The quadratic Wasserstein metric for inverse data matching
Björn Engquist, Kui Ren, Yunan Yang
Abstract
Abstract This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ( W 2 ) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W 2 distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that, for some finite-dimensional problems, the W 2 distance leads to optimization problems that have better convexity than the classical L 2 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> distances, making it a more preferred distance to use when solving such inverse matching problems.