An intrinsic Cramér-Rao bound on Lie groups

28/10/2015
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Abstract

In his 2005 paper, S.T. Smith proposed an intrinsic Cramér- Rao bound on the variance of estimators of a parameter defined on a Riemannian manifold. In the present technical note, we consider the special case where the parameter lives in a Lie group. In this case, by choosing, e.g., the right invariant metric, parallel transport becomes very simple, which allows a more straightforward and natural derivation of the bound in terms of Lie bracket, albeit for a slightly different definition of the estimation error. For bi-invariant metrics, the Lie group exponential map we use to define the estimation error, and the Riemannian exponential map used by S.T. Smith coincide, and we prove in this case that both results are identical indeed.

An intrinsic Cramér-Rao bound on Lie groups

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The intrinsic Cramèr-Rao bound on Lie groups S. Bonnabel (Mines ParisTech) Joint work with A. Barrau (Mines ParisTech/Safran) GSI’15 Ecole Polytechnique, Paris-Saclay, October 30th, 2015 Introduction Parametric estimation consists in infering θ from a sample x1, x2, · · · , xN with xi ∼ p(x; θ), ∀1 ≤ i ≤ N Cramèr-Rao bound: Any unbiased estimator ˆθ(x1, · · · , xn) has limited accuracy on average. Its covariance is lower bounded by the CR bound1. Manifold structure for the parameter space: Estimation error implied the definition of accuracy ˜θ = ˆθ − θ may even be undefined ! 1 Rao, Calyampudi Radakrishna (1945). "Information and the accuracy attainable in the estimation of statistical parameters" Introduction Steven Smith’s insight:2 define the accuracy in an intrinsic way using a Riemannian metric over the manifold ! (see also N. Boumal) 2 Covariance, Subspace, and Intrinsic Cramèr-Rao Bounds, S.T. Smith, 2005. Introduction Steven Smith’s insight:2 define the accuracy in an intrinsic way using a Riemannian metric over the manifold ! (see also N. Boumal) Our approach: if θ ∈ G belongs to Lie group, then use the Lie group counterpart of meaningless error θ − ˆθ to assess estimator accuracy Lie group natural estimation error = θ−1 · ˆθ Our result: we lower bound this natural error. 2 Covariance, Subspace, and Intrinsic Cramèr-Rao Bounds, S.T. Smith, 2005. Outline 1) Classical Cramèr-Rao bound 2) The Riemannian manifold case 3) Intrinsic Cramèr-Rao bound on Lie groups Classical bound derivation Parametric density p(x; θ) and unbiased estimator ˆθ(x) E(ˆθ(x) − θ) = (ˆθ(x) − θ)p(x; θ)dx = 0 Classical bound derivation Parametric density p(x; θ) and unbiased estimator ˆθ(x) E(ˆθ(x) − θ) = (ˆθ(x) − θ)p(x; θ)dx = 0 Under regularity assumptions differentiating w.r.t. θ and using Cauchy-Schwartz inequality 1 ≤ (ˆθ(x) − θ)2 p(x; θ)dx ∂Log(p) ∂θ 2 p(x; θ)dx Classical bound derivation Cramèr-Rao bound we have obtained Var(ˆθ) ≥ I(θ)−1 with I(θ) the Fisher information3 I(θ) = ∂Log(p) ∂θ 2 p(x; θ)dx = E( ∂Log(p) ∂θ 2 ) For higher dimensions θ ∈ Rd similar derivation with I(θ) a matrix. 3 Fisher. "On the foundations of theoretical statistics". 1922. Outline 1) Classical Cramèr-Rao bound 2) The Riemannian manifold case 3) Intrinsic Cramèr-Rao bound on Lie groups S.T. Smith’s insight Signal processing the parameter θ may belong to a manifold (the world is not flat) Signal processing examples S.T. Smith’s insight Signal processing the parameter θ may belong to a manifold (the world is not flat) Signal processing examples Expectations what is the average between e.g., subspaces ?? S.T. Smith’s insight Unbiasedness: Use the tangent (vector) space and exponential map to compute averages (and covariances) ! Outline 1) Classical Cramèr-Rao bound 2) The Riemannian manifold case 3) Intrinsic Cramèr-Rao bound on Lie groups Prototypical example Lie group SO(3) Lie Group of rotations/orientations SO(3) = {R ∈ R3×3 | RT R = Id, det(R) = 1} Prototypical example Lie group SO(3) Lie Group of rotations/orientations SO(3) = {R ∈ R3×3 | RT R = Id, det(R) = 1} Prototypical example Lie group SO(3) Lie Group of rotations/orientations SO(3) = {R ∈ R3×3 | RT R = Id, det(R) = 1} Lie algebra and exponential map: Any ω ∈ R3 defines a rotation matrix exp ((ω)×) with (ω)×b = ω × b Prototypical example Lie group SO(3) Lie Group of rotations/orientations SO(3) = {R ∈ R3×3 | RT R = Id, det(R) = 1} Lie algebra and exponential map: Any ω ∈ R3 defines a rotation matrix exp ((ω)×) with (ω)×b = ω × b Measuring discrepancy between rotations use ω ∈ R3 defined by ˆR = R exp((ω)×) Prototypical example Lie group SO(3) Lie Group of rotations/orientations SO(3) = {R ∈ R3×3 | RT R = Id, det(R) = 1} Lie algebra and exponential map: Any ω ∈ R3 defines a rotation matrix exp ((ω)×) with (ω)×b = ω × b Measuring discrepancy between rotations use ω ∈ R3 defined by ˆR = R exp((ω)×) Intrinsic: identical for e.g., Euler angles, quaternions etc. A motivation: Wahba’s problem Well known problem from the sixties: by measuring known noisy directions in the body frame, estimate the orientation4 yj = Rxj + Vj, 1 ≤ j ≤ N, Vj ∼ N(0, I3) i.i.d 4 Grace Wahba. A least squares estimate of satellite attitude. SIAM review. 1965 A motivation: Wahba’s problem Well known problem from the sixties: by measuring known noisy directions in the body frame, estimate the orientation4 yj = Rxj + Vj, 1 ≤ j ≤ N, Vj ∼ N(0, I3) i.i.d Applications in aeronautics. 4 Grace Wahba. A least squares estimate of satellite attitude. SIAM review. 1965 A motivation: Wahba’s problem Well known problem from the sixties: by measuring known noisy directions in the body frame, estimate the orientation5 yj = Rxj + Vj, 1 ≤ j ≤ N, Vj ∼ N(0, I3) i.i.d Applications in aeronautics. MaxLike estimator is the least squares estimator as − log p(y1, · · · , yN; R) = 1 2 N 1 yj − Rxj 2 + C. 5 Grace Wahba. A least squares estimate of satellite attitude. SIAM review. 1965 A motivation: Wahba’s problem Well known problem from the sixties: by measuring known noisy directions in the body frame, estimate the orientation5 yj = Rxj + Vj, 1 ≤ j ≤ N, Vj ∼ N(0, I3) i.i.d Applications in aeronautics. MaxLike estimator is the least squares estimator as − log p(y1, · · · , yN; R) = 1 2 N 1 yj − Rxj 2 + C. Is the MaxLike estimator efficient ?? 5 Grace Wahba. A least squares estimate of satellite attitude. SIAM review. 1965 CR derivation on SO(3) Unbiased estimators are such that ˆR(y) = R exp(ω(y)×), with ω(y)p(y; R)dy = 0 (∗) and their (intrinsic) covariance is defined by P = ω(y)ω(y)T p(y; R)dy = 0 CR derivation on SO(3) Unbiased estimators are such that ˆR(y) = R exp(ω(y)×), with ω(y)p(y; R)dy = 0 (∗) and their (intrinsic) covariance is defined by P = ω(y)ω(y)T p(y; R)dy = 0 Differentiating the equality (*) w.r.t small variations of R in the tangent space and applying Cauchy-Schwartz we get P J(R)−1 CR derivation on SO(3) Unbiased estimators are such that ˆR(y) = R exp(ω(y)×), with ω(y)p(y; R)dy = 0 (∗) and their (intrinsic) covariance is defined by P = ω(y)ω(y)T p(y; R)dy = 0 Differentiating the equality (*) w.r.t small variations of R in the tangent space and applying Cauchy-Schwartz we get P J(R)−1 − 1 12 (Tr(P)I3 − P)J(R)−1 − 1 12 J(R)−1 (Tr(P)I3 − P) assuming E ω 3 P, and with J the intrinsic FIM. Final result Final result For sufficiently peaked error distribution we have P = ω(y)ω(y)T p(y; R)dy J(R)−1 + C Final result Final result For sufficiently peaked error distribution we have P = ω(y)ω(y)T p(y; R)dy J(R)−1 + C Lie group case we obtained the general formula P Id + 1 12 P.H J−1 Id + 1 12 P.H T with H the structure tensor defined by H(X, Y, Z) = [X, [Y, Z]] and (P.H)kl = ij PijHl ijk (result up to third order terms). Conclusion Our result allows to lower bound the accuracy of any unbiased estimator on Lie groups for estimation error in the sense of group error projected onto the Lie algebra. Differences with Smith’s bound we recover his result for compact Lie groups. 6 See e.g. "Intrinsic filtering on Lie groups" Barrau and Bonnabel, 2013. See also "Extended Kalman filter on Lie groups" by G. Bourmaud, R. Megret, A. Giremus, Y. Berthoumieu, 2013. Conclusion Our result allows to lower bound the accuracy of any unbiased estimator on Lie groups for estimation error in the sense of group error projected onto the Lie algebra. Differences with Smith’s bound we recover his result for compact Lie groups. Posterior CR bounds can be obtained for filtering problems on Lie groups.6 Homegeneous spaces Can we derive a similar result ? 6 See e.g. "Intrinsic filtering on Lie groups" Barrau and Bonnabel, 2013. See also "Extended Kalman filter on Lie groups" by G. Bourmaud, R. Megret, A. Giremus, Y. Berthoumieu, 2013. Using the exponential map, the (right-invariant) intrinsic information matrix can be defined as follows for any ξ ∈ R3 ξT J(R)ξ = (1) ( d dt log p(y | exp(tξ)R)T ( d dt log p(y | exp(tξ)R)p(y | R)dy (2) and then J(R) can be recovered using the standard polarization formulas ξT Jν = 1 2 ((ξ + ν)T J(ξ + ν) − ξT Jξ − νT Jν) and d2 dt2 log p(y | exp(tξ)R)p(y | R)dy = − ( d dt log p(y | exp(tξ)R))( d dt log p(y | exp(tξ)R))dy