On the Stochastic Rank of Metric Functions

arXiv:0810.5549v3 [math.MG] 24 Apr 2009 On the Stochastic Rank of Metric Functions Nikolay Balov May 31, 2018 Abstract For a class of integral operators with kernels metric functions on manifold we find some necessary and sufficient conditions to have fi- nite rank. The problem we pose has a stochastic nature and boils down to the following alternative question. For a random sample of discrete points, what will be the probability the symmetric matrix of pairwise distances to have full rank? When the metric is an analytic function, the question finds full and satisfactory answer. As an im- portant application, we consider a class of tensor systems of equations formulating the problem of recovering a manifold distribution from its covariance field and solve this problem for representing manifolds such as Euclidean space and unit sphere. 1 Problem formulation and motivation We start with the classical Fredholm integral equation of the first kind Z V ψ(x, y)f(y)dy = g(x), x ∈U, (1) where U and V are open sets in Rn and f : U →R, g : V →R, and ψ : U × V →R are some functions. Depending on the domains more conditions on f, g and ψ may be necessary for the correct formulation of (1). Let {xi}k i=1 and {yi}k i=1 be two discrete samples of points chosen by uniform distributions on U and V respectively. Then equation (1) can be discretized by the matrix-quadrature method k X j=1 ψ(xi, yj)f(yj) = g(xi), i = 1, ..., k. (2) 1 In general, the inverse problem of solving (1) for f, is often ill-posed and approximation based on (2) will eventually result in increasingly unstable solution as k increases. Here, however, we are interested in the first potential obstacle to solve (2) - the matrix Ψ := {ψ(xi, yj)}k,k i,j=1,1 may not be of full rank. The problem is stochastic one for the points xi’s and yj’s are chosen in random fashion. In section 2 we investigated it and find some conditions for the kernel ψ, which guarantee that for any k, Ψ has full rank with probability one. We also show some necessary and sufficient conditions for analytic kernels ψ to be of finite rank. A further generalization of equation (1) takes f to be a function on n- manifold M and g and ψ to be linear operator fields on M. For a point p ∈M with Mp we denote the tangent space at p and with T 1 1 (Mp), the vector space of (1,1)-tensors (linear operators) on Mp. Let µ be a measure on M as for example the volume measure V (p) on Riemannian manifolds. Consider the equation Z M Y (p, q)f(q)dµ(q) = C(p), p ∈M, (3) such that Y (p, .) ∈T 1 1 (Mp) and C(p) ∈T 1 1 (Mp). The inverse problem here is finding f for given fields Y and C. If we know that (3) has a unique solution for f then it can be found by solving Z M tr(Y (p, q))f(q)dµ(q) = tr(C(p)), (4) an equation of type (1). The importance of the class (3) of tensor equations is that it contains the problem of recovering a distribution from its covariance field. Next, we briefly pose this problem, while more details one can find in [1]. Let M be a Riemannian manifold with metric tensor G. For any p ∈M, G(p) ∈T 2(Mp) is a co-variant 2-tensor. Let Expp : Mp →M be the exponential map at p and U(p) ⊂M be the maximal normal neighbor- hood of p, where Exp−1 p is well defined. Note that since Exp−1 p q ∈Mp, (Exp−1 p q)(Exp−1 p q)T ∈T2(Mp), a contra-variant 2-tensor. For a density func- tion f ≥0 on M, the covariance operator field of f is GΣ : M →T 1 1 (M), such that for any p ∈M GΣ(p) := Z U(p) G(p)(Exp−1 p q)(Exp−1 p q)Tf(p)dV (p). (5) 2 The problem of distribution recovering is of type (3) if we take µ = V , C = GΣ and Y (p, q) = G(p)(Exp−1 p q)(Exp−1 p q)T. Note that tr(Y (p, q)) = tr(G(p)(Exp−1 p q)(Exp−1 p q)T) = d2(p, q), is the square geodesic distance on M. Thus equation (4), specifically, is Z U(p) d2(p, q)f(q)dV (q) = g(p). (6) In the context of problem (6) we are interested in finding the rank of an integral operator of the form Lψ : f 7→ R ψ(p, q)f(q)dV (q), where ψ(p, q) = d2(p, q) is a square distance function on M. In particular, in section 3 we study the rank of the Euclidean metric d(p, q) = ||p −q|| and the rank of the standard metric on the unit sphere Sn, d(p, q) = cos−1(< p, q >), and show that while the Euclidean metric is of finite rank, the metric on the sphere is not. The last fact can be re-phrased as follows. For any discrete sample of points on a sphere, the square matrix of pairwise distances is non-singular with probability one. The problem of establishing the non-singularity of the kernel of operator (1) is important in the context of more general statistical inverse problems on manifolds, as considered in [4], [11], [12] and [14]. Let g(p) = f(p) + ǫ, where ǫ is a mean zero random variable with small variance, be a model with unknown regression function f. For a kernel ψ, the inverse problem with random noise is formulated as estimation of Lψf from observations (pi, gi). In [4], Cavalier and Tsybakov estimate f from the model g = Lψf + ǫ, which is a noised version of (1). Usually points pi are assumed

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