The Signal-Coverage Matrix: Stratifying Type and Semantic Errors in Statement Autoformalization

Headline type-correctness (TC\%) of LLM autoformalization has climbed from $\sim$53\% to $\sim$76\% in two years, yet this scalar conceals which errors each method resolves. We propose a signal-coverage matrix that crosses the Lean elaborator (pass/fail) with a semantic-equivalence judgment (equivalent/not), sorting every output into one of four cells: true success (TS), type-only (TO), semantic-only (SO), or both fail (BF). On ProofNet\# and MiniF2F-test with DeepSeek V4-Pro across Vanilla, Lean-Retry, Sample-Filter, and Stratified Autoformalization (SAF): (1) the +34 to +36 TS gain across the three elab-feedback methods is $\sim$64\% type-stratum recovery, with SO flat on net (87.5\% of original semantic errors rescued, 8 newly created). (2) The TO-to-TS rate is 23/61 for each method (Wilson 95\% CI [26.6\%, 50.3\%]), and this stratum-level recovery rate predicts $Δ$TS on held-out methods to within 2/186 and renders $Δ$TC linear in the Vanilla elab-fail rate across six (model, dataset) cells ($R^2=0.96$). (3) The two judges disagree by 26 to 37 pp on elab-feedback outputs (vs. 7 pp on Vanilla), with 30 to 56\% of symbolic-judge false negatives traceable to elaborator-forced rewrites. The persistent residual reduces to two gold-formalization errors. TC\% gains should be credited by which cell moved, not by the scalar alone.

Robust Self-Training with Closed-loop Label Correction for Learning from Noisy Labels

Training deep neural networks with noisy labels remains a significant challenge, often leading to degraded performance. Existing methods for handling label noise typically rely on either transition matrix, noise detection, or meta-learning techniques, but they often exhibit low utilization efficiency of noisy samples and incur high computational costs. In this paper, we propose a self-training label correction framework using decoupled bilevel optimization, where a classifier and neural correction function co-evolve. Leveraging a small clean dataset, our method employs noisy posterior simulation and intermediate features to transfer ground-truth knowledge, forming a closed-loop feedback system that prevents error amplification. Theoretical guarantees underpin the stability of our approach, and extensive experiments on benchmark datasets like CIFAR and Clothing1M confirm state-of-the-art performance with reduced training time, highlighting its practical applicability for learning from noisy labels.