Cull objectives: what each one optimises, measured effect, which to use¶

CullParameters::objective selects how gram-greedy ranks the feasible candidates of each cull: "unique" (the default), "minimax" or "total-loss". This note records why there are three, what separates them and what was measured when they were added. The exact computation is in the propose entry of the reference page, and the one-line contract is the header comment on CullParameters::objective. Append, do not rewrite: a later measurement gets its own dated section.

What the objective does and does not change (2026-09-29)¶

The feasibility test is the same for all three. A candidate \(i\) may be culled only if its unique information \(v_i = 1/M_{ii}\) is at most \(\tau\) and no history row is pushed past its budget (price \(W_{hi}^2/M_{ii} \le \tau - v_h\) for a row within budget, \(\le \delta = 0.01\) for a row already above \(\tau\)). So every objective respects the same budget; they differ only in which feasible view goes first. The rule is greedy and irreversible, so the order changes what stays feasible afterwards: the objectives end with different survivor sets and, in general, a different number of survivors.

Objective	Score of a feasible candidate \(i\)	Culls first
`unique`	\(v_i\)	the view the others explain best: the most redundant one
`minimax`	\(\max\big(v_i,\ \max_h (v_h + W_{hi}^2/M_{ii})\big)\), the worst unexplained view right after the cull	the view whose removal leaves the worst-off view least bad
`total-loss`	\(v_i + \sum_h W_{hi}^2/M_{ii} + \sum_{j \ne i} \big(1/M'_{jj} - 1/M_{jj}\big)\) over the other alive views	the view whose removal raises everything least, summed

unique is the rule in the paper (eq. greedy_rule) and the only one with a stated property: on a raw kernel in count-driven mode it is a greedy maximisation of the joint information of the survivors, and the score at which each view is culled rises monotonically. It looks at the candidate alone; the history enters only through the feasibility test.
minimax optimises the quantity tau bounds. Under a budget, it prefers culls that leave headroom, so later culls are less likely to be vetoed. It costs nothing extra: worst_after is already computed by the feasibility scan.
total-loss adds up two different kinds of rise. The first two terms are information lost: the candidate and the history rows are views that are no longer alive, and their unexplained information goes up. The third term is not a loss: an alive view \(j\) is still there, and a rise in its unique information only means it has become less redundant, so harder to cull next time. Summing them treats "harder to cull later" as equal to "lost now". That is a heuristic, not something that follows from the Gaussian model. It also costs an extra O(|A|) per candidate, so O(|A|²) per proposal.

In count-driven mode (\(\tau = +\infty\)) the feasibility test cannot fail, so the objective alone decides the order, and the difference between the three is at its largest.

Measured when the objectives were added (2026-09-26)¶

From the commits that introduced them (1942b4a) and fixed the history bound (7a4005a), not re-run for this note. Offline runs: kernel_demo on YOUTUBE acinipo D.npy, --clip --raw, tau 0.3.

Objective	Survivors at `1942b4a`	Survivors at `7a4005a`	Worst view
`unique`	19	19 (unchanged)	0.295 after any cull
`minimax`	19	19 (unchanged)	0.283 after any cull
`total-loss`	16	23	worst history row 0.303 → 0.29999

The three survivor sets overlapped by at most 3 views at 1942b4a: the objectives do not re-rank a few ties, they pick substantially different frames.
minimax kept the same number of views as unique with a lower worst view (0.283 vs 0.295), the headroom effect described above.
total-loss's 16 survivors at 1942b4a came with a history row at 0.303, above tau: it relied on the old bound, \(\max(\tau - v_h, \delta)\), which let a row near its budget be pushed past tau by up to the slack. With the bound fixed it keeps 23 views, more than the other two. On this sequence it is the least aggressive objective once the budget is actually enforced.
The default output is unchanged by both commits: synthetic_demo and kernel_demo (acinipo, --clip --raw) with unique are row-identical to the builds before them.

This is one sequence at one tau. No SLAM run with a non-default objective exists yet.

ETH `table_3` and the near-tau veto (2026-09-29)¶

kernel_demo on ETH table_3 vpr-lab/D.npy (1180 views, --clip --raw --tau 0.3), with a temporary counter in propose (not committed) that logged every veto by a history row within budget, and two temporary switches for the bound of those rows: a tolerance \(\varepsilon\) (v_h + price \le \tau + \varepsilon) and the old rule \(\max(\tau - v_h, \delta)\) from before 7a4005a.

Objective	Current rule	\(\varepsilon\) = 1e-6 … 1e-3	Old rule	Vetoes by rows within 1e-3 of tau
`unique`	91 kept, worst history 0.297	identical	identical	0 of 5
`minimax`	90 kept, worst 0.299	identical	identical	0 of 325
`total-loss`	142 kept, worst 0.300	139–140 kept, worst 0.300–0.301	84 kept, worst 0.326	734 of 1260

unique and minimax never meet a near-tau row: the bound of rows within budget does not matter for them.
total-loss minimises the summed rise, so it spreads small rises over many history rows and pushes them just under tau; those rows then veto. The prices they veto are real, not rounding: median 1.6e-4, 10th percentile 5.5e-6, 90th percentile 7e-4. A tolerance does not change the outcome (140 views even at 1e-3), and the old rule kept fewer views only by letting rows drift past the budget (0.326 at tau 0.3), the violation 7a4005a fixed.
Decision: keep the strict bound for rows within budget (no tolerance). On table_3, total-loss keeps 142 views where unique keeps 91, which supports keeping it experimental.

Over-budget rows after lowering tau. synthetic_demo --lower-tau T2 (400 views, tau 0.3 lowered halfway): at T2 = 0.25 and 0.2 no row goes over budget; at 0.15 six rows do, at 0.1 sixteen. In neither case does the worst history row rise afterwards (rise +0.0000 over the 3–4 later calls), and every over-budget row is back within budget by the last call, because newly inserted views explain them. So the per-cull slack \(\delta\) did not accumulate here. This is weak evidence: few calls follow the change, and longer runs (1600, 4000 views) insert nothing after the change (the trajectory only revisits known places), so they make no cull calls. A host that keeps culling after lowering tau without inserting near the old history is the untested case.

Which to use (2026-09-29)¶

unique stays the default. It is the rule the paper derives and the cheapest.
minimax when the worst-covered view matters more than the count, e.g. offline selection where a single poorly explained frame is the failure mode. On acinipo it gave the same count with more headroom.
total-loss: experimental. It mixes lost information with a rise in redundancy, and on the one measured sequence it culled least. Keep it out of experiments until that is settled.

Open questions (2026-09-29)¶

Infinite loss can still win. A feasible total-loss candidate whose downdate makes some pivot \(M'_{jj} \le 0\) gets loss \(+\infty\), but it beats the initial empty proposal (score \(+\infty\), tie broken on \(v_i\)), so it is returned when no candidate has a finite loss. Should that count as infeasible?
The early skip could apply to all three. Every score is at least \(v_i\) (exactly for minimax, up to rounding for total-loss), so a candidate with \(v_i\) at or above the running best cannot win under any objective. Today the skip is only applied for unique.
The winning score is not reported. Proposal::score never leaves propose, so for total-loss the report cannot show the loss of each cull; that would need a CulledView field.
More sequences. The comparison above is acinipo only, and raw only; the centred kernel and the count-driven mode are unmeasured.
The paper describes only unique. If minimax or total-loss goes into the experiments, eq. greedy_rule needs the objective, and total-loss needs a justification for its third term.

The objectives in the paper (2026-09-29)¶

paper/sec/03_methodology.tex now presents all three. Eq. greedy_rule is an argmin of a score \(s_i\) (\(v_i\) by default), and a new paragraph, Ranking the feasible candidates, defines the three scores in eq. objectives with the same caveats as this note: the constraints decide which views may go and the score only the order; minimax is the quantity tau bounds and costs nothing; the third term of total-loss is a rise in redundancy, not a loss, so the sum is a heuristic that pushes history rows towards tau. This answers the last open question above except for the experiments: no numbers are in the paper yet. The count-driven paragraph now restricts the joint-information property to the unique score, and states it through \(\log\det\mathbf{K}_{A\setminus\{i\}} = \log\det\mathbf{K}_{AA} - \log v_i\).