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Oriented cross-encoding: computation

This page defines the oriented cross-encoding matrix \(M\) and how library code aggregates pre-anchor encoder means into one square cell. It complements the workflow guide Oriented cross-encoding matrix.

Paper appendix: cross_encoding_matrix.tex.

Setup

Let \(\mathcal{F}:=\{f_1,\ldots,f_K\}\) be feature classes (e.g. 3SG, Fem.Nom) and \(\mathcal{G}:=\{G_1,\ldots,G_M\}\) a set of pairwise GRADIEND models (e.g. 3SG↔3PL). Each \(G\in\mathcal{G}\) is trained on an ordered pair \((a_G,b_G)\in\mathcal{F}^2\).

We evaluate each \(G\) on a shared test pool \(\mathcal{D}\). Each probe \(x\in\mathcal{D}\) is associated with a factual class \(\mathrm{fac}(x)\in\mathcal{F}\) and a counterfactual class \(\mathrm{cnf}(x)\in\mathcal{F}\). For \(f,c\in\mathcal{F}\) with \(\mathcal{D}_{f,c}\neq\emptyset\), the directed transition slice is

\[ \mathcal{D}_{f,c}=\{x\in\mathcal{D}:\mathrm{fac}(x)=f,\ \mathrm{cnf}(x)=c\}. \]

For model \(G\), the encoding mean on that contrast slice is

\[ \bar{e}_G(f,c)=\frac{1}{|\mathcal{D}_{f,c}|}\sum_{x\in\mathcal{D}_{f,c}}\mathrm{enc}_G(x), \]

where \(\mathrm{enc}_G(x)\) is the scalar encoded feature value of probe \(x\) under \(G\). In code, these are rows of encoder_df grouped by directed transition factual→counterfactual; see compute_gradiend_transition_cross_encoding_matrix for the rectangular pre-anchor matrix (GRADIEND × transition).

For a probe feature \(f_j\), let \(\mu_G(f_j)\) be the mean encoded value of \(G\) over all probes whose GRADIEND source-side class is \(f_j\):

\[ \mu_G(f_j)= \begin{cases} \operatorname{mean}\{\mathrm{enc}_G(x):\mathrm{fac}(x)=f_j\}, & \text{if }G\text{ uses factual gradients as input},\\ \operatorname{mean}\{\mathrm{enc}_G(x):\mathrm{cnf}(x)=f_j\}, & \text{if }G\text{ uses counterfactual gradients as input}. \end{cases} \]

Rows vs columns

For each entry \(M_{i,j}\) of the cross-encoding matrix \(M\), \(f_i\) is the orienting feature and \(f_j\) is the probe feature. We ask whether GRADIEND models that separate \(f_i\) from other features assign high encoder values to \(f_j\) probes.

Role Index Meaning
Row \(f_i\) Orienting feature Select GRADIENDs whose ordered pair contains \(f_i\), then align their signs toward \(f_i\)
Column \(f_j\) Probe feature Select probes whose source-side class for the evaluated GRADIEND is \(f_j\)

Define \(\mathcal{G}_{f_i}=\{G\in\mathcal{G}: f_i\in\{a_G,b_G\}\}\) and the anchor sign

\[ \operatorname{sign}_G(f_i)= \begin{cases} +1 & \text{if } f_i=a_G \\ -1 & \text{if } f_i=b_G \end{cases} \]

The cross-encoding matrix is

\[ M_{i,j} = \frac{1}{|\mathcal{G}_{f_i}|} \sum_{G\in\mathcal{G}_{f_i}} \operatorname{sign}_G(f_i)\,\mu_G(f_j). \]

Implementation: compute_anchor_aligned_encoding_matrix builds per-contribution rows in aligned_rows, then [aggregate_anchor_aligned_encoding_rows][gradiend.comparison.anchor_aligned.aggregate_anchor_aligned_encoding_rows] pivots to \(M\).

What \(M\) is (and is not)

  • \(M\) is not a similarity matrix: \(M_{f_i,f_j}\neq M_{f_j,f_i}\) in general because rows and columns aggregate different quantities. Rows aggregate models oriented toward a feature, whereas columns select the probes being encoded.
  • Diagonal \(M_{f_i,f_i}\): self-encoding strength when the orienting feature and probe feature match. It is not a perfect self-similarity score; diagonal entries are therefore typically below \(1\).
  • Off-diagonal \(M_{f_i,f_j}\): cross-encoding from orienting feature \(f_i\) to probe feature \(f_j\) (leakage / shared structure). For inverted binary pairs, \(M_{f_a,f_b}\) and \(M_{f_b,f_a}\) typically have opposite signs.

In code, alignment="auto" follows the GRADIEND source side: factual-source models use factual probe classes, and counterfactual-source models use counterfactual probe classes. Explicit alignment values are diagnostic overrides. See [encoding_view_sign_for_source][gradiend.model._source_target.encoding_view_sign_for_source] for the view sign applied before anchor aggregation.

API map

Step Function
Shared test pool encoding [build_cross_task_encoder_summary][gradiend.comparison.cross_encoding.build_cross_task_encoder_summary]
Pre-anchor GRADIEND × transition compute_gradiend_transition_cross_encoding_matrix
Oriented rows + pivot compute_anchor_aligned_encoding_matrix
Plot plot_cross_encoding_heatmap
Inspect contributions payload["aligned_rows"] on oriented payload