lin-0220

18.3 The linear predictive-state model

Let \(\theta _i\in \mathbb R^{p_i}\) parameterize foundry \(i\), and let \(R^i_{ij}\) select or transport predictions to shared tests. After choosing an orientation of the overlap graph, stack the restriction differences:

\[ A\theta = \bigl( R^i_{ij}\theta _i-R^j_{ij}\theta _j \bigr)_{(i,j)}. \]

The compatible locus is \(\ker A\). If \(u\) is a proposed predictive update, the unit SID correction solves

\[ Aa=-A(\theta +u). \]

When \(A\) has full row rank, the minimum-norm realization is

\[ a = -A^\top (AA^\top )^{-1}A(\theta +u). \]

Theorem 18.3 Compatibility and prediction decomposition

Let

\[ P_A=I-A^\top (AA^\top )^{-1}A. \]

Starting from a compatible \(\theta \), the repaired update is

\[ \theta ^+=\theta +P_Au. \]

Thus SID retains precisely the component of the predictive update tangent to the compatibility subspace. From an arbitrary state, the unit repair projects \(\theta +u\) onto \(\ker A\).

Proof ▼

Substitute the displayed correction and use \(A\theta =0\). The matrix \(P_A\) is the orthogonal projector onto \(\ker A\).

The theorem also marks a boundary: projection may remove a locally useful component of \(u\). Structural and predictive metrics must be reported separately.