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Inverse optimal transport (OT) refers to the problem of learning the cost function for OT from observed transport plan or its samples. In this paper, we derive an unconstrained convex optimization formulation of the inverse OT problem, which can be further augmented by any customizable regularization. We provide a comprehensive characterization of the properties of inverse OT, including uniqueness of solutions. We also develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for discrete OT, and the other one is a learning based algorithm that parameterizes the cost function as a deep neural network for continuous OT. The novel framework proposed in the work avoids repeatedly solving a forward OT in each iteration which has been a thorny computational bottleneck for the bi-level optimization in existing inverse OT approaches. Numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state-of-the-art methods.
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Traditional multi-view learning methods often rely on two assumptions: ($i$) the samples in different views are well-aligned, and ($ii$) their representations in latent space obey the same distribution. Unfortunately, these two assumptions may be questionable in practice, which limits the application of multi-view learning. In this work, we propose a hierarchical optimal transport (HOT) method to mitigate the dependency on these two assumptions. Given unaligned multi-view data, the HOT method penalizes the sliced Wasserstein distance between the distributions of different views. These sliced Wasserstein distances are used as the ground distance to calculate the entropic optimal transport across different views, which explicitly indicates the clustering structure of the views. The HOT method is applicable to both unsupervised and semi-supervised learning, and experimental results show that it performs robustly on both synthetic and real-world tasks.
The objective in statistical Optimal Transport (OT) is to consistently estimate the optimal transport plan/map solely using samples from the given source and target marginal distributions. This work takes the novel approach of posing statistical OT as that of learning the transport plans kernel mean embedding from sample based estimates of marginal embeddings. The proposed estimator controls overfitting by employing maximum mean discrepancy based regularization, which is complementary to $phi$-divergence (entropy) based regularization popularly employed in existing estimators. A key result is that, under very mild conditions, $epsilon$-optimal recovery of the transport plan as well as the Barycentric-projection based transport map is possible with a sample complexity that is completely dimension-free. Moreover, the implicit smoothing in the kernel mean embeddings enables out-of-sample estimation. An appropriate representer theorem is proved leading to a kernelized convex formulation for the estimator, which can then be potentially used to perform OT even in non-standard domains. Empirical results illustrate the efficacy of the proposed approach.
We design an active learning algorithm for cost-sensitive multiclass classification: problems where different errors have different costs. Our algorithm, COAL, makes predictions by regressing to each labels cost and predicting the smallest. On a new example, it uses a set of regressors that perform well on past data to estimate possible costs for each label. It queries only the labels that could be the best, ignoring the sure losers. We prove COAL can be efficiently implemented for any regression family that admits squared loss optimization; it also enjoys strong guarantees with respect to predictive performance and labeling effort. We empirically compare COAL to passive learning and several active learning baselines, showing significant improvements in labeling effort and test cost on real-world datasets.
Variational Auto-Encoders enforce their learned intermediate latent-space data distribution to be a simple distribution, such as an isotropic Gaussian. However, this causes the posterior collapse problem and loses manifold structure which can be important for datasets such as facial images. A GAN can transform a simple distribution to a latent-space data distribution and thus preserve the manifold structure, but optimizing a GAN involves solving a Min-Max optimization problem, which is difficult and not well understood so far. Therefore, we propose a GAN-like method to transform a simple distribution to a data distribution in the latent space by solving only a minimization problem. This minimization problem comes from training a discriminator between a simple distribution and a latent-space data distribution. Then, we can explicitly formulate an Optimal Transport (OT) problem that computes the desired mapping between the two distributions. This means that we can transform a distribution without solving the difficult Min-Max optimization problem. Experimental results on an eight-Gaussian dataset show that the proposed OT can handle multi-cluster distributions. Results on the MNIST and the CelebA datasets validate the effectiveness of the proposed method.
Label space expansion for multi-label classification (MLC) is a methodology that encodes the original label vectors to higher dimensional codes before training and decodes the predicted codes back to the label vectors during testing. The methodology has been demonstrated to improve the performance of MLC algorithms when coupled with off-the-shelf error-correcting codes for encoding and decoding. Nevertheless, such a coding scheme can be complicated to implement, and cannot easily satisfy a common application need of cost-sensitive MLC---adapting to different evaluation criteria of interest. In this work, we show that a simpler coding scheme based on the concept of a reference pair of label vectors achieves cost-sensitivity more naturally. In particular, our proposed cost-sensitive reference pair encoding (CSRPE) algorithm contains cluster-based encoding, weight-based training and voting-based decoding steps, all utilizing the cost information. Furthermore, we leverage the cost information embedded in the code space of CSRPE to propose a novel active learning algorithm for cost-sensitive MLC. Extensive experimental results verify that CSRPE performs better than state-of-the-art algorithms across different MLC criteria. The results also demonstrate that the CSRPE-backed active learning algorithm is superior to existing algorithms for active MLC, and further justify the usefulness of CSRPE.
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