arXiv subjects

Sitan Chen

Publications and source records attributed to Sitan Chen.

At least 19 recordsLinked to original sources

arXiv2026

Parallelism, critical windows, and separations among diffusion language models

A popular selling point of diffusion large language models (dLLMs) is their capacity for parallelism: the ability to generate sequences of text far more efficiently than autoregressive models, which require one forward pass per token. Yet among the many competing paradigms for dLLMs, from masked to uniform to Gaussian diffusion, principled understanding of how these different proposals compare in parallelism remains limited. In this work, we initiate a fine-grained comparison of the capacity for parallelism among these three leading approaches and prove the following: - Uniform and Gaussian diffusion can sample in a number of forward passes which scales with the dual total correlation of the underlying distribution, a measure of intrinsic complexity which can be much smaller than the context length. Previously, it was only known how to achieve this using masked diffusion. - For a certain family of random empirical measures, we show that $\widetildeΘ(\sqrt{d})$ forward passes are necessary and sufficient to sample using uniform or Gaussian diffusion, yet there exist approximate score oracles for which $\widetildeΩ(d)$ forward passes are needed for masked diffusion. This establishes the first provable separation in parallelism between the three prevailing dLLM paradigms. Contrary to popular intuition that masked diffusions are harder to parallelize because they must commit to token values, the latter separation instead comes from the fact that the critical windows in masked diffusion sampling are asymptotically narrower than those in uniform and Gaussian diffusion sampling.

cs.LG↗

arXiv2026

From Interface to Inference: Eliciting Any-Order Inference from Any-Order Models

Many discrete reasoning tasks, such as code generation, are inherently non-causal: programmers move between high-level structure and local details, a process we call any-order inference. For autoregressive language models, which lack a native any-order interface, non-causal abilities such as infilling and next-edit prediction require hand-designed mechanisms. Can we instead design models that natively support any-order inference? Masked diffusion models have recently emerged as compelling candidates, as their any-order training objective naturally offers an any-order prediction interface. This interface, however, does not automatically yield any-order inference. We demonstrate that this interface-inference gap stems from positional uncertainty: fixed-canvas, token-level models may know what semantic component should appear without knowing where to place it. In light of this, we propose two complementary approaches: (1) Insertion-based masked diffusion, building on FlexMDM (Kim et al, 2025), relaxes fixed-position commitments via insertions, enabling generation across non-contiguous regions. (2) Latent-space masked diffusion shifts prediction to coarser semantic segments, enabling search over latent generation orders. Empirically, we train a 7B FlexMDM for Python coding and a 125M LatentMDM for GSM8K and show that both approaches induce distinct any-order inference behaviors and improve downstream performance. We release our codebase at https://github.com/SeunggeunKimkr/genuine-any-order.

cs.LG↗

arXiv2026

Online Shadow Tomography Matching the Classical Bounds

In Online Shadow Tomography, we are given copies of an unknown $d$-dimensional quantum state $ρ$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\mathrm{Tr}(A^{(t)}ρ)$ to within $\pm ε$. This is the direct quantum generalization of the classical problem of Adaptive Data Analysis. Prior results for online Shadow Tomography were suboptimal in all three parameters $m, d, ε$, lagging behind the best known and classical rates, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. Our first algorithm is the first to achieve $o(\log^2 m)$-dependence together with $\mathrm{poly}(\log(d)/ε)$; moreover, it improves all three exponents even in the Offline Shadow Tomography setting. Our second algorithm is known to be optimal among bounds independent of $d$, and improves the best prior result by a $\sqrt{m} \log m$ factor. The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron-Stein decomposition.

quant-ph↗

arXiv2026

The log log jam in Gaussian state tomography

Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with $\log \log E$, where $E$ is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a $\log \log E$ dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn $n$-mode pure Gaussian states with $O(n^2 / ε^2)$ samples, independent of $E$. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with $O(1/ε^2)$ samples, again independent of $E$. Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.

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arXiv2026

A Provably Efficient Method for Tensor Ring Decomposition and Its Applications

We present the first deterministic, finite-step algorithm for exact tensor ring (TR) decomposition. Our method leverages blockwise simultaneous diagonalization to recover TR cores from a limited number of tensor observations, under a dimension condition requiring each mode size to be at least quadratic in the TR rank and under a genericity assumption on the cores, thereby providing both algebraic insight and practical efficiency. We extend the approach to the symmetric TR setting, where parameter complexity is significantly reduced and applications arise naturally in physics-based modeling and exchangeable data analysis. To handle noisy observations, we develop a robust recovery scheme that couples our initialization with alternating least squares, achieving faster convergence and improved accuracy compared to classical methods. As applications, we obtain new algorithms for questions in other domains where tensor ring decomposition is a key primitive, namely matrix product state tomography in quantum information and provable learning of pushforward distributions in the foundations of machine learning. These contributions advance the algorithmic foundations of TR decomposition and open new opportunities for scalable tensor network computation.

math.NA↗

arXiv2026

Stop Training for the Worst: Progressive Unmasking Accelerates Masked Diffusion Training

Masked Diffusion Models (MDMs) have emerged as a promising approach for generative modeling in discrete spaces. By generating sequences in any order and allowing for parallel decoding, they enable fast inference and strong performance on non-causal tasks. However, this flexibility comes with a training complexity trade-off: MDMs train on an exponentially large set of masking patterns, which is not only computationally expensive, but also creates a train--test mismatch between the random masks used in training and the highly structured masks induced by inference-time unmasking. In this work, we propose Progressive UnMAsking (PUMA), a simple modification of the forward masking process that aligns training-time and inference-time masking patterns, thereby focusing optimization on inference-aligned masks and speeding up training. Empirically, PUMA speeds up pretraining at the 125M scale by $\approx 2.5\times$ and offers complementary advantages on top of common recipes like autoregressive initialization. We open-source our codebase at https://github.com/JaeyeonKim01/PUMA.

cs.LG↗

arXiv2026

Fine-Tuning Masked Diffusion for Provable Self-Correction

A natural desideratum for generative models is self-correction--detecting and revising low-quality tokens at inference. While Masked Diffusion Models (MDMs) have emerged as a promising approach for generative modeling in discrete spaces, their capacity for self-correction remains poorly understood. Prior attempts to incorporate self-correction into MDMs either require overhauling MDM architectures/training or rely on imprecise proxies for token quality, limiting their applicability. Motivated by this, we introduce PRISM--Plug-in Remasking for Inference-time Self-correction of Masked Diffusions--a lightweight, model-agnostic approach that applies to any pretrained MDM. Theoretically, PRISM defines a self-correction loss that provably learns per-token quality scores, without RL or a verifier. These quality scores are computed in the same forward pass with MDM and used to detect low-quality tokens. Empirically, PRISM advances MDM inference across domains and scales: Sudoku; unconditional text (170M); and code with LLaDA (8B).

cs.LG↗

arXiv2026

Optimal Quantum State Testing Even with Limited Entanglement

In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state $ρ$, test whether it matches some target state $σ$ or is $ε$-far from it. For certifying $d$-dimensional states, $Θ(d/ε^2)$ copies of $ρ$ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all $O(d/ε^2)$ copies of $ρ$. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on $t$ copies at once, for some $1 < t \ll d/ε^2$. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements. Our main result is a smooth copy complexity upper bound for state certification as a function of $t$, which achieves a near-optimal rate at $t = d^2$. In the high-precision regime, i.e., for $ε< \frac{1}{\sqrt{d}}$, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at $t = d^2$ as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our upper bounds with smooth lower bounds that imply joint measurements on $t \geq d^{Ω(1)}$ copies are necessary to achieve optimal rates for certification in the high-precision regime.

quant-ph↗

arXiv2026

Computation-Utility-Privacy Tradeoffs in Bayesian Estimation

Bayesian methods lie at the heart of modern data science and provide a powerful scaffolding for estimation in data-constrained settings and principled quantification and propagation of uncertainty. Yet in many real-world use cases where these methods are deployed, there is a natural need to preserve the privacy of the individuals whose data is being scrutinized. While a number of works have attempted to approach the problem of differentially private Bayesian estimation through either reasoning about the inherent privacy of the posterior distribution or privatizing off-the-shelf Bayesian methods, these works generally do not come with rigorous utility guarantees beyond low-dimensional settings. In fact, even for the prototypical tasks of Gaussian mean estimation and linear regression, it was unknown how close one could get to the Bayes-optimal error with a private algorithm, even in the simplest case where the unknown parameter comes from a Gaussian prior. In this work, we give the first efficient algorithms for both of these problems that achieve mean-squared error $(1+o(1))\mathrm{OPT}$ and additionally show that both tasks exhibit an intriguing computational-statistical gap. For Bayesian mean estimation, we prove that the excess risk achieved by our method is optimal among all efficient algorithms within the low-degree framework, yet is provably worse than what is achievable by an exponential-time algorithm. For linear regression, we prove a qualitatively similar lower bound. Our algorithms draw upon the privacy-to-robustness framework of arXiv:2212.05015, but with the curious twist that to achieve private Bayes-optimal estimation, we need to design sum-of-squares-based robust estimators for inherently non-robust objects like the empirical mean and OLS estimator. Along the way we also add to the sum-of-squares toolkit a new kind of constraint based on short-flat decompositions.

cs.DS↗

arXiv2026

High-accuracy and dimension-free sampling with diffusions

Diffusion models have shown remarkable empirical success in sampling from rich multi-modal distributions. Their inference relies on numerically solving a certain differential equation. This differential equation cannot be solved in closed form, and its resolution via discretization typically requires many small iterations to produce \emph{high-quality} samples. More precisely, prior works have shown that the iteration complexity of discretization methods for diffusion models scales polynomially in the ambient dimension and the inverse accuracy $1/\varepsilon$. In this work, we propose a new solver for diffusion models relying on a subtle interplay between low-degree approximation and the collocation method (Lee, Song, Vempala 2018), and we prove that its iteration complexity scales \emph{polylogarithmically} in $1/\varepsilon$, yielding the first ``high-accuracy'' guarantee for a diffusion-based sampler that only uses (approximate) access to the scores of the data distribution. In addition, our bound does not depend explicitly on the ambient dimension; more precisely, the dimension affects the complexity of our solver through the \emph{effective radius} of the support of the target distribution only.

cs.LG↗

arXiv2025

Optimal Inference Schedules for Masked Diffusion Models

A major bottleneck of standard auto-regressive large language models is that their inference process is inherently sequential, resulting in very long and costly inference times. To circumvent this, practitioners proposed a class of language models called diffusion language models, of which the masked diffusion model (MDM) is the most successful. The MDM is able to sample tokens out-of-order and, ostensibly, many tokens at once and in parallel. However, there is very limited rigorous understanding of how much parallel sampling these models can perform without noticeable degradation in their sampling performance. Prior work of Li and Cai obtained some preliminary bounds, but these are not tight for many natural classes of distributions. In this work, we give a new, exact characterization of the expected divergence between the true distribution and the sampled distribution, for any distribution and any unmasking schedule for the sampler, showing an elegant connection to the theory of univariate function approximation. By leveraging this connection, we then attain a number of novel lower and upper bounds for this problem. While the connection to function approximation in principle gives the optimal unmasking schedule for any distribution, we show that it is in general impossible to compete with it without strong a priori knowledge of the distribution, even in seemingly benign settings. However, we also demonstrate new upper bounds and new sampling schedules in terms of well-studied information-theoretic properties of the base distribution, namely, its total correlation and dual total correlation, which show that in some natural settings, one can sample in $O(log n)$ steps without any visible loss in performance, where $n$ is the total sequence length.

cs.LG↗

arXiv2025

Sublinear iterations can suffice even for DDPMs

SDE-based methods such as denoising diffusion probabilistic models (DDPMs) have shown remarkable success in real-world sample generation tasks. Prior analyses of DDPMs have been focused on the exponential Euler discretization, showing guarantees that generally depend at least linearly on the dimension or initial Fisher information. Inspired by works in log-concave sampling (Shen and Lee, 2019), we analyze an integrator -- the denoising diffusion randomized midpoint method (DDRaM) -- that leverages an additional randomized midpoint to better approximate the SDE. Using a recently-developed analytic framework called the "shifted composition rule", we show that this algorithm enjoys favorable discretization properties under appropriate smoothness assumptions, with sublinear $\widetilde{O}(\sqrt{d})$ score evaluations needed to ensure convergence. This is the first sublinear complexity bound for pure DDPM sampling -- prior works which obtained such bounds worked instead with ODE-based sampling and had to make modifications to the sampler which deviate from how they are used in practice. We also provide experimental validation of the advantages of our method, showing that it performs well in practice with pre-trained image synthesis models.

cs.LG↗

arXiv2025

Quantum Probe Tomography

Characterizing quantum many-body systems is a fundamental problem across physics, chemistry, and materials science. While significant progress has been made, many existing Hamiltonian learning protocols demand digital quantum control over the entire system, creating a disconnect from many real-world settings that provide access only through small, local probes. Motivated by this, we introduce and formalize the problem of quantum probe tomography, where one seeks to learn the parameters of a many-body Hamiltonian using a single local probe access to a small subsystem of a many-body thermal state undergoing time evolution. We address the identifiability problem of determining which Hamiltonians can be distinguished from probe data through a new combination of tools from algebraic geometry and smoothed analysis. Using this approach, we prove that generic Hamiltonians in various physically natural families are identifiable up to simple, unavoidable structural symmetries. Building on these insights, we design the first efficient end-to-end algorithm for probe tomography that learns Hamiltonian parameters to accuracy $\varepsilon$, with query complexity scaling polynomially in $1/\varepsilon$ and classical post-processing time scaling polylogarithmically in $1/\varepsilon$. In particular, we demonstrate that translation- and rotation-invariant nearest-neighbor Hamiltonians on square lattices in one, two, and three dimensions can be efficiently reconstructed from single-site probes of the Gibbs state, up to inversion symmetry about the probed site. Our results demonstrate that robust Hamiltonian learning remains achievable even under severely constrained experimental access.

quant-ph↗

arXiv2025

Selective Underfitting in Diffusion Models

Diffusion models have emerged as the principal paradigm for generative modeling across various domains. During training, they learn the score function, which in turn is used to generate samples at inference. They raise a basic yet unsolved question: which score do they actually learn? In principle, a diffusion model that matches the empirical score in the entire data space would simply reproduce the training data, failing to generate novel samples. Recent work addresses this question by arguing that diffusion models underfit the empirical score due to training-time inductive biases. In this work, we refine this perspective, introducing the notion of selective underfitting: instead of underfitting the score everywhere, better diffusion models more accurately approximate the score in certain regions of input space, while underfitting it in others. We characterize these regions and design empirical interventions to validate our perspective. Our results establish that selective underfitting is essential for understanding diffusion models, yielding new, testable insights into their generalization and generative performance.

cs.LG↗

arXiv2025

Efficient Pauli channel estimation with logarithmic quantum memory

Here we revisit one of the prototypical tasks for characterizing the structure of noise in quantum devices: estimating every eigenvalue of an $n$-qubit Pauli noise channel to error $ε$. Prior work [14] proved no-go theorems for this task in the practical regime where one has a limited amount of quantum memory, e.g. any protocol with $\le 0.99n$ ancilla qubits of quantum memory must make exponentially many measurements, provided it is non-concatenating. Such protocols can only interact with the channel by repeatedly preparing a state, passing it through the channel, and measuring immediately afterward. This left open a natural question: does the lower bound hold even for general protocols, i.e. ones which chain together many queries to the channel, interleaved with arbitrary data-processing channels, before measuring? Surprisingly, in this work we show the opposite: there is a protocol that can estimate the eigenvalues of a Pauli channel to error $ε$ using only $O(\log n/ε^2)$ ancilla and $\tilde{O}(n^2/ε^2)$ measurements. In contrast, we show that any protocol with zero ancilla, even a concatenating one, must make $Ω(2^n/ε^2)$ measurements, which is tight. Our results imply, to our knowledge, the first quantum learning task where logarithmically many qubits of quantum memory suffice for an exponential statistical advantage. Our protocol can be naturally extended to a protocol that learns the eigenvalues of Pauli terms within any subset $A$ of a Pauli channel with $O(\log\log(|A|)/ε^2)$ ancilla and $\tilde{O}(n^2/ε^2)$ measurements.

quant-ph↗

arXiv2025

Any-Order Flexible Length Masked Diffusion

Masked diffusion models (MDMs) have recently emerged as a promising alternative to autoregressive models over discrete domains. MDMs generate sequences in an any-order, parallel fashion, enabling fast inference and strong performance on non-causal tasks. However, a crucial limitation is that they do not support token insertions and are thus limited to fixed-length generations. To this end, we introduce Flexible Masked Diffusion Models (FlexMDMs), a discrete diffusion paradigm that simultaneously can model sequences of flexible length while provably retaining MDMs' flexibility of any-order inference. Grounded in an extension of the stochastic interpolant framework, FlexMDMs generate sequences by inserting mask tokens and unmasking them. Empirically, we show that FlexMDMs match MDMs in perplexity while modeling length statistics with much higher fidelity. On a synthetic maze planning task, they achieve $\approx 60 \%$ higher success rate than MDM baselines. Finally, we show pretrained MDMs can easily be retrofitted into FlexMDMs: on 16 H100s, it takes only three days to fine-tune LLaDA-8B into a FlexMDM, achieving superior performance on math (GSM8K, $58\% \to 67\%$) and code infilling performance ($52\% \to 65\%$).

cs.LG↗

arXiv2025

Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions

In recent years, masked diffusion models (MDMs) have emerged as a promising alternative approach for generative modeling over discrete domains. Compared to autoregressive models (ARMs), MDMs trade off complexity at training time with flexibility at inference time. At training time, they must learn to solve an exponentially large number of infilling problems, but at inference time, they can decode tokens in essentially arbitrary order. In this work, we closely examine these two competing effects. On the training front, we theoretically and empirically demonstrate that MDMs indeed train on computationally intractable subproblems compared to their autoregressive counterparts. On the inference front, we show that a suitable strategy for adaptively choosing the token decoding order significantly enhances the capabilities of MDMs, allowing them to sidestep hard subproblems. On logic puzzles like Sudoku, we show that adaptive inference can boost solving accuracy in pretrained MDMs from $<7$% to $\approx 90$%, even outperforming ARMs with $7\times$ as many parameters and that were explicitly trained via teacher forcing to learn the right order of decoding.

cs.LG↗

arXiv2025

Information-Computation Gaps in Quantum Learning via Low-Degree Likelihood

In a variety of physically relevant settings for learning from quantum data, designing protocols that can computationally efficiently extract information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the "standard" variant of Learning Stabilizers with Noise and for agnostically learning product states.

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