An information-theoretic analog of a result of Perelman

An informati on-theoretic analog of a result of P erelman Bro c kwa y McMillan Sedgwic k Maine, USA (e-mail: bmcmlln@h ughes.net) Octob er 29, 2018 Abstract Eac h compact manif old M of finite dimension k is differen tiable and supp orts an in t rin sic probability measure. There then exists a mea su rable transformation of M to the k -dimensional ” surface” of th e ( k + 1) − dim- ensional ball. 1 Manifolds 1.1 T op ologies, co ordinates, and measures Let k b e a p o s itive in teg e r . By definition , a given compact topo logical space M is a manifold of k dimensions .if every p oint p ∈ M has a neig hborho o d that is top olog ically eq uiv alen t to a Euclidean op en spher e , o f k dimensions, cen tere d at p. Call s uch a neighborho o d a c el l c enter e d at p, . or, less for mally , a c el l . T o a given cell C centered a t p then corresp o nds a top ologica l transforma- tion T , sp ecific to C , tha t trans forms a E uclidean k -spher e, and therefor e a lso transfers the co o rdinate axes defined ther ein, to a to p o logical space C ⊂ M . On the Euclidean space T − 1 C there exists a σ -a lgebra F E of measur able sets– the smallest σ -a lgebra that con tains every op en set. Be cause T takes op en sets to op en sets T F E is also a σ - algebra on C. Then T is what is often called a me asur able tr ansformation . It transfers the Leb esgue measure λ ( · ) defined on T − 1 C to a meas ure µ ( · ) = λ ( T − 1 · ) on C. As the image of a Euclidean spher e, a ce ll centered at p can be giv e n a quasi- Cartesian co ordinate frame, sp ecific to that cell, with origin at p and identified by its (curvilie a r) co ordina tes, say x 1 , x 2 , · · · , x k . M is compact and is cov ered b y finitely many c e lls. Such a covering is a pr op er c overing. Let C 1 , C 2 , b e distinct cells of a giv en pro per co vering, endow ed resp ectively with co or dinate frames { x r } , { y s } . Let p b e a p oint in C 1 ∩ C 2 . Becaus e M is a manifold, the co ordinates { x r } of p a r e alrea dy constrained to b e contin uous 1 functions of the co ordinates { y s } of p, . and vic e versa. The rest of the pap er hinges on the ex istence of a much stro ng er constr aint. Theorem 1 The c omp act manifold M ab ove is differen tiable in the sense t hat the { x r } ar e differ entiable fun ctions of the { y s } , and vic e versa . A pro of app ea rs in the next subsection. The compa ctness of M implies that any covering of M by c e lls contains a sub c ov ering by finitely man y ce lls . Such a s ub covering, by definition, is a pr op er c overing. A p oint in M with c o ordinates { x r } is o ften ca lle d a ve ctor, and M then treated a s a linea r spa ce. T ha t vector space is a Riema n nian manifold. On it exists its tensor calculus , a p ow er ful to ol, somewhat complex. The arguments here do no t require a linea r structure on M and can then for tunately ignore the tensor calculus. 1.2 The in trinsic measure of v olume Let C ∈ M be a cell c e nt er ed at a p oint p. C is the image under a top olo g- ical transfor mation o f an op en Euclidean s pher e. C is what is often ca lle d a me asur able sp ac e ,– it supp orts a σ -a lg ebra F C of mea surable sets,– the small- est σ -a lgebra that con tains ev ery o p en set. On that σ - algebra there exists a well-defined measure– a ” lo cal” volume defined on a cell of a k -dimensio nal differentiable manifold . In ter ms of the lo cal co ordina tes { x r } that volume is defined by its lo cal density dx 1 dx 2 · · · dx k . This volume elemen t is a function of po sition in the cell on which it is defined. The measure, say µ ( σ ) , of a set σ ∈ F C is the integral over σ of that density function. F ur thermore, at that same p oint, but in an ov er la pping cell, ther e is a volume element, say dy 1 dy 2 · · · dy k . These tw o differential volume elements are e qu al ,– they ar e in fact the volume element of the (Euclidean) tang ent space to M at that p oint. They a re simply trans fer red from the Euclidean pre-ima ge o f the cell at is sue. They therefore do no t v anish at any p oint of M . It follows that there exists on the whole of M a σ -algebra of measurable subsets with a measure µ ( · ) defined ther e o n. That σ -alg ebra is the smalles t σ -alg ebra th a t contains every op en set. (Being an algebr a, it then also contains every closed set.) Since every proper cov er ing is made up of finitely many cells, ea ch of finite volume, µ ( M ) < ∞ . O ne ma y then c ho o se to normalize it to the v alue µ ( M ) = 1 and ca ll µ a pr o bability . Whether o r not nor malized, the µ de fined by this co nstruction is an intrinsic me asur e on M , or int r ins ic pr ob ability if nor malized. 1.2.1 Summary: R -measures By definition an R -measure on M is a measure µ that enjoys the following prop erties : 2 • 0 < µ ( M ) < ∞ . • µ is s m o oth in that each p oint of M is a s et of measur e zero . • If A is a no n- empt y op en subset of M then µ ( A ) > 0 . • It follows that the discrete subsets of M constitute the totality of n ull sets of µ. • It then further follows that any tw o R -mea sures on M a re c omp atible in that each is a bsolutely contin uous with resp ect to the other . Theorem 2 The int rinsic me asure on a given differ entiable k -manifold M is a top olo gic al invariant and is an R -me asur e . Pr o of. First addr ess invari anc e. L et T b e a c ontinuous tr ansformation fr om M onto M ′ = T M . T maps op en sets to op en sets and is ther efor e also a me asur able tr ansformation. If µ is a me asur e on M then µ ′ ( A ) = µ ( T − 1 A ) is a me asu r e on M ′ . It is then e asy t o verify that al l five bul lets ab ove apply to M ′ 2 En trop y Consider a finite pa rtition of the spa ce M into pairwise- dis joint measurable sets. Say π is one such partition: π : M = ⊕ car d ( π ) k =1 C k . (1) With this par tition Claude Shanno n [CS] asso ciates the quantit y H ( π ) = 1 card ( π ) P C ∈ π µ ( C ) log 1 µ ( C ) . (2) He then considers a r efining sequence of partition π 1 ≻ π 2 ≻ · · · and shows that the qua ntit ies H ( π n ) co nv erge in probability to a limit H ( M ) that is indepe n- dent o f the chosen sequence . He calls H ( M ) the entr opy of the measur e µ. It was shown in [Mc1] that these H ( π n ) also conv er ge in £ 1 mean. Stronger con- vergence theor ems were so on pr ov ed by others . By a b o ut 19 80, with the work of Kolmogor ov and colleagues, and finally of D.S. Ornstein, Shannon’s full the- ory of communication be c a me a closed b o ok .That theory includes much mo re than is rep or ted o n here; sp ecifically it also pres ent s w ha t is usually known as Shannon’s Co ding Theo r em. 3 2.0.2 Nomenclature During development of his theor y , Shannon was reluctant to use the term ” en- tropy”. A t the ur g ing of colleag ues, he finally relented. As he fear e d, the term ”entrop y” spawned m uch nonsense. At a meeting of the American Physical Soci- ety in 195 0 one member of the large a udience announced that ”Cla ude Shannon has proved that a heat eng ine can do ma thematical log ic.” (I was there. I heard it. I r ecognized the sp eaker but fortunately no longer reca ll his name.) 2.1 Simple prop erties ( 1 ) Let M 1 , M 2 , b e distinct instance s of the gener ic compa ct differentiable man- ifold M , not necessar ily of the same dimensio n. It fo llows that M 1 ⊗ M 2 is also an instance. By calcula tion, directly from (2), the en tropy o f their c a rtesian pro duct is H ( M 1 ⊗ M 2 ) = H ( M 1 ) + H ( M 2 ) . (3) ( 2 ) Return to the g eneric compact a nd differentiable manifold M . Let µ b e an R -measur e on M and let π b e the finite pa rtition (1). Define Λ( µ, π ) = P C ∈ π µ ( C ) log 1 µ ( C ) − µ ( M ) log 1 µ ( M ) . (4) Let α > 0 . One calcula tes that Λ ( αµ, π ) = α Λ( µ, π ) . Consequently , if α, β , a r e non-negative num b ers and µ 1 , µ 2 , are R -measure s on M then Λ( αµ 1 + β µ 2 , π ) = α Λ( µ 1 , π ) + β Λ( µ 2 , π ) , (5) the prop er ty of strict conv exity . (See ¶ 4 later.) Let π 1 ≻ π 2 ≻ · · · be a refining sequence of finite partition of M into measurable sets. By Sha nnon’s theor em, if µ is a pr obability measure o n M then lim n →∞ ( card ( π n )) − 1 Λ( µ, π n ) = H ( µ ) . It follows that Theorem 3 ( 1 ) H ( µ ) , as a function of pr ob ability me asu r es µ define d on M , is strictly c onvex. ( 2 ) Ther e exists a u nique pr ob ability me asur e µ ∞ on M that maximizes H ( · ) . It is char acterize d by the pr op erty t hat for al l me asur able subsets A, B , in M µ ∞ ( A ∩ B ) = µ ∞ ( A ) · µ ∞ ( B ) . (6) ( 3 ) Every pr ob ability me asur e µ on M is c omp atible with µ ∞ , and in p articular is absolutely c ontinu ous with r esp e ct to µ ∞ . Pro of. ( 1 ) simply r ep eats (5). The pro of of ( 2 ) inv olves a n excurs io n into the theory of p oint proces ses on a space such as M , undertaken in the next section. 4 3 The Poi sson Pro cess on M A p oint pr o c ess on M is a random pro cess of whic h the generic r andom v ar ia ble γ is a dis c rete subset of M ; in the present ca s e, that discreteness implies that a random set is als o a finite subset of M . It is a theorem of Thomas Kurtz [TK] that any p oint pr o cess is characterized by its avoida n c e function E ( A ) = P r ob { γ | A ∩ γ = ∅} = Pr ob { ca r d ( A ∩ γ ) = 0 } . . All that one needs to know here is that if µ is a measure on M then the function E ( A ) = e − µ ( A ) is a v a lid av oida nce function. It is the av o idance function of a p oint pr o cess that is the co un ter part on M of the Poisson pro cess on the real line. This latter is the discr ete homo gene ous chaos o f Norb ert Wiener [NW]. On any space that is lo cally compact and metrizable, the P ois s on pro cess is tha t unique pr o cess for which, for each tw o measurable sets A, B , the r andom sets A ∩ γ , B ∩ γ , are s tatistically indep endent whenever A ∩ B = ∅ . Co nclusions ( 2 ) and ( 3 ) ab ove ar e explicit in each of Lemma 43 and Theorem 48 of [Mc2 ]  4 P erelman’s result, discussion Perelman shows that the Ricci flow car ries a given compact k -manifold M to a termina l k - manifold that has a maximum entropy . The arguments ab ov e demonstrate a differ e nt w ay to make a simila r a sso ciation. In each case the en- tropy in volved is that of Shannon and of s ta tistical mechanics. The maxentropic manifold in ea ch is a featureles s ma nifold that is top olog ically the k -dimensional ”surface” of a ( k + 1) − dimens io nal ba ll . Subsection 2.1 mentions conv exity , a matter of no conseq ue nce to Theorem 3 ab ov e. There is a literally mo nstrous ge dankenexp erimente, attributed to Einstein, showing that some simple thermo dynamic prop er ties, combined with the prop er ty of convexit y , make the en tr opy H ( · ) a unique functional. In [J vN], von Neumann des crib es E instein’s arg umen t,– with some ev idence of distaste,– without, to my reading, fully closing the iss ue of uniqueness. Perelman’s Ricci flow clinc hes that latter iss ue. Theorem 3 a bove does a lso, but, as an exis tence theorem, lacks the inev itable forc e of Perelman’s cons tr uctive Ricci flow . 4.1 Credits I thank Pr of. Auber t Daignea ult for cor rections, suggestions, and patience . Thanks also are due Aaron F. McMillan for a revie w of mo dern differential geometry . 4.2 Bibliograph y [CES] Shanno n, Claude E. A mathematic al the ory of c ommunic ation , Bell Sys- tem T echnical Jour nal, v .27 , pp 379 -423, pp 623 -656, July-O ctob er 1948. 5 [GP] Grisha Perelman The ent r opy formula for t he Ric ci flow and its ge ometric applic ations, Arxiv ( Math/ DG) [JvN] John von Neumann, Quantenme chanik, Dov er NY, 194 5. (r e pr inted from Springer, pre-19 40.) [Mc1] B r o ckw ay McMilla n, The b asic t he or ems of information the ory, Annals of Mathematical Statistics, V 24 pp 196 -219, June 1953 . [Mc2] · · · , A taxonomy for r andom sets, International Journa l of Pure and Ap- plied Mathematics, v 34 , No.3, pp 3 47-39 6. Addendum (for the author’s conv enience during the s ubmission pro cess .) Grisha Perelman < p erelma n@math.sunysb.edu > 6