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Running on Zero
Running on Zero
| ## 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: | |
| $$y_j = \sqrt{w_j}\,(U^\top e_j)^\top z + g_j^\top u + \varepsilon_j,\qquad g_j = (I-UU^\top)e_j .$$ | |
| 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 | |
| $$I(z; y_j \mid y_S) = \tfrac12 \log \frac{\mathrm{Var}(y_j \mid y_S)}{\mathrm{Var}(y_j \mid y_S, z)},$$ | |
| 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. | |