Two-sample Test using Projected Wasserstein Distance: Breaking the Curse of Dimensionality

We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. In particular, we aim to circumvent the curse of dimensionality in Wasserstein distance: when the dimension is high, it has diminishing testing power, which is inherently due to the slow concentration property of Wasserstein metrics in the high dimension space. A key contribution is to couple optimal projection to find the low dimensional linear mapping to maximize the Wasserstein distance between projected probability distributions. We characterize the theoretical property of the finite-sample convergence rate on IPMs and present practical algorithms for computing this metric. Numerical examples validate our theoretical results.Comment: 10 pages, 3 figures. Accepted in ISIT-2

Sinkhorn Distributionally Robust Optimization

We study distributionally robust optimization (DRO) with Sinkhorn distance -- a variant of Wasserstein distance based on entropic regularization. We derive convex programming dual reformulation for a general nominal distribution. Compared with Wasserstein DRO, it is computationally tractable for a larger class of loss functions, and its worst-case distribution is more reasonable for practical applications. To solve the dual reformulation, we develop a stochastic mirror descent algorithm using biased gradient oracles and analyze its convergence rate. Finally, we provide numerical examples using synthetic and real data to demonstrate its superior performance.Comment: 56 pages, 8 figure

Two-sample Test with Kernel Projected Wasserstein Distance

We develop a kernel projected Wasserstein distance for the two-sample test, an essential building block in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. This method operates by finding the nonlinear mapping in the data space which maximizes the distance between projected distributions. In contrast to existing works about projected Wasserstein distance, the proposed method circumvents the curse of dimensionality more efficiently. We present practical algorithms for computing this distance function together with the non-asymptotic uncertainty quantification of empirical estimates. Numerical examples validate our theoretical results and demonstrate good performance of the proposed method.Comment: 49 pages, 10 figures, 4 table

Decomposing the age effect on risk tolerance

Postprint.The importance of investment portfolio allocation has become more apparent since the onset of the late 2000s Great Recession. Individual willingness to take financial risks affects portfolio decisions and investment returns among other factors. Previous research found that people of different ages have dissimilar levels of risk tolerance but the effects of generation, period, and aging were confounded. Using the 1998 to 2007 Survey of Consumer Finances cross-sectional datasets, this study uses an analytical method to separate such effects on financial risk tolerance. Aging and period effects on financial risk tolerance were statistically significant. Implications for researchers and financial planning practitioners and educators are provided.Includes bibliographical references

Majorana Fermions on Zigzag Edge of Monolayer Transition Metal Dichalcogenides

Majorana fermions, quantum particles with non-Abelian exchange statistics, are not only of fundamental importance, but also building blocks for fault-tolerant quantum computation. Although certain experimental breakthroughs for observing Majorana fermions have been made recently, their conclusive dection is still challenging due to the lack of proper material properties of the underlined experimental systems. Here we propose a new platform for Majorana fermions based on edge states of certain non-topological two-dimensional semiconductors with strong spin-orbit coupling, such as monolayer group-VI transition metal dichalcogenides (TMD). Using first-principles calculations and tight-binding modeling, we show that zigzag edges of monolayer TMD can host well isolated single edge band with strong spin-orbit coupling energy. Combining with proximity induced s-wave superconductivity and in-plane magnetic fields, the zigzag edge supports robust topological Majorana bound states at the edge ends, although the two-dimensional bulk itself is non-topological. Our findings points to a controllable and integrable platform for searching and manipulating Majorana fermions.Comment: 12 pages, 7 figure