Uniqueness of the Fisher Rao metric on the space of smooth densities Peter W. ichor University of Vienna, Austria www.mat.univie.ac.at/ michor CES-Seminar April 6, 2018, Brno Devoted to the memory of Thomas Friedrich (12. Oktober 1949 in Leipzig 27. Februar 2018 in arburg)
Based on: [T.Friedrich: Lecture in Winterschool in Srni, 1990 or 1991] [T.Friedrich. Die Fisher-Information und symplektische Strukturen. ath. Nachr., 153, 273 296, 1991] [.Bauer,.Bruveris, P.ichor: Uniqueness of the Fisher Rao metric on the space of smooth densities, Bull. London ath. Soc. 48, 3 (2016), 499-506, arxiv:1411.5577] [.Bruveris, P.ichor: Geometry of the Fisher-Rao metric on the space of smooth densities, athematische Nachrichten??, arxiv:1607.04550] [.Bruveris, P. ichor, A.Parusinski, A. Rainer: oser s Theorem for manifolds with corners, Proc. AS., arxiv:1604.07787] The infinite dimensional geometry used here is based on: [Andreas Kriegl, Peter W. ichor: The Convenient Setting of Global Analysis. athematical Surveys and onographs, Volume: 53, Amer. ath. Soc., 1997] Wikipedia [https://en.wikipedia.org/wiki/convenient vector space]
Abstract For a smooth compact manifold, any weak Riemannian metric on the space of smooth positive densities which is invariant under the right action of the diffeomorphism group Diff () is of the form G µ (α, β) = C 1 (µ()) α β µ µ µ + C 2(µ()) α for smooth functions C 1, C 2 of the total volume µ() = µ. In this talk the result is extended to compact smooth manifolds with corners (for example, a simplex), and the full proof is given (keeping the (partial) tradition of naturality questions in CES). β
The Fisher Rao metric on the space Prob() of probability densities is of importance in the field of information geometry. Restricted to finite-dimensional submanifolds of Prob(), so-called statistical manifolds, it is called Fisher s information metric [Amari: Differential-geometrical methods in statistics, 1985]. The Fisher Rao metric is invariant under the action of the diffeomorphism group. A uniqueness result was established [Čencov: Statistical decision rules and optimal inference, 1982, p. 156] for Fisher s information metric on finite sample spaces and [Ay, Jost, Le, Schwachhöfer, 2014] extended it to infinite sample spaces. See also [Ay, Jost, Le, Schwachhöfer: Information Geometry, 2017]. The Fisher Rao metric on the infinite-dimensional manifold of all positive smooth probability densities was studied in [Friedrich: Die Fisher-Information und symplektische Strukturen, 1991], including the computation of its curvature.
anifolds with corners A manifold with corners (recently also called a quadrantic manifold) is a smooth manifold modelled on open subsets of R m 0. Assume it is connected and second countable; then it is paracompact and it admits smooth partitions of unity. Any manifold with corners is a submanifold with corners of an open manifold of the same dim. Restriction C ( ) C () is a surjective continuous linear map which admits a continuous linear section.thus C () is a topological direct summand in C ( ) and the same holds for the dual spaces: The space of distributions D (), which we identity with C (), is a direct summand in D ( ). It consists of all distributions with support in. We do not assume that is oriented, but eventually, that is compact. Diffeomorphisms of map the boundary to itself and map the boundary q of corners of codimension q to itself; q is a submanifold of codimension q in ; in general q has finitely many connected components. We shall consider as stratified into the connected components of all q for q > 0.
The space of densities Let m be a smooth manifold, possibly with corners. Let (U α, u α ) be a smooth atlas for it. The volume bundle (Vol(), π, ) of is the 1-dimensional vector bundle (line bundle) which is given by the following cocycle of transition functions: ψ αβ : U αβ = U α U β R \ {0} = GL(1, R), ψ αβ (x) = det d(u β u 1 α )(u α (x)) = 1 det d(u α u 1 β )(u β(x)). Vol() is a trivial line bundle over. But there is no natural trivialization. There is a natural order on each fiber. Since Vol() is a natural bundle of order 1 on, there is a natural action of the group Diff() on Vol(), given by det(t ϕ Vol() 1 ) ϕ Vol(). ϕ
If is orientable, then Vol() = Λ m T. If is not orientable, let be the orientable double cover of with its deck-transformation τ :. Then Γ(Vol()) is isomorphic to the space {ω Ω m ( ) : τ ω = ω}. These are the formes impaires of de Rham. See [ 2008, 13.1] for this. Sections of the line bundle Vol() are called densities. The space Γ(Vol()) of all smooth sections is a Fréchet space in its natural topology; see [Kriegl-, 1997]. For each section α of Vol() of compact support the integral α is invariantly defined as follows: Let (U α, u α ) be an atlas on with associated trivialization ψ α : Vol() Uα R, and let f α be a partition of unity with supp(f α ) U α. Then we put µ = f α µ := f α (uα 1 (y)).ψ α (µ(uα 1 (y))) dy. α U α α u α(u α) The integral is independent of the choice of the atlas and the partition of unity.
The Fisher Rao metric Let m be a smooth compact manifold without boundary. Let Dens + () be the space of smooth positive densities on, i.e., Dens + () = {µ Γ(Vol()) : µ(x) > 0 x }. Let Prob() be the subspace of positive densities with integral 1. For µ Dens + () we have T µ Dens + () = Γ(Vol()) and for µ Prob() we have T µ Prob() = {α Γ(Vol()) : α = 0}. The Fisher Rao metric on Prob() is defined as: G FR µ (α, β) = α β µ µ µ. It is invariant for the action of Diff() on Prob(): ( (ϕ ) G FR) µ FR (α, β) = Gϕ µ(ϕ α, ϕ β) = = ( α µ ϕ )( β µ ϕ ) ϕ µ = α β µ µ µ.
ain Theorem. [BB, 2016] for without boundary Let be a connected smooth compact manifold with corners, of dimension 2. Let G be a smooth (equivalently, bounded) bilinear form on Dens + () which is invariant under the action of Diff(). Then α β G µ (α, β) = C 1 (µ()) µ µ µ + C 2(µ()) α β for smooth functions C 1, C 2 of the total volume µ(). To see that this theorem implies the uniqueness of the Fisher Rao metric, note that if G is a Diff()-invariant Riemannian metric on Prob(), then we can equivariantly extend it to Dens + () via G µ (α, β) = G µ µ() ( ( ) µ ( ) µ α α µ(), β β µ() ).
Relations to right-invariant metrics on diffeom. groups Let µ 0 Prob() be a fixed smooth probability density. In [Khesin, Lenells, isiolek, Preston, 2013] it has been shown, that the degenerate, H 1 -metric 1 2 divµ 0 (X ). div µ 0 (X ).µ 0 on X() is invariant under the adjoint action of Diff(, µ 0 ). Thus the induced degenerate right invariant metric on Diff() descends to a metric on Prob() = Diff(, µ 0 )\ Diff() via Diff() ϕ ϕ µ 0 Prob() which is invariant under the right action of Diff(). This is the Fisher Rao metric on Prob(). In [odin, 2014], the H 1 -metric was extended to a non-degenerate metric on Diff(), also descending to the Fisher Rao metric.
Corollary. Let dim() 2. If a weak right-invariant (possibly degenerate) Riemannian metric G on Diff() descends to a metric G on Prob() via the right action, i.e., the mapping ϕ ϕ µ 0 from (Diff(), G) to (Prob(), G) is a Riemannian submersion, then G has to be a multiple of the Fisher Rao metric. Note that any right invariant metric G on Diff() descends to a metric on Prob() via ϕ ϕ µ 0 ; but this is not Diff()-invariant in general.
Invariant metrics on Dens + (S 1 ). Dens + (S 1 ) = Ω 1 +(S 1 ), and Dens + (S 1 ) is Diff(S 1 )-equivariantly isomorphic to the space of all Riemannian metrics on S 1 via Φ = ( ) 2 : Dens + (S 1 ) et(s 1 ), Φ(fdθ) = f 2 dθ 2. On et(s 1 ) there are many Diff(S 1 )-invariant metrics; see [Bauer, Harms,, 2013]. For example Sobolev-type metrics. Write g et(s 1 ) in the form g = gdθ 2 and h = hdθ 2, k = kdθ 2 with g, h, k C (S 1 ). The following metrics are Diff(S 1 )-invariant: Gg l (h, k) = S 1 h g. (1 + g ) n ( k g ) g dθ ; here g is the Laplacian on S 1 with respect to the metric g. The pullback by Φ yields a Diff(S 1 )-invariant metric on Dens + (): α ( G µ (α, β) = 4 S 1 µ. 1 + Φ(µ)) ( ) n β µ. µ For n = 0 this is 4 times the Fisher Rao metric. For n 1 we get many Diff(S 1 )-invariant metrics on Dens + (S 1 ) and on Prob(S 1 ).
oser s theorem for manifolds with corners [BPR18] Let be a compact smooth manifold with corners, possibly non-orientable. Let µ 0 and µ 1 be two smooth positive densities in Dens + () with µ 0 = µ 1. Then there exists a diffeomorphism ϕ : such that µ 1 = ϕ µ 0. If and only if µ 0 (x) = µ 1 (x) for each corner x 2 of codimension 2, then ϕ can be chosen to be the identity on. This result is highly desirable even for a simplex. The proof is essentially contained in [Banyaga1974], who proved it for manifolds with boundary.
Aside: Geometry of the Fisher-Rao metric α β G µ (α, β) = C 1 (µ()) µ µ µ + C 2(µ()) α β This metric will be studied in different representations. Dens + () R C (, R >0 ) Φ R >0 S C>0 W Id (W, W + ) S C>0. We fix µ 0 Prob() and consider the mapping R : Dens + () C (, R >0 ), R(µ) = f = µ µ 0. The map R is a diffeomorphism and we will denote the induced metric by G = ( R 1) G; it is given by the formula G f (h, k) = 4C 1 ( f 2 ) h, k + 4C 2 ( f 2 ) f, h f, k, and this formula makes sense for f C (, R) \ {0}. The map R is inspired by [B. Khesin, J. Lenells, G. isiolek, S. C. Preston: Geometry of diffeomorphism groups, complete integrability and geometric statistics. Geom. Funct. Anal., 23(1):334-366, 2013.]
Proof of the ain Theorem Let us fix a basic probability density µ 0. By the oser s theorem for manifolds with corners, there exists for each µ Dens + () a diffeomorphism ϕ µ Diff() with ϕ µµ = µ()µ 0 =: c.µ 0 where c = µ() = µ > 0. Then ( (ϕ µ ) G ) µ (α, β) = G ϕ µ µ (ϕ µα, ϕ µβ) = G c.µ0 (ϕ µα, ϕ µβ). Thus it suffices to show that for any c > 0 we have α β G cµ0 (α, β) = C 1 (c). µ 0 + C 2 (c) α β µ 0 µ 0 for some functions C 1, C 2 of the total volume c = µ(). Both bilinear forms are still invariant under the action of the group Diff(, cµ 0 ) = Diff(, µ 0 ) = {ψ Diff() : ψ µ 0 = µ 0 }.
The bilinear form ( α T µ0 Dens + () T µ0 () Dens + (α, β) G cµ0 µ 0, β ) µ 0 µ 0 µ 0 can be viewed as a bilinear form C () C () (f, g) G c (f, g). We will consider now the associated bounded linear mapping Ǧ c : C () C () = D (). (1) The Lie algebra X(,, µ 0 ) of Diff(, µ 0 ) consists of vector fields X which are tangent to each boundary component q with 0 = div µ 0 (X ) := L X µ 0 µ 0.
On an oriented open subset U, each density is an m-form, m = dim(), and div mu 0 (X ) = di X µ 0. The mapping ˆι µ0 : X(U) Ω m 1 (U) given by X i X µ 0 is an isomorphism, and also ˆι µ0 : X(U, U) Ω m 1 (U, U) = = {α Ω m 1 () : j q α = 0 for all q 1} is an isomorphism onto the space of differential forms that pull back to 0 on each boundary stratum. The Lie subalgebra X(U, U, µ 0 ) of divergence free vector fields corresponds to the space of closed (m 1)-forms. Denote by X exact (,, µ 0 ) the set (not a vector space) of exact divergence free vector fields X = ˆι 1 µ 0 (dω), where ω Ωc m 2 (U, U) for an oriented open subset U.
(2) If for f C () and a connected open set U we have (L X f ) U = 0 for all X X exact (,, µ 0 ), then f U is constant. Since we shall need some details later on, we prove this well-known fact. Let x U \ U. For every tangent vector X x T x we can find a vector field X X exact (,, µ 0 ) such that X (x) = X x ; to see this, choose a chart (U x, u) near x such that U x U \ U and µ 0 U x = du 1 du m, and choose g Cc (U x ), such that g = 1 near x. Then X := ˆι 1 µ 0 d(g.u 2.du 3 du m ) X exact (,, µ 0 ) and X = u 1 near x. So we can produce a basis for T x and even a local frame near x. Thus L X f U = 0 for all X X exact (,, µ 0 ) implies df U\ U = 0, thus df = 0 and f is constant on U. (2 ) Similarly, if x q and X x T x ( q ) we can find X X exact (,, µ 0 ) with X (x) = X x.
(3) If for a distribution A D () and a connected open set U we have L X A U = 0 for all X X exact (,, µ 0 ), then A U = Cµ 0 U for some constant C, meaning A, f = C f µ 0 for all f Cc (U). Because L X A, f = A, L X f, the invariance L X A U = 0 implies A, L X f = 0 for all f Cc (U). Clearly, (L X f )µ 0 = 0. For each x U let U x U be an open oriented chart which is diffeomorphic to R q 0 Rm q. Let g Cc (U x ) satisfy gµ 0 = 0; we will show that A, g = 0. The integral over gµ 0 is zero, so the compact cohom. class [gµ 0 ] Hc m (U x, U x ) = R vanishes; see [BPR2018, section 8]. Thus there exists α Ωc m 1 (U x, U x ) Ω m 1 (, ) with dα = gµ 0. Since U x is diffeomorphic to R q 0 Rm q, we can write α = j f jdβ j with β j Ω m 2 (U x, U x ) and f j Cc (U x ). Choose h Cc (U x ) with h = 1 on j supp(f j), so that α = j f jd(hβ j ) and hβ j Ω m 2 (, ). Then the vector fields X j = ˆι 1 µ 0 d(hβ j ) lie in X exact (,, µ 0 ) and we have the identity j f j.i Xi µ 0 = α.
This means j (L X j f j )µ 0 = j L X j (f j µ 0 ) = j di X j (f j µ 0 ) = ( ) d j f j.i Xj µ 0 = dα = gµ 0 or j L X j f j = g, leading to A, g = j A, L Xj f j = j L Xj A, f j = 0. So A, g = 0 for all g Cc (U x ) with gµ 0 = 0. Finally, choose a function ϕ with support in U x and ϕµ 0 = 1. Then for any f Cc (U x ), the function defined by g = f ( f µ 0).ϕ in C () satisfies gµ 0 = 0 and so A, f = A, g + A, ϕ f µ 0 = C x f µ 0, with C x = A, ϕ. Thus A U x = C x µ 0 U x. Since U is connected, the constants C x are all equal: Choose ϕ Cc (U x U y ) with ϕµ0 = 1. Thus (3) is proved.
(4) The operator Ǧc : C () D () has the following property: If for f C () and a connected open U the restriction f U is constant, then we have Ǧ(f ) U = C U (f )µ 0 U for some constant C U (f ). For x U choose g C () with g = 1 near \ U and g = 0 on a neighborhood V of x. Then for any X X exact (,, µ 0 ), that is X = ˆι 1 µ 0 (dω) for some ω Ω m 2 c (W, W ) where W is an oriented open set, let Y = ˆι 1 µ 0 (d(gω)). The vector field Y X exact (,, µ 0 ) equals X near \ U and vanishes on V. Since f is constant on U, L X f = L Y f. For all h C () we have L X Ǧ c (f ), h = Ǧ c (f ), L X h = G c (f, L X h) = G c (L X f, h) = Ǧ c (L X f ), h, since G c is invariant. Thus also L X Ǧ c (f ) = Ǧ c (L X f ) = Ǧ c (L Y f ) = L Y Ǧ c (f ). Now Y vanishes on V and therefore so does L X Ǧ c (f ). By (3) we have Ǧc(f ) V = C V (f )µ 0 V for some C V (f ) R. Since U is connected, all the constants C V (f ) have to agree, giving a constant C U (f ), depending only on U and f. Thus (4) follows.
By the Schwartz kernel theorem, Ǧ c has a kernel Ĝ c, which is a distribution (generalized function) in Ĝ c D ( ) = (D() D()) = = (D() D()) = L(D(), D ()) Ǧc where one needs first the completed inductive or ɛ-tensorproduct, and then the projective one. Note the defining relations G c (f, g) = Ǧ c (f ), g = Ĝ c, f g. oreover, Ĝ c is invariant under the diagonal action of Diff(, µ 0 ) on. In view of the tensor product in the defining relations, the infinitesimal version of this invariance is: L X 0+0 X Ĝ c = 0 for all X X(,, µ 0 ).
(5) There exists a constant C 2 = C 2 (c) such that the distribution Ĝ c C 2 µ 0 µ 0 is supported on the diagonal of. Namely, if (x, y) is not on the diagonal, then there exist open neighborhoods U x of x and U y of y in such that U x U y is disjoint to the diagonal, or U x U y =. Choose any functions f, g C () with supp(f ) U x and supp(g) U y. Then f ( \ U x ) = 0, so by (4), Ǧ c (f ) ( \ U x ) = C \Ux (f ).µ 0. Therefore, G c (f, g) = Ĝc, f g = Ǧc(f ), g = Ǧ c (f ) ( \ U x ), g ( \ U x ), since supp(g) U y \ U x, = C \Ux (f ) gµ 0 By applying the argument for the transposed bilinear form Gc T (g, f ) = G c (f, g), which is also Diff(, µ 0 )-invariant, we get G c (f, g) = Gc T (g, f ) = C (g) f µ \U y 0.
Fix two functions f 0, g 0 with the same properties as f, g and additionally f 0µ 0 = 1 and g 0µ 0 = 1. Then we get C \Ux (f ) = C \U y (g 0 ) f µ 0, and so G c (f, g) = C (g \U y 0 ) = C \Ux (f 0 ) f µ 0 f µ 0 gµ 0 gµ 0. Since dim() 2 and is connected, the complement of the diagonal in is also connected, and thus the constants C \Ux (f 0 ) and C \U y (g 0 ) cannot depend on the functions f 0, g 0 or the open sets U x and U y as long as the latter are disjoint. Thus there exists a constant C 2 (c) such that for all f, g C () with disjoint supports we have G c (f, g) = C 2 (c) f µ 0 gµ 0 Since C c (U x U y ) = C c (U x ) C c (U y ), this implies claim (5).
Now we can finish the proof. We may replace Ĝ c D ( ) by Ĝ c C 2 µ 0 µ 0 and thus assume without loss that the constant C 2 in (5) is 0. Let (U, u) be an oriented chart on such that µ 0 U = du 1 du m, and let Ũ be an extension of U to a smooth manifold without boundary with an extension of the chart mapping u. The distribution Ĝ c U U D (U U) D (Ũ Ũ) has support contained in the diagonal and is of finite order k. By [Hörmander I, 1983, Theorem 5.2.3], the corresponding operator Ǧ c : Cc (U) D (U) is of the form Ĝ c (f ) = α k A α. α f for A α D (U), so that G(f, g) = Ǧ c (f ), g = α A α, ( α f ).g. oreover, the A α in this representation are uniquely given, as is seen by a look at [Hörmander I, 1983, Theorem 2.3.5].
For x U choose an open set U x with x U x U x U, and choose X X exact (,, µ 0 ) with X U x = u i (tangential to the boundary), as in (2 ). For functions f, g Cc (U x ) we then have, by the invariance of G c, 0 = G c (L X f, g) + G c (f, L X g) = Ĝ c U U, L X f g + f L X g = A α, ( α u i f ).g + ( α f )( u i g) α = α A α, u i (( α f ).g) = α u i A α, ( α f ).g. Since the corresponding operator has again a kernel distribution which is supported on the diagonal, and since the distributions in the representation are unique, we can conclude that u i A α U x = 0 for each α, and each i such that u i is tangential to the boundary.
To see that this implies that A α U x = C α µ 0 U x, let f Cc (U x ) with f µ 0 = 0. Then, as in (3), there exists ω Ω m 1 c (U x, U x ) with dω = f µ 0. We have ω = i ω i.du 1 du i du m (only those i with u i tangential to the boundary have ω i 0), and so f = i ( 1)i+1 u i ω i with ω i Cc (U x ). Thus A α, f = i ( 1) i+1 A α, u i ω i = i ( 1) i u i A α, ω i = 0. Hence A α, f = 0 for all f Cc (U x ) with zero integral and as in the proof of (3) we can conclude that A α U x = C α µ 0 U x.
But then G c (f, g) = U x (Lf ).gµ 0 for the differential operator L = α k C α α with constant coefficients on U x. Now we choose g Cc (U x ) such that g = 1 on the support of f. By the invariance of G c we have again 0 = G c (L X f, g)+g c (f, L X g) = L(L X f ).gµ 0 + L(f ).L X g.µ 0 U x U x = L(L X f )µ 0 + 0 U x for each X X(,, µ 0 ). Thus the distribution f U x L(f )µ 0 vanishes on all functions of the form L X f, and by (3) we conclude that L( ).µ 0 = C x.µ 0 in D (U x ), or L = C x Id. By covering with open sets U x, we see that all the constants C x are the same. This concludes the proof of the ain Theorem.
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