METHOD.md

Method

For one query with candidate passages $d_1,\dots,d_n$:

• $r_j$: cross-encoder logit (InfoMax) or z-score fusion of cross-encoder logit and dense cosine (InfoMax+F).
• $e_j$: all-MiniLM-L6-v2 embedding, centred on the pool mean and renormalised; $q$ the query embedding, treated the same way.
• $w_j = \mathrm{minmax}(r)_j^{\gamma}$: relevance weight.

Target. A latent query-information vector $z \in \mathbb{R}^r$, with prior $z \sim \mathcal N(0,\Lambda)$, on the aspect basis $U = [,q,\ u_2,\dots,u_r,]$. Here $u_2,\dots,u_r$ are the top right-singular vectors of the relevance-weighted pool embeddings after removing the query direction, and $\Lambda = \mathrm{diag}(1, \rho, s_k^2/s_1^2)$.

Observations. Each passage is a noisy linear measurement of the target:

$$y_j=\sqrt{w_j}(U^{\mathrm{\top }}{e}_j{)}^{\mathrm{\top }}z+g_j^{\mathrm{\top }}u+{\epsilon }_j,g_j=(I-U{U}^{\mathrm{\top }})e_j\mathrm{.}y\_j\; =\; \backslash sqrt\{w\_j\}\backslash ,(U^\backslash top\; e\_j)^\backslash top\; z\; +\; g\_j^\backslash top\; u\; +\; \backslash varepsilon\_j,\backslash qquad\; g\_j\; =\; (I-UU^\backslash top)e\_j\; .$$

The shared nuisance $u \sim \mathcal N(0,\beta I)$ correlates the noise of passages with similar off-query content (duplicates, same source), and $\varepsilon_j \sim \mathcal N(0,\sigma^2)$.

Selection. Greedy maximisation of the conditional mutual information

$$I(z;y_j\mid y_S)=\frac{1}{2}\log \frac{\mathrm{V}\mathrm{a}\mathrm{r}(y_j\mid y_S)}{\mathrm{V}\mathrm{a}\mathrm{r}(y_j\mid y_S,z)},I(z;\; y\_j\; \backslash mid\; y\_S)\; =\; \backslash tfrac12\; \backslash log\; \backslash frac\{\backslash mathrm\{Var\}(y\_j\; \backslash mid\; y\_S)\}\{\backslash mathrm\{Var\}(y\_j\; \backslash mid\; y\_S,\; z)\},$$

computed exactly by Schur complements. No probabilities from a language model are needed: in the linear-Gaussian model the posterior entropy depends only on which passages are selected.

Anchoring. The first pick is the top-1 of the base ranking $r$. At each later step, the base ranking's next item $b$ is replaced by the max-gain item $j^\star$ only if $\mathrm{gain}(j^\star) > (1+\delta),\mathrm{gain}(b)$. With $\delta=\infty$ the selector returns the base ranking exactly.

Guarantee. None is claimed. Correlated noise breaks submodularity of the objective, so the greedy rule has no worst-case approximation bound.