Geometric Observation

Abstract

We derive geometric observation. Starting from the Donsker-Varadhan bridge, we pass to quotient-visible precision and obtain the geometry in closed form. The intrinsic visible object is $\Phi_{C}(H)=(CH^{-1}C^{\mathsf{T}})^{-1}$, while hidden load, determinant clock, minimal hidden dimension, and canonical hidden realisation arise exactly after a ceiling or reference has been fixed.

1  Introduction

Observation induces a canonical geometry.

Let $C:\mathbf{R}^{n}\to\mathbf{R}^{m}$ be a surjective observation map and let $H\in\operatorname{Sym}_{++}(n)$ be a latent precision.

The induced visible precision is

$$\Phi_{C}(H):=(CH^{-1}C^{\mathsf{T}})^{-1}.$$ (1.1)

Applied to the Donsker-Varadhan quadratic bridge, (1.1) forces the route developed below. This quotient-visible precision is intrinsic to the observed pair $(H,C)$; hidden load and determinant clock arise only after a ceiling or reference has been fixed.

2  Bridge to visible precision

We first isolate the standard local Donsker-Varadhan bridge in reduced coordinates. This produces the latent quadratic form. The new step is to pass that form through the canonical visible map (1.1); from there the visible calculus and the geometry are forced.

Let $x\in\mathbf{R}^{n}$ be the reduced empirical-measure coordinate and $y\in\mathbf{R}^{n}$ the reduced current coordinate. Assume

$$\mathcal{L}(x,y)=-y^{\mathsf{T}}Sy+x^{\mathsf{T}}(S-\widehat{J})y+O(\|(x,y)\|^{3}),$$ (2.1)

with $S\in\operatorname{Sym}_{++}(n)$ and $\widehat{J}^{\mathsf{T}}=-\widehat{J}$. Define the local Donsker-Varadhan rate by

$$I_{DV}(x)=\sup_{y}\mathcal{L}(x,y)=\frac{1}{2}x^{\mathsf{T}}H_{DV}x+O(\|x\|^{3}),$$ (2.2)

and set

$$H_{0}:=\frac{1}{2}S.$$ (2.3)

Theorem 2.1 (Donsker–Varadhan bridge).

Under (2.1) and (2.2),

$$H_{DV}=H_{0}+\Delta_{DV},\qquad\Delta_{DV}=\frac{1}{4}\,\widehat{J}\,H_{0}^{-1}\widehat{J}^{\mathsf{T}}\succeq 0.$$ (2.4)

Equivalently,

$$\Delta_{DV}=\frac{1}{4}\bigl(\widehat{J}H_{0}^{-1/2}\bigr)\bigl(\widehat{J}H_{0}^{-1/2}\bigr)^{\mathsf{T}}.$$ (2.5)

Proof in Appendix A.

Now let $C:\mathbf{R}^{n}\to\mathbf{R}^{m}$ be surjective. The passage from latent precision to visible precision is canonical: for $z\in\mathbf{R}^{m}$, define the constrained quadratic energy

$$\mathcal{E}_{H}(z):=\inf\{x^{\mathsf{T}}Hx:Cx=z\}.$$ (2.6)

Proposition 2.2 (Visible precision).

For every $H\in\operatorname{Sym}_{++}(n)$,

$$\mathcal{E}_{H}(z)=z^{\mathsf{T}}\Phi_{C}(H)z\qquad(z\in\mathbf{R}^{m}),$$ (2.7)

with $\Phi_{C}(H)$ given by (1.1). If $C=C_{2}C_{1}$ with $C_{1}:\mathbf{R}^{n}\to\mathbf{R}^{k}$

and $C_{2}:\mathbf{R}^{k}\to\mathbf{R}^{m}$ both surjective, then

$$\Phi_{C}(H)=\Phi_{C_{2}}(\Phi_{C_{1}}(H)).$$ (2.8)

In a visible-hidden block splitting,

$$H=\begin{pmatrix}H_{vv}&H_{vh}\\ H_{hv}&H_{hh}\end{pmatrix},\qquad\Phi_{C}(H)=H_{vv}-H_{vh}H_{hh}^{-1}H_{hv}.$$ (2.9)

Proof in Appendix A.

The first variation and the first nonlinear correction are then forced by (1.1). Define the canonical lift and complementary projector by

$$L_{C,H}:=H^{-1}C^{\mathsf{T}}\Phi_{C}(H),\qquad P_{C,H}:=I-L_{C,H}C.$$ (2.10)

Here $L_{C,H}$ is the canonical lift of a visible covector back to the latent space, and $P_{C,H}$ projects onto the latent directions not represented by that lift. Let $\Phi:=\Phi_{C}(H)$, $L:=L_{C,H}$, and $P:=P_{C,H}$.

Theorem 2.3 (Local visible calculus).

For $\Delta\in\operatorname{Sym}(n)$,

$$\Phi_{C}(H+t\Delta)=\Phi+tV-t^{2}Q+O(t^{3}),$$ (2.11)

where

$$V=L^{\mathsf{T}}\Delta L,\qquad Q=L^{\mathsf{T}}\Delta P\,H^{-1}\Delta L.$$ (2.12)

Moreover,

$$Q=\bigl(H^{-1/2}P^{\mathsf{T}}\Delta L\bigr)^{\mathsf{T}}\bigl(H^{-1/2}P^{\mathsf{T}}\Delta L\bigr)\succeq 0.$$ (2.13)

If

$$\kappa_{C}(H):=-\log\det\Phi_{C}(H),$$ (2.14)

then

$$D^{2}\kappa_{C}(H)[\Delta,\Delta]=\lVert\Phi^{-1/2}V\Phi^{-1/2}\rVert_{F}^{2}+2\operatorname{Tr}(\Phi^{-1}Q).$$ (2.15)

Proof in Appendix A.

Remark.

The operator $Q$ measures the failure of the perturbation to remain closed on the canonical visible lift. By (2.13),

$$Q=0\iff P^{\mathsf{T}}\Delta L=0,$$

so $Q$ vanishes exactly when the perturbation closes on the canonical visible lift to second order.

We now apply Theorem 2.3 to a small perturbative family. Assume

$$\widehat{J}(\varepsilon)=\varepsilon\widehat{J}_{1}+O(\varepsilon^{3}).$$ (2.16)

We consider perturbative families with this parity. Then Theorem 2.1 gives

$$H_{DV}(\varepsilon)=H_{0}+\varepsilon^{2}\Delta_{2}+O(\varepsilon^{4}),\qquad\Delta_{2}:=\frac{1}{4}\widehat{J}_{1}H_{0}^{-1}\widehat{J}_{1}^{\mathsf{T}}\succeq 0.$$ (2.17)

Define the visible branch

$$X_{\varepsilon}:=\Phi_{C}(H_{DV}(\varepsilon))-\Phi_{C}(H_{0}).$$ (2.18)

Corollary 2.4 (Quadratic onset and quartic defect).

Let $L_{0}:=L_{C,H_{0}}$ and $P_{0}:=P_{C,H_{0}}$. Then

$$X_{\varepsilon}=\varepsilon^{2}V_{2}-\varepsilon^{4}Q_{4}+O(\varepsilon^{6}),$$ (2.19)

with

$$V_{2}=L_{0}^{\mathsf{T}}\Delta_{2}L_{0},\qquad Q_{4}=L_{0}^{\mathsf{T}}\Delta_{2}P_{0}H_{0}^{-1}\Delta_{2}L_{0}\succeq 0.$$ (2.20)

Thus the visible response appears at order $\varepsilon^{2}$, while the first hidden defect appears at order $\varepsilon^{4}$.

Proof in Appendix A.

3  The geometry

Fix a visible ceiling $T\succeq 0$ and let $S:=\operatorname{Ran}(T)$. Let $X$ satisfy $0\prec X|_{S}\leq T|_{S}$ on $S$. Define the hidden-load operator by

Theorem 3.1 (Hidden-load geometry).

The operator $\Lambda$ is positive semidefinite and

Conversely, every $\Lambda\succeq 0$ on $S$ determines a unique $X$ by (3.2). The canonical hidden realisation is

Then $\mathcal{H}_{\mathrm{can}}(T,X)\succ 0$, and its visible Schur complement is $X|_{S}$. If $\Lambda=BB^{\mathsf{T}}$ has rank $r$, then

This is a minimal hidden realisation, unique up to $B\mapsto BQ$ with $Q\in O(r)$. Finally,

$$\operatorname{rank}(\Lambda)=\operatorname{rank}(T-X),\qquad\log\operatorname{pdet}(T)-\log\det(X|_{S})=\log\det(I_{S}+\Lambda).$$ (3.5)

Proof in Appendix A.

Remark.

The support-preserving visible class beneath a fixed ceiling is exactly parametrised by the positive cone $\Lambda\succeq 0$. The quantity $\operatorname{rank}(\Lambda)=\operatorname{rank}(T-X)$ is the minimal hidden dimension, so the minimal realisation already defines an intrinsic model-selection criterion once the ceiling has been fixed. It is equally important to state the boundary: there is no canonical nontrivial ceiling determined by $(H,C)$ alone. Thus $\Phi_{C}(H)$ is the intrinsic visible object, whereas hidden load and determinant clock become exact only relative to a declared ceiling or reference.

The hidden loads compose exactly.

Proposition 3.2 (Transport law and determinant clock).

Let $\Lambda,M\succeq 0$ on $S$ and set

$$\Pi=(I_{S}+\Lambda)^{-1},\qquad\Xi=(I_{S}+M)^{-1}.$$ (3.6)

Define the sequential visible product by

$$\Pi\circ\Xi:=\Pi^{1/2}\Xi\Pi^{1/2}.$$ (3.7)

Then

The scalar clock

$$\tau(\Lambda):=\log\det(I_{S}+\Lambda)$$ (3.9)

is additive under this transport:

$$\tau(\Lambda_{\mathrm{tot}})=\tau(\Lambda)+\tau(M).$$ (3.10)

Proof in Appendix A.

Remark.

Equation (3.8) is the algebra of sequential observation, while the clock $\tau$ is its scalar entropy-type coordinate. If $\Lambda_{t}=tA+O(t^{2})$, then

$$\tau(\Lambda_{t})=t\,\operatorname{Tr}(A)+O(t^{2}).$$ (3.11)

Accordingly, stagewise hidden births add at first order and interact only at higher order.

The local calculus of Section 2 is the infinitesimal form of this geometry.

Theorem 3.3 (Local hidden return as hidden-load birth).

Let

$$X_{t}:=\Phi_{C}(H+t\Delta)-\Phi_{C}(H)=tV-t^{2}Q+O(t^{3})$$ (3.12)

on $S:=\operatorname{Ran}(V)$, and set $T_{t}:=tV$. Then the hidden load of $X_{t}$ beneath the ceiling $T_{t}$ satisfies

Proof in Appendix A.

Remark.

Theorem 3.3 identifies the local hidden return exactly: $Q$ is the infinitesimal generator of the nonlinear hidden-load cone after ceiling normalisation.

Remark.

The later coupled field layer preserves the same hidden-load cone and the same transport algebra on every support-stable stratum. On such a stratum, Proposition 3.2 and Theorem 3.3 continue with infinitesimal generator

$$A_{\mathrm{cpl}}:=-\tfrac{1}{2}V_{S}^{-1/2}(D_{t}^{\alpha}V)_{S}V_{S}^{-1/2},$$

while the present theorem is the anchored fixed-observer case $A=V_{S}^{-1/2}Q_{S}V_{S}^{-1/2}$.

Combining Corollary 2.4 with Theorem 3.3 gives the final identification.

Corollary 3.4 (The quartic defect is the first hidden birth).

For the bridge family (2.17), after ceiling normalisation by $T_{\varepsilon}:=\varepsilon^{2}V_{2}$ on $S:=\operatorname{Ran}(V_{2})$,

Thus the quartic visible defect is exactly the first hidden-load birth.

Proof in Appendix A.

Equations (2.4), (1.1), (2.11), (3.2), and (3.14) complete the route from the bridge to the full geometry of observation. The transport law points toward sequential observation, the clock toward entropy, and the minimal realisation toward model selection.

4  Conclusion

We record the geometry here and go no further, with one worked example in Appendix B. To unfold its implications and the identities it brings into view would require a much longer treatment. In the Gaussian case alone, the hidden-load operator identifies the determinant clock with twice the mutual information, the minimal hidden dimension with the number of active canonical correlations, and the gauge freedom of the minimal realisation with the orthogonal indeterminacy of factor analysis.

Later work sharpens the picture. First, for fixed observer $C$, the visible map $H\mapsto\Phi_{C}(H)$ admits a global split chart

$$H=T_{K}\operatorname{diag}(\Phi,H_{hh})T_{K}^{\mathsf{T}},\qquad\log\det H=\log\det\Phi+\log\det H_{hh},$$

and in that fixed-observer sector the Fisher-Rao geometry is flat in the Ehresmann sense. Second, the same quartic obstruction $Q=L^{\mathsf{T}}\Delta P\,H^{-1}\Delta L$ from Theorem 2.3 reappears as the exact forcing term in the fixed-observer visible dynamics,

$$\Phi^{\prime\prime}=\Phi^{\prime}\Phi^{-1}\Phi^{\prime}-Q,$$

so static leakage and dynamic forcing are two faces of the same visible-hidden defect. Third, field work shows that support-stable transport is exact only interval wise: genuine support birth or death forces canonical restart rather than a single globally smooth observer-covariant law.

Appendix A Proofs

Proof of Theorem 2.1

Complete the square in $y$:

$$\mathcal{L}(x,y)=-\Bigl(y-\frac{1}{2}S^{-1}(S+\widehat{J})x\Bigr)^{\mathsf{T}}S\Bigl(y-\frac{1}{2}S^{-1}(S+\widehat{J})x\Bigr)+\frac{1}{4}x^{\mathsf{T}}(S+\widehat{J})^{\mathsf{T}}S^{-1}(S+\widehat{J})x+O(\|(x,y)\|^{3}).$$

Taking the supremum over $y$ gives

$$I_{DV}(x)=\frac{1}{4}x^{\mathsf{T}}(S+\widehat{J})^{\mathsf{T}}S^{-1}(S+\widehat{J})x+O(\|x\|^{3}).$$

Since $\widehat{J}^{\mathsf{T}}=-\widehat{J}$,

$$(S+\widehat{J})^{\mathsf{T}}S^{-1}(S+\widehat{J})=(S-\widehat{J})S^{-1}(S+\widehat{J})=S-\widehat{J}S^{-1}\widehat{J}=S+\widehat{J}S^{-1}\widehat{J}^{\mathsf{T}}.$$

Using $H_{0}=\tfrac{1}{2}S$, hence $H_{0}^{-1}=2S^{-1}$, yields (2.4). The Gram form (2.5) is immediate.

Proof of Proposition 2.2

A Lagrange multiplier calculation for (2.6) gives the minimiser

$$x=H^{-1}C^{\mathsf{T}}(CH^{-1}C^{\mathsf{T}})^{-1}z,$$

which yields (2.7). For (2.8),

$$z^{\mathsf{T}}\Phi_{C}(H)z=\inf_{Cx=z}x^{\mathsf{T}}Hx=\inf_{C_{2}u=z}\inf_{C_{1}x=u}x^{\mathsf{T}}Hx=\inf_{C_{2}u=z}u^{\mathsf{T}}\Phi_{C_{1}}(H)u=z^{\mathsf{T}}\Phi_{C_{2}}(\Phi_{C_{1}}(H))z.$$

The block formula is the Schur complement form of (1.1).

Proof of Theorem 2.3

Write $M(t):=C(H+t\Delta)^{-1}C^{\mathsf{T}}$. The resolvent expansion gives

$$M(t)=\Phi^{-1}-t\,\Phi^{-1}V\Phi^{-1}+t^{2}\,\Phi^{-1}(V\Phi^{-1}V+Q)\Phi^{-1}+O(t^{3}),$$

which inverts to (2.11). Identity (2.13) follows from $P=I-LC$ and $HL=C^{\mathsf{T}}\Phi$ after regrouping terms. Substituting (2.11) into $-\log\det$ yields (2.15).

Proof of Corollary 2.4

Apply Theorem 2.3 with $H=H_{0}$, $t=\varepsilon^{2}$, and $\Delta=\Delta_{2}$.

Proof of Theorem 3.1

Set $\Pi:=(T|_{S})^{-1/2}(X|_{S})(T|_{S})^{-1/2}$. Then $0\prec\Pi\leq I_{S}$, so $\Lambda=\Pi^{-1}-I_{S}\succeq 0$ and $\Pi=(I_{S}+\Lambda)^{-1}$, which gives (3.2). Conversely, every $\Lambda\succeq 0$ produces such a $\Pi$ and hence such an $X$.

For (3.3), the Schur complement is

$$T|_{S}-(T|_{S})^{1/2}\Lambda^{1/2}(I_{S}+\Lambda)^{-1}\Lambda^{1/2}(T|_{S})^{1/2}=(T|_{S})^{1/2}(I_{S}+\Lambda)^{-1}(T|_{S})^{1/2}.$$

If $\Lambda=BB^{\mathsf{T}}$, the same calculation gives (3.4), and the orthogonal gauge is the usual freedom of Gram factorisation. Finally,

$$T-X=T^{1/2}\Lambda(I_{S}+\Lambda)^{-1}T^{1/2},$$

which gives the rank identity, and the determinant identity follows by taking determinants on $S$.

Proof of Proposition 3.2

Since $\Pi^{-1/2}=(I_{S}+\Lambda)^{1/2}$ and $\Xi^{-1}=I_{S}+M$,

$$(\Pi\circ\Xi)^{-1}=\Pi^{-1/2}\Xi^{-1}\Pi^{-1/2}=(I_{S}+\Lambda)^{1/2}(I_{S}+M)(I_{S}+\Lambda)^{1/2},$$

which is (3.8). Taking determinants gives (3.10).

Proof of Theorem 3.3

On $S$,

$$X_{t,S}=t\,V_{S}^{1/2}\bigl(I_{S}-t\,V_{S}^{-1/2}Q_{S}V_{S}^{-1/2}+O(t^{2})\bigr)V_{S}^{1/2}.$$

Hence

$$X_{t,S}^{-1}=t^{-1}V_{S}^{-1/2}\bigl(I_{S}+t\,V_{S}^{-1/2}Q_{S}V_{S}^{-1/2}+O(t^{2})\bigr)V_{S}^{-1/2}.$$

Conjugating by $T_{t,S}^{1/2}=t^{1/2}V_{S}^{1/2}$ gives

$$\Lambda_{t}=T_{t,S}^{1/2}X_{t,S}^{-1}T_{t,S}^{1/2}-I_{S}=t\,V_{S}^{-1/2}Q_{S}V_{S}^{-1/2}+O(t^{2}),$$

which is (3.13).

Proof of Corollary 3.4

Apply Theorem 3.3 with $t=\varepsilon^{2}$, $V=V_{2}$, and $Q=Q_{4}$.

Appendix B Gaussian coarse-graining and canonical correlations

In the Gaussian equilibrium case, the hidden-load geometry reduces to canonical correlation structure in closed form.

Let $(v,h)$ be a centred joint Gaussian vector with block precision

$$H=\begin{pmatrix}H_{vv}&H_{vh}\\ H_{hv}&H_{hh}\end{pmatrix},\qquad C=(I_{m}\;\;0),$$

where $H_{vv}\in\operatorname{Sym}_{++}(m)$ and $H_{hh}\in\operatorname{Sym}_{++}(r)$. The visible precision is the Schur complement

$$X:=\Phi_{C}(H)=H_{vv}-H_{vh}H_{hh}^{-1}H_{hv},$$

and the natural ceiling is $T:=H_{vv}$. Define

$$R:=H_{vv}^{-1/2}H_{vh}H_{hh}^{-1}H_{hv}H_{vv}^{-1/2}.$$

Then $0\preceq R\prec I_{m}$ and

$$X=T^{1/2}(I_{m}-R)T^{1/2}.$$

Therefore the hidden load is

$$\Lambda=T^{1/2}X^{-1}T^{1/2}-I_{m}=(I_{m}-R)^{-1}-I_{m}=R(I_{m}-R)^{-1}.$$

The nonzero eigenvalues of $R$ coincide with the squared canonical correlations $\rho_{1}^{2},\dots,\rho_{q}^{2}$ between the visible and hidden sectors, where $q:=\operatorname{rank}(R)$. Hence the nonzero hidden-load eigenvalues are

$$\lambda_{k}=\frac{\rho_{k}^{2}}{1-\rho_{k}^{2}},\qquad k=1,\dots,q.$$

Thus each hidden-load eigenvalue is the odds ratio associated with a squared canonical correlation.

The determinant clock becomes

$$\tau(\Lambda)=\log\det(I_{m}+\Lambda)=-\sum_{k=1}^{q}\log(1-\rho_{k}^{2}).$$

For a joint Gaussian vector,

$$I(v;h)=-\frac{1}{2}\sum_{k=1}^{q}\log(1-\rho_{k}^{2}),$$

so in this equilibrium setting

$$\tau(\Lambda)=2I(v;h).$$

The rank theorem gives

$$\operatorname{rank}(\Lambda)=\operatorname{rank}(R)=q,$$

so the hidden-load rank is the number of nonzero canonical correlations, equivalently the minimal latent dimension in the canonical hidden realisation.

For sequential Gaussian coarse-grainings, the transport law reads

$$I+\Lambda_{\mathrm{tot}}=(I+\Lambda_{1})^{1/2}(I+\Lambda_{2})(I+\Lambda_{1})^{1/2},\qquad\tau(\Lambda_{\mathrm{tot}})=\tau(\Lambda_{1})+\tau(\Lambda_{2}).$$

Thus the stagewise Gaussian clocks add exactly, while the operator-valued transport records how the directional hidden loads compose.

This is the static equilibrium case; the load is carried entirely by correlation. The general construction retains the same geometry and adds the dynamical hidden load induced by the Donsker–Varadhan bridge.