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We consider a regression problem, where the correspondence between the input and output data is not available. Such shuffled data are commonly observed in many real world problems. Take flow cytometry as an example: the measuring instruments are unable to preserve the correspondence between the samples and the measurements. Due to the combinatorial nature of the problem, most of the existing methods are only applicable when the sample size is small, and are limited to linear regression models. To overcome such bottlenecks, we propose a new computational framework --- ROBOT --- for the shuffled regression problem, which is applicable to large data and complex models. Specifically, we propose to formulate regression without correspondence as a continuous optimization problem. Then by exploiting the interaction between the regression model and the data correspondence, we propose to develop a hypergradient approach based on differentiable programming techniques. Such a hypergradient approach essentially views the data correspondence as an operator of the regression model, and therefore it allows us to find a better descent direction for the model parameters by differentiating through the data correspondence. ROBOT is quite general, and can be further extended to an inexact correspondence setting, where the input and output data are not necessarily exactly aligned. Thorough numerical experiments show that ROBOT achieves better performance than existing methods in both linear and nonlinear regression tasks, including real-world applications such as flow cytometry and multi-object tracking. 

This paper proposes a new meta-learning method – named HARMLESS (HAwkes Relational Meta LEarning method for Short Sequences) for learning heterogeneous point process models from short event sequence data along with a relational net- work. Specifically, we propose a hierarchical Bayesian mixture Hawkes process model, which naturally incorporates the relational information among sequences into point process modeling. Compared with existing methods, our model can capture the underlying mixed-community patterns of the relational network, which simultaneously encourages knowledge sharing among sequences and facilitates adaptive learning for each individual sequence. We further propose an efficient stochastic variational meta expectation maximization algorithm that can scale to large problems. Numerical experiments on both synthetic and real data show that HARMLESS outperforms existing methods in terms of predicting the future events. 

Optimal Transport (OT) naturally arises in many machine learning applications, yet the heavy computational burden limits its wide-spread uses. To address the scalability issue, we propose an implicit generative learning-based framework called SPOT (Scalable Push-forward of Optimal Transport). Specifically, we approximate the optimal transport plan by a pushforward of a reference distribution, and cast the optimal transport problem into a minimax problem. We then can solve OT problems efficiently using primal dual stochastic gradient-type algorithms. We also show that we can recover the density of the optimal transport plan using neural ordinary differential equations. Numerical experiments on both synthetic and real datasets illustrate that SPOT is robust and has favorable convergence behavior. SPOT also allows us to efficiently sample from the optimal transport plan, which benefits downstream applications such as domain adaptation. 

Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regularized variations with large regularization parameter will degradate the performance in several important machine learning applications, and small regularization parameter will fail due to numerical stability issues with existing algorithms. We address this challenge by developing an Inexact Proximal point method for exact Optimal Transport problem (IPOT) with the proximal operator approximately evaluated at each iteration using projections to the probability simplex. The algorithm (a) converges to exact Wasserstein distance with theoretical guarantee and robust regularization parameter selection, (b) alleviates numerical stability issue, (c) has similar computational complexity to Sinkhorn, and (d) avoids the shrinking problem when apply to generative models. Furthermore, a new algorithm is proposed based on IPOT to obtain sharper Wasserstein barycenter. 

Optimal Transport (OT) naturally arises in many machine learning applications, yet the heavy computational burden limits its wide-spread uses. To address the scalability issue, we propose an implicit generative learning-based framework called SPOT (Scalable Push-forward of Optimal Transport). Specifically, we approximate the optimal transport plan by a pushforward of a reference distribution, and cast the optimal transport problem into a minimax problem. We then can solve OT problems efficiently using primal dual stochastic gradient-type algorithms. We also show that we can recover the density of the optimal transport plan using neural ordinary differential equations. Numerical experiments on both synthetic and real datasets illustrate that SPOT is robust and has favorable convergence behavior. SPOT also allows us to efficiently sample from the optimal transport plan, which benefits downstream applications such as domain adaptation.