome these limitations, we propose a self-modulating nonparametric Bayesian factorization model. We use the latent factors to construct mutually governed, general random point processes, which can capture various short-term/long-term, excitation/inhibition effects, so as to encode the complex temporal dependencies into factor representations. In addition, our model couples with a latent Gaussian process to estimate and fuse nonlinear yet static relationships between the entities. For efficient inference, we derive a fully decomposed model evidence lower bound to dispense with the huge kernel matrix and costly summations inside the rate and log rate functions. We then develop an efficient stochastic optimization algorithm. We show the advantage of our method in four real-world applications.

Gaussian process regression networks (GPRN) are powerful Bayesian models for multi-output regression, but their inference is intractable. To address this issue, existing methods use a fully factorized structure (or a mixture of such structures) over all the outputs and latent functions for posterior approximation, which, however, can miss the strong posterior dependencies among the latent variables and hurt the inference quality. In addition, the updates of the variational parameters are inefficient and can be prohibitively expensive for a large number of outputs. To overcome these limitations, we propose a scalable variational inference algorithm for GPRN, which not only captures the abundant posterior dependencies but also is much more efficient for massive outputs. We tensorize the output space and introduce tensor/matrix-normal variational posteriors to capture the posterior correlations and to reduce the parameters. We jointly optimize all the parameters and exploit the inherent Kronecker product structure in the variational model evidence lower bound to accelerate the computation. We demonstrate the advantages of our method in several real-world applications.

Bayesian optimization (BO) is a popular framework for optimizing black-box functions. In many applications, the objective function can be evaluated at multiple fidelities to enable a trade-off between the cost and accuracy. To reduce the optimization cost, many multi-fidelity BO methods have been proposed. Despite their success, these methods either ignore or over-simplify the strong, complex correlations across the fidelities. While the acquisition function is therefore easy and convenient to calculate, these methods can be inefficient in estimating the objective function. To address this issue, we propose Deep Neural Network Multi-Fidelity Bayesian Optimization (DNN-MFBO) that can flexibly capture all kinds of complicated relationships between the fidelities to improve the objective function estimation and hence the optimization performance. We use sequential, fidelity-wise Gauss-Hermite quadrature and moment-matching to compute a mutual information-based acquisition function in a tractable and highly efficient way. We show the advantages of our method in both synthetic benchmark datasets and real-world applications in engineering design.

Despite the success of the recent nonlinear tensor decomposition models based on Gaussian processes (GPs), they lack an effective way to deal with streaming data, which are important for many applications. Using the standard streaming variational Bayes framework or the recent streaming sparse GP approximations will lead to intractable model evidence lower bounds; although we can use stochastic gradient descent for incremental updates, they are unreliable and often yield poor estimations. To address this problem, we propose Streaming Nonlinear Bayesian Tensor Decomposition (SNBTD) that can conduct high-quality, closed-form and iteration-free updates upon receiving new tensor entries. Specifically, we use random Fourier features to build a sparse spectrum GP decomposition model to dispense with complex kernel/matrix operations and to ease posterior inference. We then extend the assumed-density-filtering framework by approximating all the data likelihoods in a streaming batch with a single factor to perform one-shot updates. We use conditional moment matching and Taylor approximations to fulfill efficient, analytical factor calculation. We show the advantage of our method on four real-world applications.

While most Gaussian processes (GP) work focus on learning single-output functions, many applications, such as physical simulations and gene expressions prediction, require estimations of functions with many outputs. The number of outputs can be much larger than or comparable to the size of training samples. Existing multi-output GP models either are limited to low-dimensional outputs and restricted kernel choices, or assume oversimplified low-rank structures within the outputs. To address these issues, we propose HOGPR, a High-Order Gaussian Process Regression model, which can flexibly capture complex correlations among the outputs and scale up to a large number of outputs. Specifically, we tensorize the high-dimensional outputs, introducing latent coordinate features to index each tensor element (i.e., output) and to capture their correlations. We then generalize a multilinear model to a hybrid of a GP and latent GP model. The model is endowed with a Kronecker product structure over the inputs and the latent features. Using the Kronecker product properties and tensor algebra, we are able to perform exact inference over millions of outputs. We show the advantage of the proposed model on several real-world applications.

Kernel support vector machines (SVMs) deliver state-of-the-art results in many real-world nonlinear classification problems, but the computational cost can be quite demanding in order to maintain a large number of support vectors. Linear SVM, on the other hand, is highly scalable to large data but only suited for linearly separable problems. In this paper, we propose a novel approach called low-rank linearized SVM to scale up kernel SVM on limited resources. Our approach transforms a nonlinear SVM to a linear one via an approximate empirical kernel map computed from efficient kernel low-rank decompositions. We theoretically analyze the gap between the solutions of the approximate and optimal rank- k kernel map, which in turn provides guidance on the sampling scheme of the Nyström approximation. Furthermore, we extend it to a semisupervised metric learning scenario in which partially labeled samples can be exploited to further improve the quality of the low-rank embedding. Our approach inherits rich representability of kernel SVM and high efficiency of linear SVM. Experimental results demonstrate that our approach is more robust and achieves a better tradeoff between model representability and scalability against state-of-the-art algorithms for large-scale SVMs.

Link prediction is a fundamental problem in network modeling. A family of link prediction approaches is to treat network data as an exchangeable array whose entries can be explained by random functions (e.g., block models and Gaussian processes) over latent node factors. Despite their powerful ability in modeling missing links, these models tend to have a large computational complexity and thus are hard to deal with large networks. To address this problem, we develop a novel variational random function model by defining latent Gaussian processes on exchangeable arrays. This model not only inherits the ability of Gaussian process to describe the nonlinear interactions between nodes, but also enjoys significant reduction on computational complexity. To further make the model scalable to large network data, we develop an efficient key-value-free strategy under the map-reduce framework to tremendously reduce the inference time. Experimental results on large network data have demonstrated both the efficacy and efficiency of the proposed method over state-of-the-arts methods in network modeling.

Resource flexing is the notion of allocating resources on-demand as workload changes. This is a key advantage of Virtualized Network Functions (VNFs) over their non-virtualized counterparts. However, it is difficult to balance the timeliness and resource efficiency when making resource flexing decisions due to unpredictable workloads and complex VNF processing logic.

Recurrent Neural Networks (RNNs) are powerful sequence modeling tools. However, when dealing with high dimensional inputs, the training of RNNs becomes computational expensive due to the large number of model parameters. This hinders RNNs from solving many important computer vision tasks, such as Action Recognition in Videos and Image Captioning. To overcome this problem, we propose a compact and flexible structure, namely Block-Term tensor decomposition, which greatly reduces the parameters of RNNs and improves their training efficiency. Compared with alternative low-rank approximations, such as tensortrain RNN (TT-RNN), our method, Block-Term RNN (BT-RNN), is not only more concise (when using the same rank), but also able to attain a better approximation to the original RNNs with much fewer parameters. On three challenging tasks, including Action Recognition in Videos, Image Captioning and Image Generation, BT-RNN outperforms TT-RNN and the standard RNN in terms of both prediction accuracy and convergence rate. Specifically, BT-LSTM utilizes 17,388 times fewer parameters than the standard LSTM to achieve an accuracy improvement over 15.6% in the Action Recognition task on the UCF11 dataset.

Tensor decomposition is a fundamental tool for multiway data analysis. While most decomposition algorithms operate a collection of static data and perform batch processes, many applications produce data in a streaming manner - every time a subset of entries are generated, and previously seen entries cannot be revisited. In such scenarios, traditional decomposition approaches will be inappropriate, because they cannot provide timely updates when new data come in, and they need to access the whole dataset many times for batch optimization. To address this issue, we propose POST, a PrObabilistic Streaming Tensor decomposition algorithm, which enables real-time updates and predictions upon receiving new tensor entries, and supports dynamic growth of all the modes. Compared with the state-of-the-art streaming decomposition approach MAST, POST is more flexible in that it can handle arbitrary orders of streaming entries, and hence is more widely applicable. In addition, as a Bayesian inference algorithm, POST can quantify the uncertainty of the latent embeddings via their posterior distributions, and the confidence levels of the missing entry value predictions. On several real-world datasets, POST exhibits better or comparable predictive performance than MAST and other static decomposition algorithms.

Tensor decompositions are fundamental tools for multiway data analysis. Existing approaches, however, ignore the valuable temporal information along with data, or simply discretize them into time steps so that important temporal patterns are easily missed. Moreover, most methods are limited to multilinear decomposition forms, and hence are unable to capture intricate, nonlinear relationships in data. To address these issues, we formulate event-tensors, to preserve the complete temporal information for multiway data, and propose a novel Bayesian nonparametric decomposition model. Our model can (1) fully exploit the time stamps to capture the critical, causal/triggering effects between the interaction events, (2) flexibly estimate the complex relationships between the entities in tensor modes, and (3) uncover hidden structures from their temporal interactions. For scalable inference, we develop a doubly stochastic variational Expectation-Maximization algorithm to conduct an online decomposition. Evaluations on both synthetic and real-world datasets show that our model not only improves upon the predictive performance of existing methods, but also discovers interesting clusters underlying the data.

Multiway data, described by tensors, are common in real-world applications. For example, online advertising click logs can be represented by a three-mode tensor (user, advertisement, context). The analysis of tensors is closely related to many important applications, such as click-through-rate (CTR) prediction, anomaly detection and product recommendation. Despite the success of existing tensor analysis approaches, such as Tucker, CANDECOMP/PARAFAC and infinite Tucker decompositions, they are either not enough powerful to capture complex hidden relationships in data, or not scalable to handle real-world large data. In addition, they may suffer from the extreme sparsity in real data, i.e., when the portion of nonzero entries is extremely low; they lack of principled ways to discover other patterns — such as an unknown number of latent clusters — which are critical for data mining tasks such as anomaly detection and market targeting. To address these challenges, I used nonparametric Bayesian techniques, such as Gaussian processes (GP) and Dirichlet processes (DP), to model highly nonlinear interactions and to extract hidden patterns in tensors; I derived tractable variational evidence lower bounds, based on which I developed scalable, distributed or online approximate inference algorithms. Experiments on both simulation and real-world large data have demonstrated the effect of my propoaed approaches.

Gaussian processes (GPs) are powerful non-parametric function estimators. However, their applications are largely limited by the expensive computational cost of the inference procedures. Existing stochastic or distributed synchronous variational inferences, although have alleviated this issue by scaling up GPs to millions of samples, are still far from satisfactory for real-world large applications, where the data sizes are often orders of magnitudes larger, say, billions. To solve this problem, we propose ADVGP, the first Asynchronous Distributed Variational Gaussian Process inference for regression, on the recent large-scale machine learning platform, PARAMETER SERVER. ADVGP uses a novel, flexible variational framework based on a weight space augmentation, and implements the highly efficient, asynchronous proximal gradient optimization. While maintaining comparable or better predictive performance, ADVGP greatly improves upon the efficiency of the existing variational methods. With ADVGP, we effortlessly scale up GP regression to a real-world application with billions of samples and demonstrate an excellent, superior prediction accuracy to the popular linear models.

We consider a different setting from regular multitask learning, where data from different tasks share no common instances and no common feature dictionary, while the features can be semantically correlated and all tasks share the same class space. For example, in the two tasks of identifying terrorism information from English news and Arabic news respectively, one associated dataset could be news from Cable News Network (CNN), and the other be news crawled from websites of Arabic countries. Intuitively, these two tasks could help each other, although they share no common feature space. This new setting has brought obstacles to traditional multi-task learning algorithms and multi-view learning algorithms. We argue that these different data sources can be co-trained together by exploring the latent semantics among them. To this end, we propose a new graphical model based on sparse Gaussian Conditional Random Fields (GCRF) and Hilbert-Schmidt Independence Criterion (HSIC). In additional to output the prediction accuracy for each single task, it can also model (1) the dependency between the latent feature spaces of different tasks, (2) the dependency of the category spaces, and (3) the dependency between the latent feature space and the category space in each task. To make the model inference effective, we have provided an efficient variational EM algorithm. Experiments on both synthetic data sets and real-world data sets have indicated the feasibility and effectiveness of the proposed framework.

MOTIVATION: High-throughput mRNA sequencing (RNA-Seq) is a powerful tool for quantifying gene expression. Identification of transcript isoforms that are differentially expressed in different conditions, such as in patients and healthy subjects, can provide insights into the molecular basis of diseases. Current transcript quantification approaches, however, do not take advantage of the shared information in the biological replicates, potentially decreasing sensitivity and accuracy. RESULTS: We present a novel hierarchical Bayesian model called Differentially Expressed Isoform detection from Multiple biological replicates (DEIsoM) for identifying differentially expressed (DE) isoforms from multiple biological replicates representing two conditions, e.g. multiple samples from healthy and diseased subjects. DEIsoM first estimates isoform expression within each condition by (1) capturing common patterns from sample replicates while allowing individual differences, and (2) modeling the uncertainty introduced by ambiguous read mapping in each replicate. Specifically, we introduce a Dirichlet prior distribution to capture the common expression pattern of replicates from the same condition, and treat the isoform expression of individual replicates as samples from this distribution. Ambiguous read mapping is modeled as a multinomial distribution, and ambiguous reads are assigned to the most probable isoform in each replicate. Additionally, DEIsoM couples an efficient variational inference and a post-analysis method to improve the accuracy and speed of identification of DE isoforms over alternative methods. Application of DEIsoM to an hepatocellular carcinoma (HCC) dataset identifies biologically relevant DE isoforms. The relevance of these genes/isoforms to HCC are supported by principal component analysis (PCA), read coverage visualization, and the biological literature. AVAILABILITY AND IMPLEMENTATION: The software is available at https://github.com/hao-peng/DEIsoM. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Tensor decomposition methods are effective tools for modelling multidimensional array data (i.e., tensors). Among them, nonparametric Bayesian models, such as Infinite Tucker Decomposition (InfTucker), are more powerful than multilinear factorization approaches, including Tucker and PARAFAC, and usually achieve better predictive performance. However, they are difficult to handle massive data due to a prohibitively high training cost. To address this limitation, we propose Distributed infinite Tucker (DinTucker), a new hierarchical Bayesian model that enables local learning of InfTucker on subarrays and global information integration from local results. We further develop a distributed stochastic gradient descent algorithm, coupled with variational inference for model estimation. In addition, the connection between DinTucker and InfTucker is revealed in terms of model evidence. Experiments demonstrate that DinTucker maintains the predictive accuracy of InfTucker and is scalable on massive data: On multidimensional arrays with billions of elements from two real-world applications, DinTucker achieves significantly higher prediction accuracy with less training time, compared with the state-of-the-art large-scale tensor decomposition method, GigaTensor.

Sparse learning has received tremendous amount of interest in high-dimensional data analysis due to its model interpretability and the low-computational cost. Among the various techniques, adaptive l1-regularization is an effective framework to improve the convergence behaviour of the LASSO, by using varying strength of regularization across different features. In the meantime, the adaptive structure makes it very powerful in modelling grouped sparsity patterns as well, being particularly useful in high-dimensional multi-task problems. However, choosing an appropriate, global regularization weight is still an open problem. In this paper, inspired by the annealing technique in material science, we propose to achieve "annealed sparsity" by designing a dynamic shrinking scheme that simultaneously optimizes the regularization weights and model coefficients in sparse (multi-task) learning. The dynamic structures of our algorithm are twofold. Feature-wise (spatially), the regularization weights are updated interactively with model coefficients, allowing us to improve the global regularization structure. Iteration-wise (temporally), such interaction is coupled with gradually boosted l1-regularization by adjusting an equality norm-constraint, achieving an annealing effect to further improve model selection. This renders interesting shrinking behaviour in the whole solution path. Our method competes favorably with state-of-the-art methods in sparse (multi-task) learning. We also apply it in expression quantitative trait loci analysis (eQTL), which gives useful biological insights in human cancer (melanoma) study.

Tensor factorization is a powerful tool to analyse multi-way data. Recently proposed nonlinear factorization methods, although capable of capturing complex relationships, are computationally quite expensive and may suffer a severe learning bias in case of extreme data sparsity. Therefore, we propose a distributed, flexible nonlinear tensor factorization model, which avoids the expensive computations and structural restrictions of the Kronecker-product in the existing TGP formulations, allowing an arbitrary subset of tensor entries to be selected for training. Meanwhile, we derive a tractable and tight variational evidence lower bound (ELBO) that enables highly decoupled, parallel computations and high-quality inference. Based on the new bound, we develop a distributed, key-value-free inference algorithm in the MapReduce framework, which can fully exploit the memory cache mechanism in fast MapReduce systems such as Spark. Experiments demonstrate the advantages of our method over several state-of-the-art approaches, in terms of both predictive performance and computational efficiency.

The analysis and diagnosis of Alzheimer’s disease (AD) can be based on genetic variations, e.g., single nucleotide polymorphisms (SNPs) and phenotypic traits, e.g., Magnetic Resonance Imaging (MRI) features. We consider two important and related tasks: i) to select genetic and phenotypical markers for AD diagnosis and ii) to identify associations between genetic and phenotypical data. While previous studies treat these two tasks separately, they are tightly coupled because underlying associations between genetic variations and phenotypical features contain the biological basis for a disease. Here we present a new sparse Bayesian approach for joint association study and disease diagnosis. In this approach, common latent features are extracted from different data sources based on sparse projection matrices and used to predict multiple disease severity levels; in return, the disease status can guide the discovery of relationships between data sources. The sparse projection matrices not only reveal interactions between data sources but also select groups of biomarkers related to the disease. Moreover, to take advantage of the linkage disequilibrium (LD) measuring the non-random association of alleles, we incorporate a graph Laplacian type of prior in the model. To learn the model from data, we develop an efficient variational inference algorithm. Analysis on an imaging genetics dataset for the study of Alzheimer’s Disease (AD) indicates that our model identifies biologically meaningful associations between genetic variations and MRI features, and achieves significantly higher accuracy for predicting ordinal AD stages than the competing methods.

Supervised dimensionality reduction has shown great advantages in finding predictive subspaces. Previous methods rarely consider the popular maximum margin principle and are prone to overfitting to usually small training data, especially for those under the maximum likelihood framework. In this paper, we present a posterior-regularized Bayesian approach to combine Principal Component Analysis (PCA) with the max-margin learning. Based on the data augmentation idea for max-margin learning and the probabilistic interpretation of PCA, our method can automatically infer the weight and penalty parameter of max-margin learning machine, while finding the most appropriate PCA subspace simultaneously under the Bayesian framework. We develop a fast mean-field variational inference algorithm to approximate the posterior. Experimental results on various classification tasks show that our method outperforms a number of competitors.

In the analysis and diagnosis of many diseases, such as the Alzheimer's disease (AD), two important and related tasks are usually required: i) selecting genetic and phenotypical markers for diagnosis, and ii) identifying associations between genetic and phenotypical features. While previous studies treat these two tasks separately, they are tightly coupled due to the same underlying biological basis. To harness their potential benefits for each other, we propose a new sparse Bayesian approach to jointly carry out the two important and related tasks. In our approach, we extract common latent features from different data sources by sparse projection matrices and then use the latent features to predict disease severity levels; in return, the disease status can guide the learning of sparse projection matrices, which not only reveal interactions between data sources but also select groups of related biomarkers. In order to boost the learning of sparse projection matrices, we further incorporate graph Laplacian priors encoding the valuable linkage disequilibrium (LD) information. To efficiently estimate the model, we develop a variational inference algorithm. Analysis on an imaging genetics dataset for AD study shows that our model discovers biologically meaningful associations between single nucleotide polymorphisms (SNPs) and magnetic resonance imaging (MRI) features, and achieves significantly higher accuracy for predicting ordinal AD stages than competitive methods.

Multiway data analysis deals with multiway arrays, i.e., tensors, and the goal is twofold: predicting missing entries by modeling the interactions between array elements and discovering hidden patterns, such as clusters or communities in each mode. Despite the success of existing tensor factorization approaches, they are either unable to capture nonlinear interactions, or computationally expensive to handle massive data. In addition, most of the existing methods lack a principled way to discover latent clusters, which is important for better understanding of the data. To address these issues, we propose a scalable nonparametric tensor decomposition model. It employs Dirichlet process mixture (DPM) prior to model the latent clusters; it uses local Gaussian processes (GPs) to capture nonlinear relationships and to improve scalability. An efficient online variational Bayes Expectation-Maximization algorithm is proposed to learn the model. Experiments on both synthetic and real-world data show that the proposed model is able to discover latent clusters with higher prediction accuracy than competitive methods. Furthermore, the proposed model obtains significantly better predictive performance than the state-of-the-art large scale tensor decomposition algorithm, GigaTensor, on two large datasets with billions of entries.

MOTIVATION: By capturing various biochemical interactions, biological pathways provide insight into underlying biological processes. Given high-dimensional microarray or RNA-sequencing data, a critical challenge is how to integrate them with rich information from pathway databases to jointly select relevant pathways and genes for phenotype prediction or disease prognosis. Addressing this challenge can help us deepen biological understanding of phenotypes and diseases from a systems perspective. RESULTS: In this article, we propose a novel sparse Bayesian model for joint network and node selection. This model integrates information from networks (e.g. pathways) and nodes (e.g. genes) by a hybrid of conditional and generative components. For the conditional component, we propose a sparse prior based on graph Laplacian matrices, each of which encodes detailed correlation structures between network nodes. For the generative component, we use a spike and slab prior over network nodes. The integration of these two components, coupled with efficient variational inference, enables the selection of networks as well as correlated network nodes in the selected networks. Simulation results demonstrate improved predictive performance and selection accuracy of our method over alternative methods. Based on three expression datasets for cancer study and the KEGG pathway database, we selected relevant genes and pathways, many of which are supported by biological literature. In addition to pathway analysis, our method is expected to have a wide range of applications in selecting relevant groups of correlated high-dimensional biomarkers. AVAILABILITY: The code can be downloaded at www.cs.purdue.edu/homes/szhe/software.html. CONTACT: [email protected].

To improve web search effectiveness and help personalized search applications, it is important to understand users' search process, especially the underlying information goal transitions and satisfaction judgment on result pages. Unlike previous work modeling the two types of hidden information separately, the paper proposes to simultaneously model them based on users' full search process, including both queries and clicks. Thus, a full model can be built up and the dependences between them can be leveraged. Specially, we employ a hierarchical conditional random field (HCRF) for learning and prediction, with fruitful search activity features proposed and leveraged. Experimental results show that our approach reaches a high overall precision(87%) and significantly outperforms the baseline methods. Moreover, our model is applied in a re-ranking application and shows that it can benefit personalized web search.