Covariance Fields

Let M be a Riemannian n-manifold with metric tensor G and volume form V ,

∘ ------ dVx(p) = Gx (p)dx,
with respect to coordinate system x about p ∈ M. If a continuous random variable on M is given by distribution function in the form
dF (p) = f(p)dV (p).
then the covariance of F with respect to q ∈ M is defined to be
∫ -→ -→ Σ (q) = (qp)(qp)′dF (p), M
where -→ qp is the tangent vector at q obtained by the inverse exponential map exp q-1. Σ(q) is a contra-variant 2-tensor on the tangent space M q. Collectively, Σ is a tensor field on M, which we call covariance field.

Recall that the metric G is a co-variant tensor field on M. For any q ∈ M, the product G(q)Σ(q) is a linear operator on the tangent space Mq, which trace is

∫ 2 tr(G (q)Σ (q)) = d (q,p)dV (p), M
the expected value of the square distance from q to random position on M according to F. GΣ is a field of linear operators on M that can be used for obtaining some properties of distribution F. For example,
μint = argmin {tr(G (q)Σ(q))|, q ∈ M }
is the so called intrinsic mean of M. More details on the definition and various properties of covariance fields can be found in [1].

Distribution Representations

Given a distribution F on M and its covariance operator field GΣ, one can obtain a family of representations of F, by applying a similarity invariant h on it. Similarity invariants are functions that takes as arguments linear operators, such as tr, det and tr2ln, to mention a few. Are such representations true ones? With other words, can one recovers F from h(GΣ)? It turns out that the answer depends on the geometry of M, which is defined by the metric G. The answer is positive when the operators GΣ have no finite rank, or equivalently, the metric G is of full rank. We direct to [4] for more precise exposition of this interesting relationship between geometry and distribution representations.

Fact 1: A distribution in Euclidean space cannot be recovered from its covariance field.

In canonical coordinates, the metric tensor in the real plane is G(x,y) = I2. The rank of a k × k matrix of square pairwise distance between k points in the real plane is always less or equal of 4. As a result, for a distribution in ℝ2, if you know the covariance at one point, you know it for all points. The same is true in higher dimensions also.

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Figure 1: Two different distributions on the real plane with identical covariance fields.

Fact 2: The covariance field in non-Euclidean space, in most cases, determines completely the underlying distribution.

For example, the metric tensor in the hyperbolic plane is G(x,y) = 1y2I2 and the rank of a k × k matrix of squared pairwise distance between k points in the hyperbolic plane is k (w.p. 1).

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Figure 2: The same two distributions on the hyperbolic plane have different covariance fields.

As we mentioned above, the question of recovering a distribution from its covariance field is contingent on the properties of the geodesic metric d on M. Let (U,φ), U ⊂ ℝn be a local chart on M. Define ψ(x,y) = d(φ(x),φ(y)), x,y ∈ U. Then we claim that recovering is possible only when ψ does not have a finite rank.

For example, if ψ(x,y) = h(< x,y >) for function h that is analytic about the origin and for which h(s)(0)⁄=0 for infinitely many s, then ψ will not have a finite rank. By applying this fact for h = cos, we obtain that the rank of the metric on the unit sphere Sn is not finite and recovering distributions from their covariance fields is always possible on Sn (w.p.1).

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Figure 3: On the unit 2-sphere their covariance fields are also different.

References

[1]   Balov, N. (2008) Covariance fields, http://arxiv.org/abs/0807.4690v2.

[2]   Balov, N. (2008) Covariance of centered distributions on manifold, http://arxiv.org/abs/0805.0732v1.

[3]   Balov, N. (2008) Comparing and interpolating distributions on manifold, http://arxiv.org/abs/0807.0782v2.

[4]   Balov, N. (2008) On the Stochastic Rank of Metric Functions, http://arxiv.org/abs/0810.5549v2.

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