2019/02/12 by Amir-Hossein Bateni, Bateni, Amir-Hossein, Arnak S. Dalalyan +1 · 1 citation
Mathematics · Computer Science · Engineering · #Advanced Statistical Methods and Models #Machine Learning and Algorithms #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1902.04650
We consider the problem of estimating the mean of a distribution supported by\nthe k-dimensional probability simplex in the setting where an \ε\nfraction of observations are subject to adversarial corruption. A simple\nparticular example is the problem of estimating the distribution of a discrete\nrandom variable. Assuming that the discrete variable takes k values, the\nunknown parameter boldsymbol \θ is a k-dimensional vector belonging to\nthe probability simplex. We first describe various settings of contamination\nand discuss the relation between these settings. We then establish minimax\nrates when the quality of estimation is measured by the total-variation\ndistance, the Hellinger distance, or the mathbb L2-distance between two\nprobability measures. We also provide confidence regions for the unknown mean\nthat shrink at the minimax rate. Our analysis reveals that the minimax rates\nassociated to these three distances are all different, but they are all\nattained by the sample average. Furthermore, we show that the latter is\nadaptive to the possible sparsity of the unknown vector. Some numerical\nexperiments illustrating our theoretical findings are reported.\n