Conceptual map
- III-02.01Smooth state spaces
- III-02.02Restricted derivatives
- III-02.03Distributions and frames
- III-02.04The role of a metric
- III-02.05Coordinates for distribution states
1. State spaces, fibers, and admissible directions
A derivative has economic content only along feasible state directions. Constraints determine the tangent space, an observation map determines the hidden directions, and a metric determines the size assigned to each perturbation.
Let \(\mathcal{M}\) be a \(d\)-dimensional smooth state manifold, let \(h:\mathcal{M}\to\mathbb{R}^{k}\) be an observable report, and let \(\tau:\mathcal{M}\to\mathbb{R}^{r}\) be a target. For \(y\) in the range of \(h\), the observational fiber is \(\mathcal{M}_{y}=\{x\in\mathcal{M}:h(x)=y\}\).
If \(Dh_{x}\) has rank \(k\), the tangent space to the fiber through \(x\) is \(V_{x}\) = \(\operatorname{ker}(Dh_{x})\) \(\subset\) \(T_{x}\mathcal{M}\). The restriction \(D\tau_{x}|V_{x}\) measures local target movement that the report leaves hidden.
A metric is a positive-definite bilinear form \(G_{x}\) on \(T_{x}M\). It fixes units for state perturbations. A target weight \(W\) fixes units in \(\mathbb{R}^{r}\). Both objects are part of any magnitude or spectral claim.
2. Coordinate-invariant response spectra
Choose a full-column-rank frame \(E\) whose columns span an admissible tangent space. Define \(A\) = \(D\tau_{x}E\) and \(M\) = \(E^{\top}G_{x}E\). The squared response-to-cost ratios are generalized Rayleigh quotients.
Let \(E^{\prime}=ER\) for an invertible coordinate matrix \(R\). The generalized eigenvalues of (\(A^{\top}WA\), \(M\)) equal those of (\(A^{\prime}^{\top}WA^{\prime}\), \(M^{\prime}\)). Corresponding generalized eigenspaces map by \(v=Rv^{\prime}\), so their physical tangent subspaces \(Ev\) are unchanged. An individual eigenvector needs a normalization, and a repeated eigenvalue identifies a subspace rather than a unique direction.
Proof. Under the frame change, \(A^{\prime}=AR\) and \(M^{\prime}=R^{\top}MR\). Substitution in (1) shows \(R^{\prime}(v^{\prime})=R(Rv^{\prime})\). Invertibility of \(R\) makes this a bijection over nonzero coordinate vectors, so the stationary values and generalized spectrum coincide. ∎
An ordinary singular-value calculation silently sets \(M=I\). Rescaling a coordinate then changes the implied metric and can change every reported singular value.
3. Worked probability-simplex example
Take \(x\)=(0.2,0.3,0.5) in the interior of the three-category simplex. Feasible perturbations satisfy \(1^{\top}\delta x=0\). Use the frame
Let the target be \(\tau(x)=(x_{1}-x_{3},x_{2}-x_{3})\). Then \(A=\begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}\). The Fisher–Rao metric in ambient coordinates is \(\operatorname{diag}(1/x_{i})\), so the restricted metric is
With \(W=I\), solve \(A^{\top}Av=\lambda Mv\). The two generalized eigenvalues are approximately 1.13135 and 0.23865. Replacing \(E\) by \(E\operatorname{diag}(10,0.1)\) changes the ordinary singular values of \(A\) by orders of magnitude. Transforming \(M\) at the same time leaves 1.13135 and 0.23865 unchanged.
4. Boundary cases
The vector (1,0,0) changes total probability and therefore lies outside the simplex tangent space. An ambient derivative evaluated in that direction answers a perturbation question for an enlarged model. The admissible state model supplies the linear restriction \(1^{\top}\delta x=0\) before any spectrum is computed.
At a boundary point with a zero probability, the Fisher–Rao coordinate expression diverges. A boundary analysis must use a compatible tangent cone, a different chart, or a limiting argument. The interior calculation does not extend by substituting a zero coordinate into (3).
5. Implementation and exercises
- Construct a basis or frame for the null space of the active linearized constraints.
- Restrict the target derivative and metric with the same frame.
- Solve a symmetric generalized eigenvalue problem after a Cholesky whitening of the restricted metric.
- Repeat under an invertible frame change and compare physical directions.
Exercises
- Verify (3) by direct multiplication and reproduce the two generalized eigenvalues.
- Use \(R=\operatorname{diag}(10,0.1)\), compute the naive singular values, and verify Proposition 1 numerically.
- Replace the Fisher–Rao metric with a declared quadratic loss and describe how the leading direction changes.
Partial solutions
1. Equation (3) follows from adding the third ambient metric weight, 2, to every entry generated by the last row of \(E\). The generalized characteristic equation is \(100\lambda ^{2}-137\lambda+27=0\). 2. Use \(A^{\prime}=AR\) and \(M^{\prime}=R^{\top}MR\); substitution returns the same polynomial in \(\lambda\). The singular values of \(AR\) alone change because that calculation assigns identity cost to the rescaled coordinates.
6. Audit checkpoint
Record the state space, active constraints, tangent frame, target derivative, state metric, target weight, and coordinate transformation. A reported spectrum has an economic interpretation only after those objects are fixed.
7. Scope and sources
The chapter uses local coordinates, regular fibers, tangent restrictions, and finite-dimensional metrics. Global manifold topology and curvature enter later only where policy transport needs them.
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Chapters 3–5.Tangent vectors, differentials, submersions, and regular level sets.
- Shun-ichi Amari, Information Geometry and Its Applications, Chapter 2.Fisher metric on finite probability models.
- Gene H. Golub and Charles F. Van Loan, Matrix Computations, 4th ed., §8.7.Symmetric-definite generalized eigenvalue problems.