Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica

Sliced Wasserstein Distance for Learning Gaussian Mixture Models

Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm

A Submodular Optimization Framework for Imbalanced Text Classification with Data Augmentation

In the domain of text classification, imbalanced datasets are a common occurrence. The skewed distribution of the labels of these datasets poses a great challenge to the performance of text classifiers. One popular way to mitigate this challenge is to augment underwhelmingly represented labels with synthesized items. The synthesized items are generated by data augmentation methods that can typically generate an unbounded number of items. To select the synthesized items that maximize the performance of text classifiers, we introduce a novel method that selects items that jointly maximize the likelihood of the items belonging to their respective labels and the diversity of the selected items. Our proposed method formulates the joint maximization as a monotone submodular objective function, whose solution can be approximated by a tractable and efficient greedy algorithm. We evaluated our method on multiple real-world datasets with different data augmentation techniques and text classifiers, and compared results with several baselines. The experimental results demonstrate the effectiveness and efficiency of our method