3. Kähler Manifold for Signal Processing
An advantage of the transfer function representation in the complex z-domain is that it is easy to test whether or not the information geometry of a given signal processing filter is a Kähler manifold. As mentioned before, choosing the coefficients in a(z; ξ) is considered as fixing the degrees of freedom in calculation without changing any geometry. By setting a(z; ξ)/a0(ξ) a constant function in ξ, the description of a statistical model becomes much simpler, and the emergence of Kähler manifolds can be easily verified. Since causal filters are our main concerns in practice, we concentrate on unilateral transfer functions. Although we will work with causal filters, the results in this section are also valid for the cases of bilateral transfer functions.
Theorem 1. For a signal filter with a finite complex cepstrum norm, the information geometry of the signal filter is a Kähler manifold.
Proof. The information manifold of a signal filter is described by the metric tensor
g with the components of the expressions,
Equation (10) and
Equation (11). Any complex manifold admits a Hermitian manifold by introducing a new metric tensor ĝ [
29]:
where
X, Y are tangent vectors at point
p on the manifold and
J is the almost complex structure, such that
With the new metric tensor ĝ, it is straightforward to verify that the information manifold is equipped with the Hermitian structure:
Based on the above metric tensor expressions, it is obvious that the information geometry of a linear system is a Hermitian manifold.
The Kähler two-form Ω of the manifold is given by
where ∧ is the wedge product. By plugging
Equation (11) into Ω, it is easy to check that the Kähler two-form is closed by satisfying
and
.
Since Kähler manifolds are defined as the Hermitian manifolds with the closed Kähler two-forms, the information geometry of a signal filter is a Kähler manifold.
An information manifold for a linear system with purely real parameters is a submanifold of a Kählerian information manifold where the metric tensor has the isometry of exchanging holomorphic- and anti-holomorphic coordinates. In addition to that, a given linear system can be described by two manifolds: one is Kähler, and another is non-Kähler. Although the dimension is doubled, working with Kähler manifolds has many advantages, which will be reiterated later.
In Theorem 1, the Hermitian condition is clearly seen after introducing the new metric tensor ĝ. It is also possible to find a condition for which the metric tensor g shows the explicit Hermitian structure. To impose the explicit Hermitian condition, the following theorem is worthwhile to mention.
Theorem 2. In the Kählerian information geometry of a signal filter, the Hermitian structure is explicit in the metric tensor if and only if ϕ0 (or f0a0) is a constant in ξ. Similarly, for the transfer function of which the highest degree in z is finite, the Hermitian condition is directly found if and only if the coefficient of the highest degree in z of the logarithmic transfer function is a constant in ξ.
Proof. Let us prove the first statement.
(⇒) If the geometry is Kähler, it should be the Hermitian manifold satisfying
for all
i and
j. This equation exhibits that
f0a0 is a constant in
ξ, because
ϕ0 = log (
f0a0).
(⇐) If
ϕ0 (or
f0a0) is a constant in
ξ, the metric tensor is found from
Equations (10) and
(11),
and these metric tensor conditions impose that the geometry is the Hermitian manifold. It is noteworthy that the non-vanishing metric tensor components are expressed only with
ϕr and
, which are functions of the impulse response functions
fr in
f(
z;
ξ), the unilateral part of the transfer function. For the manifold to be a Kähler manifold, the Kähler two-form Ω needs to be a closed two-form. The condition for the closed Kähler two-form Ω is that
and
. It is easy to verify that the metric tensor components,
Equation (14), satisfy the conditions for the closed Kähler two-form. The Hermitian manifold with the closed Kähler two-form is a Kähler manifold.
The proof for the second statement is straightforward, because it is similar to the proof of the first one by exchanging ϕr ↔ ηr. Let us assume that the highest degree in z is R. According to Lemma 3, it is possible to reduce a bilateral transfer function with finite terms along the non-causal direction to the unilateral transfer function by using the factorization of zR. After that, we need to replace η0 with ϕ0 in the proof. The two theorems are equivalent. □
Theorem 2 can be applied to submanifolds of the information manifolds. For example, a submanifold of a linear system is a Kähler manifold if and only if ϕ0 (or f0a0) is constant on the submanifold, i.e., ϕ0 is a function of the coordinates orthogonal to the submanifold.
On a Kähler manifold, the metric tensor is derived from the following equation:
where
is the Kähler potential. There exists the degree of freedom in Kähler potential up to the holomorphic and anti-holomorphic function:
. However, geometry is derived from the same relation:
. By using
Equation (15), the information on the geometry can be extracted from the Kähler potential. It is necessary to find the Kähler potential for the signal processing geometry. The following corollary shows how to get the Kähler potential for the Kählerian information manifold.
Corollary 1. For a given Kählerian information geometry, the Kähler potential of the geometry is the square of the Hardy norm of the logarithmic transfer function. In other words, the Kähler potential is the square of the complex cepstrum norm of a signal filer.
Proof. Given a transfer function
h(
z;
ξ), the non-trivial components of the metric tensor for a signal processing model are given by
Equation (9). By using integration by parts, the metric tensor component is represented by
where the latter term goes to zero by holomorphicity. When we integrate by parts with respect to the anti-holomorphic derivative once again, the metric tensor is expressed with
and the latter term is also zero, because
h(
z;
ξ) is a holomorphic function.
Finally, the metric tensor is obtained as
and by the definition of the Kähler potential,
Equation (15), the Kähler potential of the linear system geometry is given by
up to a holomorphic function and an anti-holomorphic function. The right-handed side of the above equation is known as the square of the Hardy norm for the logarithmic transfer function. It is straightforward to derive the relation between the Kähler potential and the square of the Hardy norm of the logarithmic transfer function:
Additionally, the Hardy norm of the logarithmic transfer function is also known as the complex cepstrum norm of a linear system [
25,
27].
For a given linear system, the Kähler potential of the geometry is given by
ϕr,
αr and the complex conjugates of
ϕr,
αr:
However, the geometry is not dependent on
α and
, because those are not the functions of the model parameters
ξ under fixing the degree of the freedom. By using
Equation (14), the Kähler potential is expressed with
and it is noticeable that the Kähler potential only depends on
ϕr and
, which come from the unilateral part of the transfer function decomposition. It is possible to obtain a similar expression for the finite highest upper-degree case by changing
ϕr to
ηr.
Since we assume that the complex cepstrum norm is finite, a transfer function
h(
z;
ξ) in the
H2-space also lives in the Hardy space of
This implies that the transfer function lives not only in H2, but also in exp (H2), equivalently log h in the H2-space.
From
Equation (15), the metric tensor is derived from the Kähler potential. Additionally, the metric tensor is also calculated from the
α-divergence. These facts indicate that there exists a connection between the Kähler potential and the
α-divergence.
Corollary 2. The Kähler potential is a constant term in α, up to purely holomorphic or purely anti-holomorphic functions, of the α-divergence between a signal processing filter and the all-pass filter of a unit transfer function.
Proof. After replacing the spectral density function with the transfer function, the 0-divergence between a signal filter and the all-pass filter with a unit transfer function is given by
where
. For a bilateral transfer function,
.
For non-zero
α, the
α-divergence between a signal and the white noise is also obtained as
When f0a0 is unity, a constant term in α of the α-divergence is the Kähler potential. This shows the relation between the α-divergence and the Kähler potential.
The
α-connection on a Kähler manifold is expressed with the transfer function by using
Equation (1) and
Equation (3). It is also cross-checked from the
α-divergence in the transfer function representation.
Corollary 3. The α-connection components of the Kählerian information geometry are found asand the non-trivial components of the symmetric tensor T are given by In particular, the non-vanishing 0-connection components are expressed withand the 0-connection is directly derived from the Kähler potential: Additionally, the α-connection and the (−α)-connection are dual to each other.
Proof. After plugging
Equation (1) into
Equation (3), the derivation of the
α-connection is straightforward by considering holomorphic and anti-holomorphic derivatives in the expression. The same procedure is applied to the derivation of the symmetric tensor
T.
The 0-connection is also directly derived from the Kähler potential. The proof is as follows:
To prove the
α-duality, we need to test the following relation:
where the Greek letters run from 1, ⋯,
n,
, ⋯,
. After tedious calculation, it is obvious that the above equation is satisfied regardless of combinations of the indices. Therefore, the
α-duality also exists on the Kählerian information manifolds.
The 0-connection and the symmetric tensor
T are expressed in terms of
ϕr and
,
With the degree of freedom that
ϕ0 is a constant in the model parameters
ξ, the non-trivial components of the 0-connection and the symmetric tensor
T are
and
, respectively. In this degree of freedom, the Hermitian condition on the metric tensor is obviously emergent, and it is also beneficial to check the
α-duality condition for non-vanishing components:
We can cross-check these formulae for the geometric objects of the linear system geometry with the well-known results on a Kähler manifold. First of all, the fact that the 0-connection is the Levi–Civita connection can be verified as follows:
where the last equality comes from the expression for the Levi–Civita connection on a Kähler manifold. This is well-matched to the Levi–Civita connection on a Kähler manifold.
In Riemannian geometry, the Riemann curvature tensor, corresponding to the 0-curvature tensor, is given by
where the Greek letters can be any holomorphic and anti-holomorphic indices. Similar to a Hermitian manifold, the non-vanishing components of the 0-curvature tensor on a Kähler manifold are
and its complex conjugate,
i.e., the components with three holomorphic indices and one anti-holomorphic index (and the complex conjugate component). The non-trivial components of the Riemann curvature tensor are represented by
because the 0-connection components with the mixed indices are vanishing.
Taking index contraction on holomorphic upper and lower indices in the Riemann curvature tensor, the 0-Ricci tensor is found as
where
is the determinant of the metric tensor. This result is consistent with the expression of the Ricci tensor on a Kähler manifold. It is also straightforward to obtain the 0-scalar curvature by contracting the indices in the 0-Ricci tensor:
where Δ is the Laplace–Beltrami operator on the Kähler manifold.
The
α-generalization of the curvature tensor, the Ricci tensor and the scalar curvature is based on the
α-connection,
Equation (4). The
α-curvature tensor is given by
The
α-Ricci tensor and the
α-scalar curvature are obtained as
It is noteworthy that the
α-curvature tensor, the
α-Ricci tensor and the
α-scalar curvature on a Kähler manifold have the linear corrections in
α comparing with the quadratic corrections in
α on non-Kähler manifolds. A submanifold of a Kähler manifold is also a Kähler manifold. When a submanifold of dimension
m exists, the transfer function of a linear system can be decomposed into two parts:
where
h|| is the transfer function on the submanifold and
h⊥ is the transfer function orthogonal to the submanifold. When it is plugged into
Equation (16), the Kähler potential of the geometry is decomposed into three terms as follows:
where
contains the coordinates from the submanifold,
is for the cross-terms and
is orthogonal to the submanifold.
It is obvious that each part in the decomposition of the Kähler potential provides the metric tensors for submanifolds,
where an uppercase index is for the coordinates on the submanifold and a lowercase index is for the coordinates orthogonal to the submanifold. As we already know, the induced metric tensor for the submanifold is derived from
, the Kähler potential of the submanifold. Based on this decomposition, it is also possible to use
as the Kähler potential of the submanifold, because it endows the same metric with
. However, the Riemann curvature tensor and the Ricci tensors include the mixing terms from embedding in the ambient manifold, because the inverse metric tensor contains the orthogonal coordinates by the Schur complement. In statistical inference, connections, tensors and scalar curvature play important roles. If those corrections are negligible, dimensional reduction to the submanifolds is meaningful from the viewpoints not only of Kähler geometry, but also of statistical inference.
The benefits of introducing a Kähler manifold as an information manifold are as follows. First of all, on a Käher manifold, the calculation of geometric objects, such as the metric tensor, the
α-connection and the Ricci tensor, is simplified by using the Kähler potential. For example, the 0-connection on a non-Kähler manifold is given by
demanding three-times more calculation steps than the Kähler case,
Equation (18). Additionally, the Ricci tensor on a Kähler manifold is directly derived from the determinant of the metric tensor. Meanwhile, the Ricci tensor on a non-Kähler manifold needs more procedures. In the beginning, the connection should be calculated from the metric tensor. Additionally, then, the Riemann curvature is obtained after taking the derivatives on the connection and considering quadratic terms of the connection. Finally, the Ricci tensor on the non-Kähler manifold is found by the index contraction on the curvature tensor indices.
Secondly, α-corrections on the Riemann curvature tensor, the Ricci tensor and the scalar curvature on the Kähler manifold are linear in α. Meanwhile, there exist the quadratic α-corrections in non-Kähler cases. The α-linearity makes it much easier to understand the properties of α-family.
Moreover, submanifolds in Kähler geometry are also Kähler manifolds. When a statistical model is reducible to its lower-dimensional models, the information geometry of the reduced statistical model is a submanifold of the geometry. If the ambient manifold is Kähler, the dimensional reduction also provides a Kähler manifold as the information geometry of the reduced model, and the submanifold is equipped with all of the properties of the Kähler manifold.
Lastly, finding the superharmonic priors suggested by Komaki [
15] is more straightforward in the Kähler setup, because the Laplace–Beltrami operator on a Kähler manifold is of the more simplified form compared to that in non-Kähler cases. For a differentiable function
ψ, the Laplace–Beltrami operator on a Kähler manifold is given by
comparing with the Laplace–Beltrami operator on a non-Kähler manifold:
where
is the determinant of the metric tensor. On a Kähler manifold, the partial derivatives only act on the superharmonic prior functions. Meanwhile, the contributions from the derivatives acting on
and
gij should be considered in the non-Kähler cases. This computational redundancy is not on the Kähler manifold.