When Can We Trust the Matching Principle? Robust Deployment Geometry Under Finite-Sample and Model Uncertainty

Vishal Rajput

cs.LGOct 2, 2026

When Can We Trust the Matching Principle? Robust Deployment Geometry Under Finite-Sample and Model Uncertainty

Authors: Vishal Rajput

Organizations: Independent researcher Leuven, Belgium

Abstract

Match only geometry you can identify; otherwise spread the penalty. We quantify that decision by the trust ratio tau = epsilon / gamma (estimation uncertainty over spectral separation). Under the linear-quadratic Matching response, oracle-relative drift between estimated and oracle projector matching scales as tau^2 for probes in the chosen top-r deployment subspace -- O(tau^2) in the Davis-Kahan separation region tau < 1/2, with practical usefulness depending on constants. Confidence-Calibrated Matching (CCM) turns tau into a policy -- directional when tau is small, progressively isotropic when not -- with thresholds from calibration, not from the theorem (match sits in the separation region; soft is mostly heuristic). Experiments show both regimes, including UCI HAR embeddings where always-match is worse than abstain on every cell.

Figures & tables

Figure 1: Exp. 6 flagship panel: practical vs. ambient trust ratio (left) and accuracy arms (right) for Office-31, Fashion rotation, and UCI HAR. HAR is the failure-avoidance cell—always-match hurts; CCM abstains. Fashion is the identifiable positive control. Not a domain-adaptation SOTA table.

Figure 2: Exp. 2 trust curve on the synthetic spike grid: $\tau$ versus subspace error (left) and LQ drift gap to oracle projector match (right). Correlations: $0.83$ and $0.91$ . Supporting evidence that $\tau$ organises the theorem quantities—not the flagship claim.

$\gamma$	$n$	regime	naive $\widehat{\Sigma}$	iso $\mu I$	CCM	LW
$0.08$	$80$	even	$\mathbf{0.425}$	$0.779$	$0.653$	$0.537$
$0.08$	$400$	soft	$0.441$	$0.774$	$\mathbf{0.112}$	$0.471$
$0.08$	$1600$	match	$0.439$	$0.767$	$\mathbf{0.013}$	$0.444$
$0.25$	$80$	soft	$\mathbf{0.318}$	$0.759$	$0.374$	$0.359$
$0.25$	$400$	soft	$0.335$	$0.755$	$\mathbf{0.070}$	$0.341$
$0.25$	$1600$	match	$0.331$	$0.768$	$\mathbf{0.009}$	$0.333$

Table 1: Exp. 3 dual cost

J

(lower better;

d=32

30

reps). Bold: best among naive / iso / CCM. Regime

=

modal per-rep CCM decision; cell-mean CCM need not equal the pure policy named by the mode (e.g.

\gamma{=}0.08

n{=}80

19/30

even-spread and

11/30

soft, hence CCM

0.653\neq

iso

0.779

Figure 3: Exp. 3: regime frequencies (left) and dual cost versus fixed policies (right) on the gap $\times n$ grid.

$d$	$n/d$	regime	naive	CCM	iso
$16$	$2.5$	soft	$\mathbf{0.374}$	$0.562$	$0.751$
$16$	$12.5$	soft	$0.347$	$\mathbf{0.122}$	$0.754$
$16$	$50$	match	$0.350$	$\mathbf{0.009}$	$0.757$
$32$	$2.5$	soft	$\mathbf{0.312}$	$0.466$	$0.765$
$32$	$12.5$	soft	$0.328$	$\mathbf{0.076}$	$0.752$
$32$	$50$	match	$0.335$	$\mathbf{0.009}$	$0.750$

Table 2: Exp. 5 dual cost at gap

\gamma=0.25

(means over

20

reps). CCM wins whenever

n/d

is not tiny.

Robust Estimation

Robust estimation infers reliable parameters and states despite outliers, heavy-tailed noise, data contamination, or model uncertainty.

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Reliable semi-dense matching is essential for modern geometric vision systems. Designed under a coarse-to-fine paradigm, it achieves an optimal balance between performance and computational cost. However, existing methods often struggle to provide well-quantified uncertainties, where catastrophic coarse-assignment failures are ignored, leading to truncated error distributions and severely misjudged geometric estimations. In this paper, we propose a lightweight, post-hoc overall uncertainty estimation framework that introduces a two-component calibrated Laplace mixture model with only 9 learnable parameters. The objective is to explicitly capture both the sharp local refinement noise and the broader tail of coarse-assignment failures. We introduce the Coarse-success posterior Refit (CoRe) method, a geometric refitting module that utilizes the posterior probability of coarse-assignment success as soft correspondence weights. Extensive experiments show that our method consistently improves downstream geometric accuracy across various pretrained-only matchers and robust estimators with minimal computational overhead. Our code is available at https://github.com/khoavpt/Probabilistic-matching.

Abstract

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