From the Kalman Filter to the Particle Filter: A Geometrical Perspective of the Curse of Dimensionality

Abstract

The aim of this contribution is to provide a description of the difference between Kalman filter and particle filter when the state space is of high dimension. In the Gaussian framework, KF and PF give the same theoretical result. However, in high dimension and using finite sampling for the Gaussian distribution, the PF is not able to reproduce the solution produced by the KF. This discrepancy is highlighted from the convergence property of the Gaussian law toward a hypersphere: in high dimension, any finite sample of a Gaussian law lies within a hypersphere centered in the mean of the Gaussian law and of radius square-root of the trace of the covariance matrix. This concentration of probability suggests the use of norm as a criterium that discriminates whether a forecast sample can be compatible or not with a given analysis state. The contribution illustrates important characteristics that have to be considered for the high dimension but does not introduce a new approach to face the curse of dimensionality.

Producing realistic climate data with generative adversarial networks

“…12 shows that the five eigenvectors are similar. One should note that the SVD algorithm used from Pedregosa et al (2011) suffers from sign indeterminacy, meaning that the signs of SVD components depend on the random state and the algorithm. For this reason, we consider the dot product close to both 1 and −1.…”

“…For a normal law in the high dimension space Z = R m , i.e., with m larger than 10, the distributions of the samples are all located in a spherical shell of radius √ m and of thickness on order 1 √ 2 (see, e.g., Pannekoucke et al, 2016). Because the covariance matrix I m is a diagonal of constant variance, no direction of R m is privileged, leading to an isotropic distribution of the direction of the sampled vectors: their unit directions uniformly cover the unit sphere.…”

Comment on npg-2021-6

“…with m larger than 10, the distribution of the samples are all located in a spherical shell of radius √ m and of thickness of order 1 √ 2 (see e.g. Pannekoucke et al (2016)). Because the covariance matrix I m is a diagonal of constant variance, no direction of R m is privileged leading to an isotropic distribution of the direction of the sampled vectors: their unit directions uniformly covers the unit sphere.…”