CurveMark: Detecting AI-Generated Text via Probabilistic Curvature and Dynamic Semantic Watermarking

. 2025 Jul 24;27(8):784. doi: 10.3390/e27080784

Algorithm 2 Information-Theoretic Feature Extraction and Detection

Require: text, LM, EMB, N,

τ

, NGram, Classifier, k
Ensure: is_watermarked, confidence

1:
// Probability Curvature Features
2:
$P_{o r i g} \leftarrow log P (t e x t | L M)$
3:
$C \leftarrow []$ ▹ Curvature values
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for $i = 1$ to N do
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$t e x t_{i} \leftarrow SemanticPerturb (t e x t)$ ▹ Random synonym replacement, preserve structure
6:
$P_{i} \leftarrow log P (t e x t_{i} | L M)$
7:
$C$ .append( $P_{o r i g} - P_{i}$ )
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end for
9:
$μ_{C}, σ_{C}, {skew}_{C}, {kurt}_{C} \leftarrow Statistics (C)$
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// Information-Theoretic Features
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$H_{a v g} \leftarrow AverageEntropy (t e x t, L M)$
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$I_{m u t u a l} \leftarrow MutualInformation (t e x t_{w o r d s}, t e x t_{c h a r s})$
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$D_{K L} \leftarrow KLDivergence (P_{t e x t}, P_{r e f e r e n c e})$
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$PPL \leftarrow 2^{H_{a v g}}$ ▹ Perplexity from entropy
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// Watermark Detection Features
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$Λ \leftarrow 0$ ▹ Log-likelihood ratio
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$g r e e n_s c o r e s \leftarrow []$
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for each token $t_{i}$ in text do
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$N_{τ} (t_{i - 1}) \leftarrow SemanticNeighbors (t_{i - 1}, τ)$
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$GreenList \leftarrow Partition (N_{τ} (t_{i - 1}), h) [0]$
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if $t_{i} \in GreenList$ then
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$Λ \leftarrow Λ + log (| GreenList | / | N_{τ} |)$
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$g r e e n_s c o r e s$ .append( $cos (e_{t_{i}}, Mean (GreenList))$ )
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else
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$Λ \leftarrow Λ - log (1 - | GreenList | / | N_{τ} |)$
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end if
27:
end for
28:
$ρ_{o b s e r v e d} \leftarrow len (g r e e n_s c o r e s) / len (t e x t)$
29:
// Feature Aggregation
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$f_{c u r v e} \leftarrow [μ_{C}, σ_{C}, {skew}_{C}, {kurt}_{C}]$
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$f_{i n f o} \leftarrow [H_{a v g}, I_{m u t u a l}, D_{K L}, PPL]$
32:
$f_{w a t e r m a r k} \leftarrow [Λ, ρ_{o b s e r v e d}, Mean (g r e e n_s c o r e s), Std (g r e e n_s c o r e s)]$
33:
$f e a t u r e s \leftarrow [f_{c u r v e}, f_{i n f o}, f_{w a t e r m a r k}]$
34:
// Classification with Confidence
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$p (watermarked | f e a t u r e s) \leftarrow Classifier (f e a t u r e s)$
36:
$c o n f i d e n c e \leftarrow 2 \cdot | p (watermarked | f e a t u r e s) - 0.5 |$
37:
$i s_w a t e r m a r k e d \leftarrow p (watermarked | f e a t u r e s) > 0.5$
38:
return $i s_w a t e r m a r k e d$ , $c o n f i d e n c e$