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Tuan Tran

Publications and source records attributed to Tuan Tran.

At least 19 recordsLinked to original sources
arXiv2026

TextSeal2: Distortion-Free LLM Watermarking with Copula Mixing and Memoryless Races

We present TextSeal2, a distortion-free watermark for large language models that keeps its signal under speculative decoding: tokens that replace rejected drafts remain watermarked with the same keys, and the acceptance rate is unchanged. It also preserves output diversity and adds no measurable inference latency. Its predecessor, TextSeal, restores diversity in the Gumbel-max scheme by routing each token to one of two keys via a coin flip. This is suboptimal because the detector averages the scores from both keys without knowing which key was used. TextSeal2 instead merges two or more keys into a single virtual key and couples it to fresh noise through AutoMark's Gaussian copula. Any number of keys, including publicly shared keys, can be merged this way without diluting detection. Under speculative decoding, TextSeal2 verifies keyed drafts with one more exponential race, run on the leftovers of the draft races. By memorylessness, this race reproduces block verification, so the output follows the target model exactly. We evaluate TextSeal2 on three LLMs. At equal output token diversity, TextSeal2 detects 8 to 22 percentage points more 100-token answers than TextSeal at a false-positive rate of $10^{-3}$, and is comparable to textGrain while requiring no post-logits optimization. Under speculative decoding, it detects 18 and 22 percentage points more 200-token answers than TextSeal and textGrain, respectively, at the same acceptance rate.

cs.CR↗
arXiv2026

Anti-concentration and Graham's rearrangement problem in finite abelian groups

Let $S$ be a subset of $n$ elements of a group $G$. An ordering $x_1,\ldots,x_n$ of its elements is valid if the partial sums $x_1,x_1+x_2,\ldots,x_1+\cdots+x_n$ are pairwise distinct. This notion has a long history in combinatorial group theory, dated back to Gordon's works in early 1960. Alspach-Liversidge conjecture asserts that any subset of the nonzero elements of a finite abelian group $G$ has a valid ordering. This generalizes Graham's conjecture, which considers the case when the group $G$ is $\mathbb{F}_p$, the cyclic group of prime order. Graham's conjecture has been solved recently, as a combination of results by Kravitz, Bedert and Kravitz, Müyesser and Pokrovskiy, Bedert, Bucić, Kravitz, Montgomery, and Müyesser, and Pham and Sauermann. Beside this, the only other case where Alspach-Liversidge conjecture has been confirmed is $G=\mathbb{F}_2^k$, thanks to a recent result by Bedert, Bucić, Kravitz, Montgomery, and Müyesser. We prove the Alspach-Liversidge conjecture in its full generality. Our proof combines ideas from many previous papers. In particular, we will use the probabilistic approach and the absorption method. Our new main technical ingredient is a Fourier-moment estimate, which is of independent interest. In particular, we can use it to show that the distribution of certain kinds of random sums in an abelian group has an almost uniform distribution.

math.CO↗
arXiv2026

On colorful Helly numbers and the growth of Tverberg numbers

We establish new bounds on colorful Helly and Tverberg numbers in abstract convexity spaces with Radon number $r$. We show that the colorful Helly number is at most $(r-1)2^r$, improving Holmsen--Lee's bound $r^{r^{O(\log r)}}$ and matching the known lower bound $2^{r-1}-1$ up to a factor $O(r)$. Furthermore, we prove that the Tverberg number is $r_k=O(r^{\lceil \log_2 r\rceil }k)$, reducing Pálvölgyi's coefficient $r^{r^{r^{O(\log r)}}}$ to a quasipolynomial function of $r$. For $S_4$-separable spaces, we prove $r_k=O(r^2k)$. In the other direction, we construct a sequence of $S_4$-separable spaces with Tverberg number $r_k=Ω(r^2k)$. This shows that the weak Eckhoff conjecture, asserting that $r_k=O(rk)$ in any abstract convexity space, fails already in separable spaces. The same construction refutes the analogue of Reay's conjecture in separable spaces.

math.CO↗
arXiv2026

Watermark Basins: Black-Box Detection of Distilled LLMs

Proprietary language models and their reasoning traces are vulnerable to API-based distillation, where an adversary trains a student model on the teacher model's outputs. To obtain evidence of distillation, model providers can watermark the teacher's outputs, aiming to leave a radioactive signal in the student's outputs. Existing black-box detectors search for such radioactivity using token-level statistics of the suspect model's outputs. However, token-level signals are not always detectable, for instance when the distillation traces are sampled at a low temperature. In this work, we discover the watermark basin, a persistent signal that enables more reliable watermark detection: during fine-tuning, watermarked teacher traces embed a distinct, key-specific geometric signal in parameter space that lies orthogonal to the ordinary task-specific fine-tuning subspace. Building on this, we introduce Watermark Basin Alignment (WM-Basin), a black-box watermark detector that queries the suspect model through its API, without access to its weights or logits. WM-Basin maps generated text to gradient representations using a fixed clean probe model, projects out the ordinary fine-tuning task directions, and tests whether the residual gradient aligns with the watermark basin induced by the watermark key. We evaluate WM-Basin extensively across diverse teacher models and student architectures (from 135M to 8.2B parameters) distilled on mathematical reasoning. At a strict 0.1% false-positive rate, WM-Basin detects 79% of distilled models at n-gram=1 and 45% at n-gram=3, substantially improving over the state-of-the-art token-level baseline (29% and 1.3%, respectively). Because the watermark signal resides in a persistent basin, it survives post-distillation modifications including quantization, pruning, and reinforcement-learning post-training.

cs.CR↗
arXiv2026

The threshold for fractional clique decompositions of random hypergraphs, via matrix scaling

Let $δ^*_{k,r}$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\ge2$, $r\ge k+1$ and $\varepsilon>0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $(δ^*_{k,r}+\varepsilon)n$ admits, with high probability, a fractional $K_r^{(k)}$-decomposition after retaining each edge independently with probability $p \ge C(\log n/n^{r-k})^{1/(\binom{r}{k}-1)}$. The bound on $p$ is sharp up to a constant factor, confirming the conjectured threshold order in the graph case. The proof develops a probabilistic matrix-scaling approach.

math.CO↗
arXiv2026

Stability of the Courtade-Kumar inequality

We prove dimension-independent stability for the Courtade-Kumar inequality: a Boolean function $f:\{-1,1\}^n\to\{-1,1\}$ whose information is close to the dictator value is close in probability to a signed dictator. The correlation dependence is sharp in order near zero and, for increasing functions, also at the noiseless endpoint.

cs.IT↗
arXiv2026

Fractional expectation thresholds and the "second" Kahn-Kalai conjecture

We show that the uniform probability measure on copies of a nonempty graph $H$ in $K_n$ is $Cq_H\log(2e(H))$-spread, where $q_H$ is its graphic expectation threshold. Consequently, the fractional expectation threshold of $H$ is at most $Cq_H\log(2e(H))$. We remove the logarithmic factor for trees and for graphs whose average degree is at least the logarithm of their maximum degree. This proves the ``second'' Kahn-Kalai conjecture for these two classes, which encompass most of the standard families studied in random graph containment problems.

math.CO↗
arXiv2026

Dictators are most informative

We prove the Courtade-Kumar conjecture: among all Boolean functions $f\colon \{-1,1\}^n\to\{-1,1\}$, a dictator retains the most information about a uniformly random input observed through independent binary noise.

cs.IT↗
arXiv2026

Quantum expanders and dimension-free commutator bounds

We prove that every traceless real or complex matrix $A$ can be written as $A=BC-CB$, with factors of the same size and over the same field satisfying $\|B\|\,\|C\|\le K\|A\|$, where $K$ is an absolute constant. The proof combines an approximate-rank dichotomy with stable commutator representations obtained from quantum expansion.

math.FA↗
arXiv2026

Three trees suffice for a constant stretch in minor-free graphs

In this short note, we show that $H$-minor-free graphs have a tree cover with $3$ trees and constant stretch for any fixed graph $H$. The number of trees matches the recent lower bound by Chen, Tan, and Xu who showed that a toroidal grid requires at least $3$ trees for constant stretch. Our result is obtained by establishing a connection between tree covers and Assouad--Nagata dimension and then invoking the recent dimension bound for minor-free metrics by Liu.

cs.DS↗
arXiv2026

A Picture Says Thousands of Words - Harnessing Dermal Exposure Data from Images through Hybrid Deep Learning for Enhanced Safety Assessment

This study developed a hybrid computer vision method to quantify exposed skin from images for dermal exposure assessment. Using 170 indoor-painting images, Mask R-CNN first identified human subjects and removed background interference; a color-based algorithm then segmented exposed skin. The resulting exposed-skin-to-body pixel ratios showed approximately 80% agreement with human estimates. The approach demonstrates a scalable way to extract semi-quantitative exposure information from images, with future extensions to body-part recognition, PPE detection, and video-based exposure analysis.

cs.CV↗
arXiv2026

Beating the Ahlswede--Khachatrian bound for the Erdős--Frankl--Pach problem

In the 1980s, Erdős and, independently, Frankl and Pach conjectured that, for sufficiently large $n$, every $(d+1)$-uniform family on $\{1,\ldots,n\}$ with VC-dimension $d$ has size at most $\binom{n-1}{d}$, the size of a star. Ahlswede and Khachatrian disproved this conjecture in 1997 by giving a family of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. This value has since been widely believed to be best possible, and Mubayi and Zhao explicitly conjectured its optimality in 2007. Very recently, Wang, Xu and Zhang proved their conjecture for $d=2$ and $n\ge 7$, providing further support for this belief. Surprisingly, we show that the Mubayi-Zhao conjecture is false for every $d\ge 3$ by constructing families larger than the Ahlswede--Khachatrian bound. Our constructions suggest that the answer to the Erdős--Frankl--Pach problem depends delicately on both $n$ and $d$.

math.CO↗
arXiv2026

Tower heights for color-avoiding Ramsey numbers of monotone paths

Ramsey numbers of monotone paths in ordered hypergraphs form a natural higher-uniformity extension of the classical Erdős--Szekeres theorems, and their tower height was determined by Moshkovitz and Shapira. A color-avoiding variant, initiated by Loh and further developed by Gowers and Long and by Mulrenin, Pohoata, and Zakharov, asks for monotone paths whose edges use only a bounded number of colors rather than a single color. For integers $q>p$, let $A_k(n;q,p)$ be the least integer $N$ such that every $q$-coloring of the ordered complete $k$-uniform hypergraph on $\{1,\ldots,N\}$ contains a monotone path of length $n$ whose edges use at most $p$ colors. We prove that, for every fixed $p$ and all sufficiently large $q$, the exact tower height of $A_k(n;q,p)$ is $\lceil (k-1)/p\rceil$. Thus the number of colors allowed on the path affects the Ramsey number at the level of tower height: allowing $p$ colors lowers the height from $k-1$ in the monochromatic problem to $\lceil (k-1)/p\rceil$. This answers questions of Mulrenin, Pohoata, and Zakharov. The upper bound follows from a simple block-compression argument. The main contribution is the matching lower bound, for which we develop a novel variant of the stepping-up method. A surprising feature of the proof is the appearance of the Morse--Hedlund theorem, a foundational result in symbolic dynamics and combinatorics on words. We establish and use a finite version of this theorem, which may be of independent interest.

math.CO↗
arXiv2026

TextSeal: A Localized LLM Watermark for Provenance & Distillation Protection

We introduce TextSeal, a state-of-the-art watermark for large language models. Building on Gumbel-max sampling, TextSeal introduces dual-key generation to restore output diversity, along with entropy-weighted scoring and multi-region localization for improved detection. It supports serving optimizations such as speculative decoding and multi-token prediction, and does not add any inference overhead. TextSeal strictly dominates baselines like SynthID-text in detection strength and is robust to dilution, maintaining confident localized detection even in heavily mixed human/AI documents. The scheme is theoretically distortion-free, and evaluation across reasoning benchmarks confirms that it preserves downstream performance; while a multilingual human evaluation (6000 A/B comparisons, 5 languages) shows no perceptible quality difference. Beyond its use for provenance detection, TextSeal is also ``radioactive'': its watermark signal transfers through model distillation, enabling detection of unauthorized use.

cs.CR↗
arXiv2026

The anti-concentration phenomenon with respect to random permutations

The anti-concentration phenomenon in probability theory has been intensively studied in recent years, with applications across many areas of mathematics. In most existing works, the ambient probability space is a product space generated by independent random variables. In this paper, we initiate a systematic study of anti-concentration when the ambient space is the symmetric group, equipped with the uniform measure. Concretely, we focus on the random sum $S_π = \sum_{i=1}^{n} w_i\, v_{π(i)}$, where $w=(w_1,\dots,w_n)$ and $v=(v_1,\dots,v_n)$ are fixed vectors and $π$ is a uniformly random permutation. The paper contains several new results, addressing both discrete and continuous anti-concentration phenomena. On the discrete side, we establish a near-optimal structural characterization of the vectors $w$ and $v$ under the assumption that the concentration probability $\sup_x P(S_π=x)$ is polynomially large. On the continuous side, we study the small-ball event $|S_π-L|\le δ$. Our results exhibit sub-gaussian decay in $L$. Our results have applications in various areas. First, we use our inverse theorems to derive and strengthen a number of previous anti-concentration bounds. In particular, we show that if both $w$ and $v$ have distinct entries, then $\sup_x P(S_π=x) \le n^{-5/2+o(1)}$. Next, we apply our new results to study random polynomials, and prove that the number of extremal points of random permutation polynomials is bounded by $O(\log n)$, extending results of S{ö}ze~\cite{Soze1, Soze2}. In the final application, we prove that random matrices whose rows are independent random permutations of a fixed non-degenerate vector are nonsingular with high probability.

math.CO↗
arXiv2026

Learning to Watermark in the Latent Space of Generative Models

Existing approaches for watermarking AI-generated images often rely on post-hoc methods applied in pixel space, introducing computational overhead and potential visual artifacts. In this work, we explore latent space watermarking and introduce DistSeal, a unified approach for latent watermarking that works across both diffusion and autoregressive models. Our approach works by training post-hoc watermarking models in the latent space of generative models. We demonstrate that these latent watermarkers can be effectively distilled either into the generative model itself or into the latent decoder, enabling in-model watermarking. The resulting latent watermarks achieve competitive robustness while offering similar imperceptibility and up to 20x speedup compared to pixel-space baselines. Our experiments further reveal that distilling latent watermarkers outperforms distilling pixel-space ones, providing a solution that is both more efficient and more robust.

cs.CV↗
arXiv2026

How Good is Post-Hoc Watermarking With Language Model Rephrasing?

Generation-time text watermarking embeds statistical signals into text for traceability of AI-generated content. We explore *post-hoc watermarking* where an LLM rewrites existing text while applying generation-time watermarking, to protect copyrighted documents, or detect their use in training or RAG via watermark radioactivity. Unlike generation-time approaches, which is constrained by how LLMs are served, this setting offers additional degrees of freedom for both generation and detection. We investigate how allocating compute (through larger rephrasing models, beam search, multi-candidate generation, or entropy filtering at detection) affects the quality-detectability trade-off. Our strategies achieve strong detectability and semantic fidelity on open-ended text such as books. Among our findings, the simple Gumbel-max scheme surprisingly outperforms more recent alternatives under nucleus sampling, and most methods benefit significantly from beam search. However, most approaches struggle when watermarking verifiable text such as code, where we counterintuitively find that smaller models outperform larger ones. This study reveals both the potential and limitations of post-hoc watermarking, laying groundwork for practical applications and future research.

cs.CR↗
arXiv2025

Pixel Seal: Adversarial-only training for invisible image and video watermarking

Invisible watermarking is essential for tracing the provenance of digital content. However, training state-of-the-art models remains notoriously difficult, with current approaches often struggling to balance robustness against true imperceptibility. This work introduces Pixel Seal, which sets a new state-of-the-art for image and video watermarking. We first identify three fundamental issues of existing methods: (i) the reliance on proxy perceptual losses such as MSE and LPIPS that fail to mimic human perception and result in visible watermark artifacts; (ii) the optimization instability caused by conflicting objectives, which necessitates exhaustive hyperparameter tuning; and (iii) reduced robustness and imperceptibility of watermarks when scaling models to high-resolution images and videos. To overcome these issues, we first propose an adversarial-only training paradigm that eliminates unreliable pixel-wise imperceptibility losses. Second, we introduce a three-stage training schedule that stabilizes convergence by decoupling robustness and imperceptibility. Third, we address the resolution gap via high-resolution adaptation, employing JND-based attenuation and training-time inference simulation to eliminate upscaling artifacts. We thoroughly evaluate the robustness and imperceptibility of Pixel Seal on different image types and across a wide range of transformations, and show clear improvements over the state-of-the-art. We finally demonstrate that the model efficiently adapts to video via temporal watermark pooling, positioning Pixel Seal as a practical and scalable solution for reliable provenance in real-world image and video settings.

cs.CV↗