June 21, Peking University. Dual Connections. Zhengchao Wan. Overview. Duality of connections. Divergence: general contrast functions

Transcription

1 Dual Peking University June 21, 2016

2 Divergences:

3 Riemannian connection Let M be a manifold on which there is given a Riemannian metric g =,. A connection satisfying Z X, Y = Z X, Y + X, Z Y (1) for all vector fields X, Y, Z T (M) is called the Riemannian connection.

4 Dual connection Giving two and on M, if for all X, Y, Z Z X, Y = Z X, Y + X, Z Y (2) holds, then we say that and are duals of each other with respect to g, and call one either the dual connection or the conjugate connection of the other. In addition, we call such a triple (g,, ) a dualistic structure on M. If is metric, then =. Hence the duality of may be considered as a ization of metric connection. In a statistical model S, (g, (α), ( α) ) is a dualistic structure.

5 Dual connection Given a local frame [x i ], from Equation (2) we have k g ij = Γ ki,j + Γ kj,i. (3) Thus, given g and, there exists a unique dual connection. In addition ( ) = holds. We also see that ( + )/2 becomes a metric connection. And conversely, if a connection has the same torsion as and if ( + )/2 is metric, then =.

6 Submanifold Letting N be a submanifold of M, consider N and N, which are respectively the projections of and onto N with respect to g. These are dual with respect to g N (the metric on N determined by g). We call (g N, N, N ) the dualistic structure on N induced by (g,, ), or the induced dualistic structure on M.

7 Covariant derivative Let γ : t γ(t) be a curve in M and let X and Y be vector fields along γ. In addition, let D t X and D t Y respectively denote the covariant derivatives of X with respect to and Y with respect to. Then from Equation (2), we see that d dt X (t), Y (t) = D tx (t), Y (t) + X (t), D t Y (t) (4) Now suppose that X is parallel with respect to, and that Y is parallel with respect to, i.e., D t X = D t Y = 0. Then X (t), Y (t) is constant on γ.

8 Parallel transform Theorem Letting P γ and P γ (:T p (M) T q (M), where p and q are boundary points of γ) respectively denote the parallel translation along γ with respect ro and, then for all X,Y T p (M) we have P γ (X ), P γ(y ) q = X, Y p. (5) This is a ization of the invariance of the inner product under parallel translation through metric discussed in Chapter 1.

9 Curvature The relationship between P γ and P γ is completely determined by Equation (5). Hence if P γ is independent of the actual curve joining p and q, and hence may be written as P γ = P p,q, then this is true of P γ also. Theorem Letting the curvature tensors of and be denoted by R and R, respectively, we have This is immediate from R = 0 R = 0. (6) R(X, Y )Z, W = R (X, Y )W, Z, X, Y, Z, X T (M). (7)

10 Measure Consider a smooth function D = D( ) : M M R satisfying for any p, q M D(p q) 0, and D(p q) = 0 iff p = q. (8) D is a distance-like measure of the separation between two points. However, it does not in satisfy the axioms of distance (symmetry and the triangle inequality).

11 Derivatives Given an arbitrary coordinate system [x i ] of M, let us represent a pair of points (p, p ) M M by a pair of coordinates ([x i ], [x i ]) and denote the partial derivatives of D(p p ) with respect to p and p by D(( i ) p p ) ( i ) x D(x p ) x = p D(( i ) p ( j ) p ) ( i ) x ( j ) y D(x y) x=p,y=p D(( i j ) p ( k ) p ) ( i ) x ( j ) x ( k ) y D(x y) x=p,y=p, etc., (9) These definitions are naturally extended to those of D((X 1 X l ) p p ), D(p (Y 1 Y m ) p ) and D((X 1 X l ) p (Y 1 Y m ) p ) for any vector fields X 1, X l, Y 1,, Y m T (M).

12 Divergence Now consider the restrictions onto the diagonal {(p, p) p M} M M and denote the induced on M by D[X 1 X l ] : p D((X 1 X l ) p p), D[ Y 1 Y m ] : p D(p (Y 1 Y m ) p ), D[X 1 X l Y 1 Y m ] : p D((X 1 X l ) p (Y 1 Y m ) p ). (10) Easily, we have D[ i ] = D[ i ] = 0, (11) D[ i j ] = D[ i j ] = D[ i j ]( g (D) ij ). (12)

13 Divergence and Riemannian metric The matrix [g (D) ij ] is positive semidefinite (it s the Hessian matrix of the minimum point). When [g (D) ij ] is strictly positive definite everywhere on M, we say that D is a or a function on M. For a D, a unique Riemannian metric g (D) =, (D) on M is defined by i, j (D) = g (D) ij, or equivalently by Using Taylor expansion, we have X, Y (D) = D[X Y ]. (13) D(p q) = 1 2 g (D) ij (q) x i x j + o( x 2 ), (14)

14 Divergence and connection We define a connection (D) with the coefficients Γ (D) ij,k by or equivalently by Γ (D) ij,k = D[ i j k ], (15) (D) X Y, Z (D) = D[XY Z] (16) It s easy to see that Γ (D) ij,k = Γ(D) ji,k Γh(D) ij = Γ h(d) ji.

15 Divergence and connection D(p q) = 1 2 g (D) ij (q) x i x j h(d) ijk (q) x i x j x k +o( x 3 ), (17) where h (D) ijk D[ i j k ] = i g (D) jk + Γ (D) jk,i. (18) Conversely, we see that g (D) and (D) are determined by the expansion (17) through Equation (18).

16 Divergence and dual connection Replace the D(p q) with its dual D (p q) = D(q p). Then we obtain g (D ) = g (D) and Γ (D ) ij,k = D[ k i j ]. (19) Theorem (D) and (D ) are dual with respect to g (D).

17 Divergence and dual connection D(p q) = D (q p) where = 1 2 g (D) ij (p) x i x j 1 6 h(d ) ijk (p) x i x j x k + o( x 3 ) h (D ) ijk (20) D[ i j k ] = i g (D) jk + Γ (D ) jk,i. (21)

18 Dual connection and We see that any induces a torsion-free dualistic structure. Conversely, any triple (g,, ) are induced from a. In fact, if we let where D(p q) 1 2 g ij(q) x i x j h ijk(q) x i x j x k, (22) h ijk i g jk + Γ jk,i = Γ ij,k + Γ ik,j + Γ jk,i, (23) then (g,, ) = (g (D), (D), (D ) ).

19 Let (g,, ) be a dualistic structure on a manifold M. If and are both symmetric (T = T = 0), then from Theorem before we see that -flatness and -flatness are equivalent. We call (M, g,, ) a dually flat space if both dual are flat.

20 Autoparallel Theorem Let (M, g,, ) be a dually flat space. If a submanifold N of M is autoparallel with respect to either or, then N is a dually flat space with respect to the dualistic structure (g N, N, N ) induced on N by (g,, ).

21 Dual coordinate Suppose (U; θ i, η j ) is a coordinate neighborhood of dually flat space (M, g,, ), where [θ i ] and [η j ] denote the affine coordinate system for and respectively. We let i θ i and j η j. i, j is constant on U since they are respectively parallel on flat manifold. Thus we can choose particular coordinate systems such that i, j = δ j i. (24) Such two systems are called mutually dual. We see then that the Euclidean coordinate system defined as i, j = δ ij (affine coordinate) is self-dual.

22 Dual coordinate Dual coordinate systems do not ly exist for a Riemannian manifold. If (M, g,, ) is a dually flat space, then dual coordinate systems exist. Conversely, if for a Riemannian manifold (M, g) there exists such coordinate systems, then and for which they are affine are determined, and (M, g,, ) is a dually flat space.

23 Dual coordinate Let the components of g with respect to [θ i ] and [η j ] be defined by g ij i, j and g ij i, j. (25) Considering j = ( j θ i ) i and i = ( i η j ) j, the Equation (24) is equivalent to η j θ i = g ij and therefore g ij g jk = δ k i. and θ i η j = g ij, (26)

24 Legendre transformations Suppose we are given mutually dual coordinate systems [θ i ] and [η j ], and consider the following partial differential equation for a function ψ : M R : i ψ = η i. (27) Rewrite this as dψ = η i dθ i, and a solution exists iff i η j = j η i. Since i η j = g ij = j η i, a solution ψ always exists. i j ψ = g ij, (28) Hessian matrix of ψ is positve definte, thus it s strictly convex of [θ 1,, θ m ].

25 Legendre transformations Similarly, a solution ϕ to i ϕ = θ i (29) exists. In fact, ϕ = θ i η i ψ is a solution. i j ϕ = g ij, (30) and hence it s a strictly convex function of [η 1,, η m ].

26 From convexity we have Legendre transformations ϕ(q) = max p M {θi (p)η i (q) ψ(p)}, (31) ψ(p) = max q M {θi (p)η i (q) ϕ(q)}. (32) Sometimes it is more natural to view these relations as ϕ(η) = max θ Θ {θi η i ψ(θ)}, (33) ψ(θ) = max η H {θi η i ϕ(η)}, (34) where ψ and ϕ are simply convex defined on convex regions Θ and H in R m.

27 Legendre transformations Those coordinate transformations expressed in Equations (27) through (32) are called Legendre transformations, and ψ and ϕ are called their potentials. Note also that Γ ij,k i j, k = i j k ψ, (35) Γ ij,k i j, k = i j k ψ, (36) which are derived from Equation (3) combined with Γ ij,k = Γ ij,k = 0.

28 Let (M, g,, ) be a dually flat space, on which we are given mutually dual affine coordinate systems {[θ i ], [η i ]}. The canonical or (g, )- is defined as D(p q) ψ(p) + ϕ(q) θ i (p)η i (q). (37) Then from Equation (31) and (32) we see that D(p q) 0 and D(p q) = 0 p = q. It is easy to verify the equations D(( i j ) p q) = g ij (p) and D(p ( i j ) q ) = g ij (q) (38) which immediately implies that D is a and induces g. Also = (D) and = (D ) since Γ ij,k = Γ ij,k = 0 due to the -affinity of [θ i ] and the -affinity of [η i ].

29 Note 1 The canonical is defined globally, though it uses locally defined charts, which is guaranteed by the following lemma. Lemma Suppose M is connected and is flat with respect to, then every two or finite points on M can be contained in a single affine chart.

30 Note 2 If given another set of dual affine coordinate systems expressed by θ j = A j i θi + B j, η j = Cj i η i + D j, (39) ψ = ψ + D j θ j + c, ϕ = ϕ + B j η j B j D j c, (40) where [A j i ] is a regular matrix and [C i j ] is its inverse, [Bj ] and [D j ] are real-valued vectors, and c is a real number, then we have ψ(p) + ϕ(q) θ i (p)η i (q) = ψ(p) + ϕ(q) θ i (p) η i (q), (41) which indicates that the canonical is well defined. On (M, g,, ), we define the (g, )- D (p q) = D(q p).

31 Example If is a Riemannian connection, the condition for dually flat reduces to being flat, and hence there exists a Euclidean coordinate system [θ i ], which is self dual (θ i = η i ), and its potential is given by ψ = ϕ = 1 2 i (θi ) 2. Hence we obtain D(p q) = 1 {(θ i (p)) 2 + (θ i (q)) 2 2θ i (p)θ i (q)} 2 i = 1 2 {d(p, q)}2, (42) where d is the Euclidean distance d(p, q) i {θi (p) θ i (q)} 2. In, D(p q) on a dually flat space is only approximately equal to 1 2 {d(p, q)}2 in the sense of Equation (14).

32 Triangular relation Theorem Let {[θ i ], [η i ]} be mutually dual affine coordinate systems of a dually flat space (M, g,, ), and let D be a on M. Then a necessary and sufficient condition for D to be the (g, )- is that for all p,q,r M the following triangular relation holds: D(p q) + D(q r) D(p r) = {θ i (p) θ i (q)}{η i (r) η i (q)}. (43)

33 Pythagorean relation Theorem Let p,q, and r be three points in M. Let γ 1 be the -geodesic connecting p and q, and let γ 2 be the -geodesic connecting q and r. If at the intersection q the curves γ 1 and γ 2 are orthogonal (with respect to the inner product g), then we have the Pythagorean relation D(p r) = D(p q) + D(q r) (44)

34 Pythagorean relation Figure: The Pythagorean relation for (g, )-s.

35 Projection Corollary Let p be a point in M and let N be a submanifold of M which is -autoparallel. Then a necessary and sufficient condition for a point q in N to satisfy D(p q) = min r N D(p r) is for the -geodesic connecting p and q to be orthogonal to N at q. The point q is called the -projection of p onto N when the geodesic connecting p and q N is orthogonal to N.

36 Projection Figure: The projection theorem of (g, )-.

37 Projection Theorem Let p be a point in M and let N be a submanifold of M. A necessary and sufficient condition for a point q N to be a stationary point of the function D(p ) : r D(p r) restricted on N (in other words, the partial derivatives with respect to a coordinate system of N are all 0) is for the -geodesic connecting p and q to be orthogonal to N at q. Corollary Given a point p in M and a positive number c, suppose that the D-sphere N = {q M D(p q) = c} forms a hypersurface in M. Then every -geodesic passing through the center p orthogonally intersects N.

38 em algorithm Given two submanifolds K and S in a dually flat M, we define a between K and S by D[K S] min D(p q) = D( p q), (45) p K,q S where D is the (g, )- of M and p K and q S are the closest pair between K and S. In order to obtain the closest pair, the following iterative algorithm is proposed.

39 em algorithm Figure: Iterated dual geodesic projections (em algorithm)

40 em algorithm Begin with an arbitrary Q t S, t = 0, 1, and search for P K that minimizes D(P Q t ) which is given by the geodesic projection of Q t to K. Let it be P t K. Then search for the point in S that minimizes D(P t Q) which is given by the dual geodesic projection of P t to S, denoted as Q t+1. Since we have D(P t 1 Q t ) D(P t Q t ) D(P t Q t+1 ), (46) the procedure converges. It is unique when S is flat and K is dual flat. Otherwise, the converging point is not necessarily unique.

41 Let f (u) be a convex function on u > 0. For each probability distributions p, q, we define ( q(x) ) D f (p q) p(x)f dx (47) p(x) and call it the.

42 Properties of Using Jensen s inequality we have ( D f (p q) f p(x) q(x) ) p(x) dx = f (1), (48) where the equality holds if p = q and, conversely, the equality implies p = q when f (u) is strictly convex at u = 1. D f is invariant when f (u) is replaced with f (u) + c(u 1) for any c R.

43 Properties of Df = D f, where f = uf (1/u). Monotonicity Let κ = {κ(y x) 0; x X, y Y} be an arbitrary transition probability distribution such that κ(y x)dy = 1, x, whereby the value of x is randomly transformed ro y according to the probability κ(y x). Denoting the distributions of y derived from p(x) and q(x) by p κ (y) and q κ (y) respectively, we have D f (p q) D f (p κ q κ ) (49)

44 Properties of Proof of monotonicity. ( q(x) D f (p q) = p(x)κ(y x)f p(x) ( q(x) = p κ (y)p κ (x y)f p(x) p κ (y)f = D f (p κ q κ ) ) dxdy ) dxdy ( p κ (x y) q(x) p(x) dx ) dy (50) The equality holds if p κ (x y) = q κ (x y) for all x and y.

45 Joint convexity The joint convexity D f (λp 1 + (1 λ)p 2 λq 1 + (1 λ)q 2 ) (51) λd f (p 1 q 1 ) + (1 λ)d f (p 2 q 2 ), 0 λ 1 follows from the convexity of pf ( q p ) ((λ 1 p 1 + λ 2 p 2 )f ( λ 1q 1 +λ 2 q 2 λ 1 p 1 +λ 2 p 2 ) = q 1 q +λ p 2 p (λ 1 p 1 + λ 2 p 2 )f ( λ 1p 1 p 2 λ 1 p 1 +λ 2 p 2 ) λ 1 p 1 f ( q 1 p 1 ) + λ 2 p 2 f ( q 2 p 2 )).

46 Assume f is strictly convex and smooth and f (1) = 0, then D f becomes a and induces the metric g (D f ) = g (f ) and the connection (D f ) = (f ).

47 α- Important examples of smooth s are given by the α- D (α) = D f (α) for a real number α, which is defined by 4 f (α) 1 α 2 {1 u(1+α)/2 } (α ±1) (u) = ulogu (α = 1) logu (α = 1). We have for α ±1 D (α) (p q) = 4 1 α 2 {1 (52) p(x) 1 α 2 q(x) 1+α 2 dx} (53)

48 and for α = ±1 D ( 1) (p q) = D (1) (q p) = α- p(x)log p(x) dx. (54) q(x) We can immediately see that the α- D (α) induces (g (f (α)), (f (α)) ) = (g, (α) ). Note that D (α) (p q) = D ( α) (q p) ly holds. In particular, D (0) (p q) is symmetric, and moreover D (0) (p q) satisfies the axioms of distance, which follows since D (0) (p q) = 2 ( p(x) q(x)) 2 dx. (55) D (0) (p q) is called the Hellinger distance.

49 Kullback The ±1- is called the Kullback or Kullback-Leibler(KL). Here we refer to D ( 1) as the KL and D (1) its dual. The KL satisfies the chain rule: D ( 1) (p q) =D ( 1) (p κ q κ ) + D ( 1) (p κ ( y) q κ ( y))p κ (y)dy. (56)

50 Expectation parameters In an family p(x; θ) = exp[c(x) + θ i F i (x) ψ(θ)], (57) the natural parameters [θ i ] form a 1-affine chart. Now if we define η i = η i (θ) E θ [F i ] = F i (x)p(x; θ)dx, (58) then η i = i ψ and i j ψ = g ij. Hence [η i ] is a (-1)-affine chart dual to [θ i ], and ψ is the potential of a Legendre transformation. We call this [η i ] the expectation parameters or the dual parameters.

51 Examples Normal Distribution η 1 = µ = θ1 2θ 2, η 2 = µ 2 + σ 2 = (θ1 ) 2 2θ 2 4(θ 2 ) 2 Poisson Distribution P(X ) for finite X η = ξ = expθ η i = p(x i ) = ξ i = expθ i 1 + n j=1 expθj

52 Entropy The dual potential ϕ is given by ϕ(η) = θ i η i ψ(θ) = E θ [logp θ C] = H(p θ ) E θ [C], where H is the entropy: H(p) p(x)logp(x)dx. In addition, we have (59) ϕ(θ) = max{θ i η i (θ) ψ(θ )}, (60) θ where the maximum is attained by θ = θ

53 The ±1- is exactly the canonical (g, (±1) )-. The triangular relation can be rewritten as D(p q) + D(q r) D(p r) = {p(x) q(x)}{logr(x) logq(x)}dx, where D = D ( 1) is the KL. (61)

54 Projection From theorems in canonical, the solutions to the minimization problems min D(p q) and min D(q p) q M q M are repectively given by the (m) -projection and (e) -projection.

55 Principle of maximum entropy Given (n + 1) C, F 1,, F n : X R, let S = {p θ θ Θ} be the n-dimensional family. Then for any θ Θ and any q P(X ) we have H(p θ ) + E pθ [C] + θ i E pθ [F i ] H(q) E q [C] θ i E q [F i ] = D(q p θ ) 0, which leads to max {H(q) + E q[c] + θ i E q [F i ]} q P(X ) = H(p θ ) + E pθ [C] + θ i E pθ [F i ] = ψ(θ). (62) (63)

56 Principle of maximum entropy Given a vector λ = (λ 1,, λ n ) R n,let M λ {q P E q [F i ] = λ i, i = 1,, n}. (64) Now assume S M λ and suppose θ λ Θ s.t. η i (θ λ ) = E pθλ [F i ] = λ i for i = 1,, n. Then we have max {H(q) + E q [C]} = H(p θλ ) + E pθλ [C] q M λ = ψ(θ λ ) θ i λ λ i = min θ Θ {ψ(θ) θi λ i }, When C = 0 it follows that max q Mλ H(q) = H(p θλ ), which is called the principle of maximum entropy. (65)

57 Boltzmann-Gibbs distribution The thermal equilibrium state which maximizes the thermodynamical entropy S(p) kh(p), where k(> 0) is Boltzmann s constant, under the constraint E q [ɛ] = ɛ on the average of the energy function ɛ, is given by the Boltzmann-Gibbs distribution p (x) = 1 Z e ɛ(x)/kt, (66) where T is the temperature and Z is the partition function. This corresponds to the previous situation by letting C = 0, n = 1, F i = ɛ, λ = ɛ, θ λ = 1/kT and ψ(θ λ ) = logz.

58 Statistical model with hidden variables Consider a statistical model M = {p(x, ξ)}, where x is divided into two parts x = (y, h) so that p(x, ξ) = p(y, h; ξ). When x is not fully observed but y is observed, h is called a hidden variable. In such a case, we estimate ξ from observed y. Actually, we want to compute the MLE of p Y (y, ξ) = p(y, h; ξ)dh. However, in many cases, the form of p(x, ξ) is simple and estimation is tractable in M, but p Y (y, ξ) is complicated and the estimation is computationally intractable.

59 Empirical distribution Consider a larger model S = {q(y, h)} consisting of all probability density of (y, h). We don t have the empirical distribution q(x) = 1 N δ(x x i ) but only an empirical distribution q Y (y) for y only. We use an arbitrary conditional distribution q(h y) and put q(y, h) = q Y (y)q(h y). (67) And we take all the candidates of observed points and consider a submanifold D = { q(y, h) q(y, h) = q Y (y)q(h y), q(h y) is arbitrary}. (68)

60 Empirical distribution D is the observed submanifold in S specified by the partially observed data y 1,, y N. By using the empirical distribution, it is written as q(y, h) = 1 N δ(y y i )q(h y i ) (69) The data submanifold D is m-flat, because it is linear with respect to q(h y i ).

61 MLE and KL- Consider the minimizer of KL- from data manifold M to the model manifold D, D[D : M] = min q Y (y)q(h y)log q Y (y)q(h y) dydh (70) p(y, h, ξ) Theorem The MLE of p Y (y, ξ) is the minimizer of the KL- from D to M. In fact, we minimize the equation above with respect to both ξ and q(h y) alternately by the em algorithm, that is, the alternating use of the e-projection and m-projection.

62 Algorithm () 1 Choose an initial parameter ξ 0. 2 E-step e-project ξ 0 to D. It can be verified that the e-projection is q(h y) = p(h y; ξ 0 ). 3 M-step Maximize a log likelihood L(ξ, ξ 0 ) = 1 p(h y; ξ 0 )logp(y N i, h, ξ)dh (71) i to obtain a new candidate ξ 1 in M. It can be verified that this is the m-projection. 4 Repeat step 2 and 3.

63 Theorem The KL- decreases monotonically by repeating the E-step and the M-step. Hence, the algorithm converges to an equilibrium. It should be noted that the m-projection is not necessarily unique unless M is e-flat. Hence, there might exist local minima. However, we often come across the family and thus there exists unique solution.