The Haar integral

By L. Nachbin, Lulu Bechtolsheim. 1965

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Efficient nonparametric n -body force fields from machine learning (2018, methodology): “A rotation-covariant matrix-valued force kernel can at this point be constructed by Haar integration [36, 37] as an integral over the SO (3) manifold [16]: This kernel can be used to infer forces on atoms using a GP regression vector formula analogous to Eq.”
Objective Priors: An Introduction for Frequentists (2011, background): “It is shown in Halmos (1950) and Nachbin (1965) that the right- and left-invariant Haar densities exist and are unique up to a multiplicative constant.”
On-the-fly machine learning force field generation: Application to melting points (2019, background): “The name SOAP relates to the fact that the dot product i B can be re-lated to the Haar integral 34 of the square of an overlap between two probability distributions as where ˆ R denotes the rotational operator, and the integral in (15) needs to be performed over all possible rotations deﬁned by…”
Probability on Trees and Networks (2017, background): “, to determine a subset of a network whose edge boundary is as small as possible relative to the product of the size of the set and the size of its complement; see Naor (2010) for a discussion of its relationship to embeddings.”
Pointwise Rotation-Invariant Network with Adaptive Sampling and 3D Spherical Voxel Convolution (2018, background): “; ) = Z()Y()Z( ) (13) where 2[0;2ˇ];2[0;ˇ];and 2[0;2ˇ], and Z/Y are rotations around Z/Y axes. Haar Measure of SO(3) The normalized Haar measure is dR= d 2ˇ dsin( ) 2 d 2ˇ : (14) The Haar measure [11,18] is invariant because it has the property that Z SO(3) f(R0R)dR= Z SO(3) f(R)dR; (15) for any R02”
Random Walks on Reductive Groups (2016, methodology): “We recall that every locally compact group H supports a left H-invariant Radon measure dh called the Haar measure (see [90]).”