Watermark Protection Columns

• Watermark Protection Columns are techniques that embed ownership signals into structured units—such as image columns or tabular column groups—to persist through transformations like shuffling, deletion, or model training.
• Key methods include column-wise stripe insertion, secret-seeded group histogram encoding, and paired-column interval biasing, each using tailored detection strategies such as pattern classification or statistical tests.
• These approaches balance trade-offs between capacity, imperceptibility, and robustness, achieving high bit accuracy (up to 99.9% in some cases) while preserving data utility under various attack scenarios.

Searching arXiv for the cited watermarking papers and closely related work on column-wise/tabular watermarking. “Watermark Protection Columns” can be understood, as an Editor’s term, as watermarking schemes that place ownership signals into column-structured units: image columns, column-groups of a table, or paired data columns whose induced statistics are intended to persist through model training, sampling, shuffling, deletion, or watermark-removal attempts. In the image setting, DiffusionShield explicitly describes a column-wise variant in which each of the $V$ columns of an image carries a vertical stripe pattern chosen from a small dictionary, with detection performed per column (Cui et al., 2023). In synthetic tabular data, TableMark encodes multi-bit identifiers through partial orders between secret-seeded column-groups (Xia et al., 14 Mar 2026), while “Adaptive and Robust Watermark for Generative Tabular Data” pairs continuous features into $(\text{key},\text{value})$ columns and biases value entries toward randomized “green” intervals (Ngo et al., 2024). A related but contrasting line of work replaces the invisibility objective with visible, hard-to-remove watermarking formulated as an inverse problem (Liu et al., 3 Jun 2025).

1. Scope, threat models, and shared design logic

Across these methods, the protected object is not an isolated pixel or a single table entry but a column-level structure that can be re-identified after downstream transformations. The motivating threats differ by modality. In generative diffusion models, the concern is that artists’ or photographers’ works may be learned without authorization and then reproduced in generated images; DiffusionShield addresses this by embedding ownership information into released images so that a generative model can learn and later reproduce the watermark (Cui et al., 2023). In synthetic tabular data, the central threats are shuffling or deletion operations, unauthorized redistribution to multiple users, and the need to preserve data utility while retaining traceability (Xia et al., 14 Mar 2026). In continuous-feature tabular generation, the stated goal is authenticity under attacks such as feature selection and numeric truncation, while maintaining negligible impact on downstream utility (Ngo et al., 2024).

A recurring design pattern is repeated, keyed structure. DiffusionShield emphasizes that a uniform watermark repeated in every training example is easy for a generative diffusion model to learn and reproduce. TableMark ties each bit to two different column-groups and keys the group-pair template. The adaptive tabular scheme regenerates green intervals from a hash of the key-column bin centers, so detection can be performed from the data itself without storing explicit interval assignments.

Scheme	Protected unit	Detection signal
DiffusionShield column-wise variant	Image columns of size $U\times 1$	Per-column pattern classification
TableMark	Secret-seeded column-groups	Partial-order comparison of group histogram pairs
Adaptive and Robust Watermark	Paired continuous columns $(K_k,V_k)$	One-proportion $z$-test on green-interval occupancy

This suggests that “column” is functioning as a stability primitive: a unit large enough to survive generative modeling or dataset manipulation, yet structured enough to permit keyed decoding.

2. Column-wise watermarking in images

DiffusionShield is introduced as a watermarking scheme tailored for generative diffusion models and is built around a globally uniform watermark that is jointly optimized with a decoder. In its original form, a textual message $M$ is converted to a binary string, grouped into $B$-nary digits, mapped to a dictionary of $B$ basic patches, tiled across the image, and injected under a small $L_\infty$ or $L_2$ constraint so that $(\text{key},\text{value})$0 remains visually indistinguishable from $(\text{key},\text{value})$1. The decoder $(\text{key},\text{value})$2 is trained jointly with the watermark patches using cross-entropy, alternating SGD updates on $(\text{key},\text{value})$3 and PGD updates on the nonzero patches under the constraint $(\text{key},\text{value})$4 (Cui et al., 2023).

The same work explicitly generalizes this blockwise design to a column-wise scheme. Instead of arranging $(\text{key},\text{value})$5 patches in a full grid, one chooses $(\text{key},\text{value})$6 columns of size $(\text{key},\text{value})$7, associates each column $(\text{key},\text{value})$8 with a vertical stripe pattern $(\text{key},\text{value})$9 selected from a small dictionary, and injects the watermark by

$U\times 1$0

The decoder then operates per column, classifying $U\times 1$1 into one of the $U\times 1$2 possible patterns, and message recovery is the sequence of those column-level classifications.

The column-wise construction inherits the central rationale of DiffusionShield: repeated structure is easy for a generative diffusion model to learn. The stated trade-offs are explicit. Robustness benefits from the fact that each column stripe is repeated in every training example, so the model “will learn it very early and reproduce it reliably.” Imperceptibility is more difficult than in the $U\times 1$3 blockwise case because vertical stripes are more noticeable, so $U\times 1$4 must be chosen smaller or smoother low-frequency stripe patterns must be used. Capacity rises to as much as $U\times 1$5 bits, but this comes at the cost of modifying higher-level visual structures. Detection complexity is simpler because partitioning is by columns, but classification must operate on entire-column inputs.

The empirical evidence reported for DiffusionShield’s blockwise system explains why the column-wise extension is plausible. On CIFAR-10 at $U\times 1$6 ($U\times 1$7), DiffusionShield achieved $U\times 1$8 bit accuracy on released images, compared with IB $U\times 1$9, FRQ $(K_k,V_k)$0–$(K_k,V_k)$1, HiDDeN $(K_k,V_k)$2, and DFD $(K_k,V_k)$3; on conditional DDPM samples it achieved $(K_k,V_k)$4–$(K_k,V_k)$5, and on unconditional samples $(K_k,V_k)$6–$(K_k,V_k)$7 even at $(K_k,V_k)$8. Its uniformity scores were $(K_k,V_k)$9, versus $z$0 for HiDDeN and $z$1 for FRQ. The paper further reports similar trends on CIFAR-100, STL-10, ImageNet-20, and under DDIM, hyperparameter changes, and mild image corruptions such as Gaussian noise, JPEG, greyscale, and low-pass filtering. Those results are for the blockwise watermark, not the stripe variant, but they establish the role of uniform repeated structure in watermark transfer through generative training.

3. Column-group histogram channels in synthetic tabular data

TableMark addresses a different setting: synthetic tabular data, where direct watermark embedding is vulnerable to column shuffling and deletion. Its watermark channel operates over secret-seeded column-groups rather than raw columns. The scheme partitions the $z$2 columns into $z$3 “column-groups” using secret-seeded clustering on column-wise statistics such as marginal histograms or pair-wise correlations, then extracts the original column-group histogram $z$4 (Xia et al., 14 Mar 2026).

To encode an $z$5-bit watermark $z$6, the method selects $z$7 distinct groups and pseudo-randomly pairs them into $z$8 group-pairs $z$9, keyed by a secret. Each bit is represented as a partial order on the watermarked histogram $M$0: if $M$1, then $M$2; if $M$3, then $M$4. Because each bit is tied to two different groups, deleting or reordering up to $M$5 groups still leaves at least $M$6 bits intact, and the decoded $M$7 may still be matched within Hamming distance $M$8.

The core optimization problem is

$M$9

together with per-bit order constraints and a refined minimum-gap condition $B$0 chosen so that the probability of a bit flip under the worst-case attack intensity remains at most $B$1. This yields quadratic constraints of the form

$B$2

plus the sign constraint

$B$3

The optimization algorithm initializes $B$4 and iteratively simplifies the constraints, solves a simplified integer program, and adjusts a tightness parameter $B$5. In the synthesis phase, the watermarked histogram $B$6 is fed into a column-conditional generative model. Extraction re-computes column-group assignments on a suspect table, forms a suspect histogram $B$7, decodes each bit by comparing $B$8 and $B$9, and matches the recovered word to a buyer database within Hamming distance $B$0.

The reported UCI “Electric” example uses $B$1 columns, $B$2 clusters, and $B$3 bits. Without deletion, $B$4 of buyers were correctly traced. If an attacker deleted up to $B$5 columns at random, approximately $B$6 of groups, the method still recovered at least $B$7 bits, yielding buyer matching at at least $B$8 bit accuracy. Against full-column shuffling, detection remained at $B$9 because the group-pair template is keyed. Compared with a single-column-LSB scheme or an un-optimized partial-order encoding without constraint gaps, the approach reduced the bit-error rate under column-drop from $L_\infty$0 to $L_\infty$1 at the same utility loss of approximately $L_\infty$2 distribution-gap, while a downstream classifier’s AUC dropped by less than $L_\infty$3 relative to non-watermarked synthetic data.

4. Paired-column watermarking with green intervals

“Adaptive and Robust Watermark for Generative Tabular Data” uses an explicitly column-paired construction for continuous tabular features. The data matrix is $L_\infty$4, and the $L_\infty$5 continuous features are partitioned into $L_\infty$6 disjoint $L_\infty$7 pairs by a subroutine $L_\infty$8. Two pairing strategies are analyzed: “Uniform pairing,” which picks a random matching, and “Feature-importance pairing,” which ranks columns by importance and pairs nearby ranks with probability proportional to $L_\infty$9 (Ngo et al., 2024).

For each key column $L_2$0, the interval $L_2$1 is partitioned into $L_2$2 equal-width bins

$L_2$3

A random half-subset of these bins on the value axis is designated “green.” The construction is keyed by the key column itself: the centers of the bins are hashed, the resulting hash seeds a pseudorandom generator, and an unbiased coin flip determines whether each bin is green or red. Embedding then modifies the paired value column $L_2$4. For each row, the algorithm finds the green bin whose center is nearest to the current value. If the value already lies in that bin it is unchanged; otherwise it is replaced by a fresh uniform sample from that nearest green bin. The key columns remain unchanged, and a generative model is trained directly on the watermarked table $L_2$5.

Detection is statistical. For each candidate key-value pair, the auditor recomputes the corresponding green intervals, counts

$L_2$6

and forms

$L_2$7

Under the null hypothesis $L_2$8 of an un-watermarked table, Lemma 5.1 states that $L_2$9 as $(\text{key},\text{value})$00, so $(\text{key},\text{value})$01 converges to $(\text{key},\text{value})$02. Because there are up to $(\text{key},\text{value})$03 tested alignments, the paper applies Bonferroni correction and rejects $(\text{key},\text{value})$04 if any $(\text{key},\text{value})$05 exceeds the $(\text{key},\text{value})$06-quantile of $(\text{key},\text{value})$07, with typical $(\text{key},\text{value})$08. Under the alternative, one correct alignment yields $(\text{key},\text{value})$09, so the detection power tends to $(\text{key},\text{value})$10 as $(\text{key},\text{value})$11 grows.

Theoretical bounds are stated for both fidelity and robustness. Theorem 4.1 gives

$(\text{key},\text{value})$12

and Corollary 4.1 states

$(\text{key},\text{value})$13

Theorem 5.1 shows that feature-importance pairing preserves at least twice as many true pairs as uniform pairing after dropping $(\text{key},\text{value})$14 columns. Theorem 5.2 quantifies the probability that truncation to $(\text{key},\text{value})$15 decimal places moves a value outside its original bin, identifying a trade-off between bin-width and fidelity.

The empirical findings align with these claims. In synthetic Gaussian experiments with $(\text{key},\text{value})$16, the kernel density estimate before and after watermarking was nearly identical, and MSE versus bin-size $(\text{key},\text{value})$17 decreased linearly as $(\text{key},\text{value})$18 increased. The $(\text{key},\text{value})$19-score grew proportionally to $(\text{key},\text{value})$20, so with $(\text{key},\text{value})$21 the reported value easily exceeded $(\text{key},\text{value})$22. Under truncation to $(\text{key},\text{value})$23 decimal places, downstream XGBoost and RandomForest accuracy degraded negligibly. When dropping $(\text{key},\text{value})$24 or $(\text{key},\text{value})$25 least-important columns, feature-importance pairing retained approximately $(\text{key},\text{value})$26 of watermarked pairs versus approximately $(\text{key},\text{value})$27 under uniform pairing, and detection remained reliable under both truncation and column dropping.

5. Visibility, removability, and the objective of protection

A common assumption in earlier watermarking is that imperceptibility is the dominant criterion. “Beyond Invisibility: Learning Robust Visible Watermarks for Stronger Copyright Protection” explicitly challenges that assumption by seeking visible watermarks that are hard to remove (Liu et al., 3 Jun 2025). Although this work is not organized around table columns or per-column decoding, it is directly relevant to the design space because it changes the optimization target from hiddenness to removal difficulty.

The formulation begins with a clean copyrighted image $(\text{key},\text{value})$28 and a candidate visible watermark $(\text{key},\text{value})$29. The watermarked observation is

$(\text{key},\text{value})$30

where $(\text{key},\text{value})$31 is a differentiable mask. Watermark removal is modeled as an inpainting inverse problem under a deep generative prior $(\text{key},\text{value})$32, with MAP reconstruction

$(\text{key},\text{value})$33

The watermark is then chosen through a bi-level problem minimizing a reconstruction-quality score $(\text{key},\text{value})$34 plus an $(\text{key},\text{value})$35 penalty $(\text{key},\text{value})$36.

Because the inner problem is NP-hard to solve exactly, the paper introduces the Harvim algorithm, which replaces $(\text{key},\text{value})$37 by an unrolled approximation $(\text{key},\text{value})$38 and back-propagates through $(\text{key},\text{value})$39 inner gradient-ascent steps. The iteration jointly updates the prior weight $(\text{key},\text{value})$40, the inpainting solution $(\text{key},\text{value})$41, and the watermark $(\text{key},\text{value})$42. In practice, the reported setting is $(\text{key},\text{value})$43–$(\text{key},\text{value})$44, $(\text{key},\text{value})$45–$(\text{key},\text{value})$46, with AdamW used to update $(\text{key},\text{value})$47; recommended parameters include $(\text{key},\text{value})$48-weight $(\text{key},\text{value})$49 and sigmoid parameters $(\text{key},\text{value})$50.

Evaluation uses PSNR, SSIM, and LPIPS, but the paper defines normalized “removal-gain” measures such as

$(\text{key},\text{value})$51

with analogous $(\text{key},\text{value})$52 and $(\text{key},\text{value})$53. Lower values mean that removal produced little gain over the already watermarked observation, hence stronger protection. On CelebA, an ImageNet subset, and a hand-drawn Cartoon set, with digit logos and initials as watermark families, the sample table reports average $(\text{key},\text{value})$54-PSNR values of $(\text{key},\text{value})$55 for random $(\text{key},\text{value})$56 and $(\text{key},\text{value})$57 for Harvim $(\text{key},\text{value})$58 under Flow-R, and $(\text{key},\text{value})$59 versus $(\text{key},\text{value})$60 under RePaint; computation time is approximately $(\text{key},\text{value})$61 seconds per image on NVIDIA A6000 for Harvim, while removal methods range from $(\text{key},\text{value})$62 to $(\text{key},\text{value})$63 seconds.

The significance for column-based watermarking is comparative rather than identical. DiffusionShield and the tabular methods prioritize imperceptibility, fidelity, or utility under structured decoding. Harvim shows that protection can instead be framed as minimizing the effectiveness of an optimal remover. A plausible implication is that column-structured schemes and hard-to-remove visible schemes address different operational regimes: one seeks covert traceability after model training or tabular manipulation, while the other seeks explicit deterrence and removal resistance.

6. Trade-offs, misconceptions, and research directions

Several trade-offs recur across the literature. First, robustness is often purchased with structure. DiffusionShield states that pattern uniformity makes the watermark easy for a generative diffusion model to learn and reproduce; the column-wise variant similarly benefits from repeated stripe patterns, but becomes vulnerable to perturbations that break column sameness, such as column-wise cropping (Cui et al., 2023). TableMark strengthens robustness by introducing minimum-gap constraints between paired column-groups, but this optimization is explicitly a trade-off between robustness and utility (Xia et al., 14 Mar 2026). The adaptive tabular method improves survivability under feature selection through feature-importance pairing, but its detection cost depends on searching over possible key-value alignments and applying Bonferroni correction (Ngo et al., 2024).

Second, capacity is not free. DiffusionShield’s column-wise scheme can encode up to $(\text{key},\text{value})$64 bits, potentially more than a grid-based layout when $(\text{key},\text{value})$65 is large, but it modifies higher-level visual structures and can be more noticeable. TableMark’s multi-bit encoding supports a large number of users because it allocates watermark bits over many column-group pairs, yet the optimization must keep the watermarked histogram near the original. The adaptive tabular method does not present capacity in the same multi-bit buyer-tracing form; instead it emphasizes detectability, fidelity, and robustness of a statistical watermark.

Third, column manipulations do not uniformly destroy watermark signals. TableMark reports $(\text{key},\text{value})$66 detection under full-column shuffling because the group-pair template is keyed and the ordering test is invariant to column order. The adaptive tabular method is explicitly designed to survive column dropping better under feature-importance pairing than under uniform pairing. By contrast, the column-wise image-stripe scheme acknowledges that attacks disrupting column structure can harm detection.

Finally, invisibility should not be treated as the only protection criterion. DiffusionShield demonstrates nearly invisible watermarking with high bit accuracy and long messages in images. The two tabular methods show that statistical distortions can remain small while detection stays reliable. Harvim shows that visible watermarks can be optimized to remain difficult to remove. The literature therefore separates at least three objectives: imperceptibility, robustness to downstream transformation, and resistance to removal. Current work suggests that “watermark protection columns” are most effective when the chosen column structure matches the anticipated attack surface: repeated vertical stripes for generative image training, keyed group histograms for shuffling and deletion in tables, or paired-column interval bias for continuous-feature authenticity testing.