arXiv subjects

Ya-Ping Hsieh

Publications and source records attributed to Ya-Ping Hsieh.

At least 19 recordsLinked to original sources

arXiv2026

Support Before Frequency in Discrete Diffusion

Discrete diffusion models are increasingly competitive for language modeling, yet it remains unclear how their denoising objectives organize learning. Although these objectives target the full data distribution, we show that the exact reverse process induces a hierarchy between coarse support information and finer frequency information. For uniform and absorbing (a.k.a. masking) diffusion, we prove that, in the small-noise regime of the final denoising steps, each single-token reverse edit decomposes into a leading scale, determined by whether it moves toward the data support (e.g., grammatically valid sentences), and a finer coefficient, determining relative probabilities within the same scale. Thus, recovering validity structure only requires learning the correct order of magnitude of reverse probabilities, whereas recovering data frequencies requires coefficient-level estimation. The separation is mechanism-dependent: uniform diffusion exhibits a trichotomy into validity-improving, validity-preserving, and validity-worsening edits, while absorbing diffusion places its leading-order mass on validity-improving moves. Experiments on a masked language diffusion model and synthetic regular-language tasks support these predictions: support-localization emerges earlier than within-support frequency ranking, and the contrast between uniform and absorbing diffusion matches the predicted rate separation. Together, our results suggest that discrete diffusion models learn data support before data frequencies.

cs.LG↗

arXiv2026

Manifold Generalization Provably Proceeds Memorization in Diffusion Models

Diffusion models often generate novel samples even when the learned score is only \emph{coarse} -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the \emph{manifold hypothesis}, this behavior can instead be explained by coarse scores capturing the \emph{geometry} of the data while discarding the fine-scale distributional structure of the population measure~$\mu_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $\mu_{\scriptscriptstyle\mathrm{data}}$ supported on a $k$-dimensional manifold is known to require the classical minimax rate $\tilde{\mathcal{O}}(N^{-1/k})$, we prove that diffusion models trained with coarse scores can exploit the \emph{regularity of the manifold support} and attain a near-parametric rate toward a \emph{different} target distribution. This target distribution has density uniformly comparable to that of~$\mu_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-\beta/(4k)}\bigr)$-neighborhood of the manifold, where $\beta$ denotes the manifold regularity. Our guarantees therefore depend only on the smoothness of the underlying support, and are especially favorable when the data density itself is irregular, for instance non-differentiable. In particular, when the manifold is sufficiently smooth, we obtain that \emph{generalization} -- formalized as the ability to generate novel, high-fidelity samples -- occurs at a statistical rate strictly faster than that required to estimate the full population distribution~$\mu_{\scriptscriptstyle\mathrm{data}}$.

cs.LG↗

arXiv2026

Verifier-Constrained Flow Expansion for Discovery Beyond the Data

Flow and diffusion models are typically pre-trained on limited available data (e.g., molecular samples), covering only a fraction of the valid design space (e.g., the full molecular space). As a consequence, they tend to generate samples from only a narrow portion of the feasible domain. This is a fundamental limitation for scientific discovery applications, where one typically aims to sample valid designs beyond the available data distribution. To this end, we address the challenge of leveraging access to a verifier (e.g., an atomic bonds checker), to adapt a pre-trained flow model so that its induced density expands beyond regions of high data availability, while preserving samples validity. We introduce formal notions of strong and weak verifiers and propose algorithmic frameworks for global and local flow expansion via probability-space optimization. Then, we present Flow Expander (FE), a scalable mirror descent scheme that provably tackles both problems by verifier-constrained entropy maximization over the flow process noised state space. Next, we provide a thorough theoretical analysis of the proposed method, and state convergence guarantees under both idealized and general assumptions. Ultimately, we empirically evaluate our method on both illustrative, yet visually interpretable settings, and on a molecular design task showcasing the ability of FE to expand a pre-trained flow model increasing conformer diversity while preserving validity.

cs.LG↗

arXiv2026

A Unified Density Operator View of Flow Control and Merging

Recent progress in large-scale flow and diffusion models raised two fundamental algorithmic challenges: (i) control-based reward adaptation of pre-trained flows, and (ii) integration of multiple models, i.e., flow merging. While current approaches address them separately, we introduce a unifying probability-space framework that subsumes both as limit cases, and enables reward-guided flow merging, allowing principled, task-aware combination of multiple pre-trained flows (e.g., merging priors while maximizing drug-discovery utilities). Our formulation renders possible to express a rich family of operators over generative models densities, including intersection (e.g., to enforce safety), union (e.g., to compose diverse models), interpolation (e.g., for discovery), their reward-guided counterparts, as well as complex logical expressions via generative circuits. Next, we introduce Reward-Guided Flow Merging (RFM), a mirror-descent scheme that reduces reward-guided flow merging to a sequence of standard fine-tuning problems. Then, we provide first-of-their-kind theoretical guarantees for reward-guided and pure flow merging via RFM. Ultimately, we showcase the capabilities of the proposed method on illustrative settings providing visually interpretable insights, and apply our method to high-dimensional de-novo molecular design and low-energy conformer generation.

cs.LG↗

arXiv2025

Flow Density Control: Generative Optimization Beyond Entropy-Regularized Fine-Tuning

Adapting large-scale foundation flow and diffusion generative models to optimize task-specific objectives while preserving prior information is crucial for real-world applications such as molecular design, protein docking, and creative image generation. Existing principled fine-tuning methods aim to maximize the expected reward of generated samples, while retaining knowledge from the pre-trained model via KL-divergence regularization. In this work, we tackle the significantly more general problem of optimizing general utilities beyond average rewards, including risk-averse and novelty-seeking reward maximization, diversity measures for exploration, and experiment design objectives among others. Likewise, we consider more general ways to preserve prior information beyond KL-divergence, such as optimal transport distances and Renyi divergences. To this end, we introduce Flow Density Control (FDC), a simple algorithm that reduces this complex problem to a specific sequence of simpler fine-tuning tasks, each solvable via scalable established methods. We derive convergence guarantees for the proposed scheme under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we validate our method on illustrative settings, text-to-image, and molecular design tasks, showing that it can steer pre-trained generative models to optimize objectives and solve practically relevant tasks beyond the reach of current fine-tuning schemes.

cs.LG↗

arXiv2025

When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis

Score-based methods, such as diffusion models and Bayesian inverse problems, are often interpreted as learning the data distribution in the low-noise limit ($\sigma \to 0$). In this work, we propose an alternative perspective: their success arises from implicitly learning the data manifold rather than the full distribution. Our claim is based on a novel analysis of scores in the small-$\sigma$ regime that reveals a sharp separation of scales: information about the data manifold is $\Theta(\sigma^{-2})$ stronger than information about the distribution. We argue that this insight suggests a paradigm shift from the less practical goal of distributional learning to the more attainable task of geometric learning, which provably tolerates $O(\sigma^{-2})$ larger errors in score approximation. We illustrate this perspective through three consequences: i) in diffusion models, concentration on data support can be achieved with a score error of $o(\sigma^{-2})$, whereas recovering the specific data distribution requires a much stricter $o(1)$ error; ii) more surprisingly, learning the uniform distribution on the manifold-an especially structured and useful object-is also $O(\sigma^{-2})$ easier; and iii) in Bayesian inverse problems, the maximum entropy prior is $O(\sigma^{-2})$ more robust to score errors than generic priors. Finally, we validate our theoretical findings with preliminary experiments on large-scale models, including Stable Diffusion.

stat.ML↗

arXiv2025

Provable Maximum Entropy Manifold Exploration via Diffusion Models

Exploration is critical for solving real-world decision-making problems such as scientific discovery, where the objective is to generate truly novel designs rather than mimic existing data distributions. In this work, we address the challenge of leveraging the representational power of generative models for exploration without relying on explicit uncertainty quantification. We introduce a novel framework that casts exploration as entropy maximization over the approximate data manifold implicitly defined by a pre-trained diffusion model. Then, we present a novel principle for exploration based on density estimation, a problem well-known to be challenging in practice. To overcome this issue and render this method truly scalable, we leverage a fundamental connection between the entropy of the density induced by a diffusion model and its score function. Building on this, we develop an algorithm based on mirror descent that solves the exploration problem as sequential fine-tuning of a pre-trained diffusion model. We prove its convergence to the optimal exploratory diffusion model under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we empirically evaluate our approach on both synthetic and high-dimensional text-to-image diffusion, demonstrating promising results.

cs.LG↗

arXiv2024

A Dynamical System View of Langevin-Based Non-Convex Sampling

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important challenges: Existing guarantees (1) typically only hold for the averaged iterates rather than the more desirable last iterates, (2) lack convergence metrics that capture the scales of the variables such as Wasserstein distances, and (3) mainly apply to elementary schemes such as stochastic gradient Langevin dynamics. In this paper, we develop a new framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as proximal, randomized mid-point, and Runge-Kutta integrators. Beyond existing methods, our framework also motivates more efficient schemes that enjoy the same rigorous guarantees.

cs.LG↗

arXiv2024

Aligned Diffusion Schrödinger Bridges

Diffusion Schrödinger bridges (DSB) have recently emerged as a powerful framework for recovering stochastic dynamics via their marginal observations at different time points. Despite numerous successful applications, existing algorithms for solving DSBs have so far failed to utilize the structure of aligned data, which naturally arises in many biological phenomena. In this paper, we propose a novel algorithmic framework that, for the first time, solves DSBs while respecting the data alignment. Our approach hinges on a combination of two decades-old ideas: The classical Schrödinger bridge theory and Doob's $h$-transform. Compared to prior methods, our approach leads to a simpler training procedure with lower variance, which we further augment with principled regularization schemes. This ultimately leads to sizeable improvements across experiments on synthetic and real data, including the tasks of predicting conformational changes in proteins and temporal evolution of cellular differentiation processes.

cs.LG↗

arXiv2023

Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm

Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics not only generalize but also offer a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of (Deb et al. 2023) or the "mean-field Schrödinger equation" of (Claisse et al. 2023).

cs.LG↗

arXiv2023

Riemannian stochastic optimization methods avoid strict saddle points

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, and are typically solved with a Riemannian stochastic gradient method (or some variant thereof). However, in many cases of interest, the resulting minimization problem is not geodesically convex, so the convergence of the chosen solver to a desirable solution - i.e., a local minimizer - is by no means guaranteed. In this paper, we study precisely this question, that is, whether stochastic Riemannian optimization algorithms are guaranteed to avoid saddle points with probability 1. For generality, we study a family of retraction-based methods which, in addition to having a potentially much lower per-iteration cost relative to Riemannian gradient descent, include other widely used algorithms, such as natural policy gradient methods and mirror descent in ordinary convex spaces. In this general setting, we show that, under mild assumptions for the ambient manifold and the oracle providing gradient information, the policies under study avoid strict saddle points / submanifolds with probability 1, from any initial condition. This result provides an important sanity check for the use of gradient methods on manifolds as it shows that, almost always, the limit state of a stochastic Riemannian algorithm can only be a local minimizer.

math.OC↗

arXiv2023

Ultrahigh Photoresponsivity of Gold Nanodisk Array/CVD MoS$_2$-based Hybrid Phototransistor

Owing to its atomically thin thickness, layer-dependent tunable band gap, flexibility, and CMOS compatibility, MoS$_2$ is a promising candidate for photodetection. However, mono-layer MoS2-based photodetectors typically show poor optoelectronic performances, mainly limited by their low optical absorption. In this work, we hybridized CVD-grown monolayer MoS$_2$ with a gold nanodisk (AuND) array to demonstrate a superior visible photodetector through a synergetic effect. It is evident from our experimental results that there is a strong light-matter interaction between AuNDs and monolayer MoS$_2$, which results in better photodetection due to a surface trap state passivation with a longer charge carrier lifetime compared to pristine MoS$_2$. In particular, the AuND/MoS$_2$ system demonstrated a photoresponsivity of $8.7 \times 10^{4}$ A/W, specific detectivity of $6.9 \times 10^{13}$ Jones, and gain $1.7 \times 10^{5}$ at $31.84 μW/cm^{2}$ illumination power density of 632 nm wavelength with an applied voltage of 4.0 V for an AuND/MoS$_2$-based photodetector. To our knowledge, these optoelectronic responses are one order higher than reported results for CVD MoS$_2$-based photodetector in the literature.

cond-mat.mtrl-sci↗

arXiv2023

A unified stochastic approximation framework for learning in games

We develop a flexible stochastic approximation framework for analyzing the long-run behavior of learning in games (both continuous and finite). The proposed analysis template incorporates a wide array of popular learning algorithms, including gradient-based methods, the exponential/multiplicative weights algorithm for learning in finite games, optimistic and bandit variants of the above, etc. In addition to providing an integrated view of these algorithms, our framework further allows us to obtain several new convergence results, both asymptotic and in finite time, in both continuous and finite games. Specifically, we provide a range of criteria for identifying classes of Nash equilibria and sets of action profiles that are attracting with high probability, and we also introduce the notion of coherence, a game-theoretic property that includes strict and sharp equilibria, and which leads to convergence in finite time. Importantly, our analysis applies to both oracle-based and bandit, payoff-based methods - that is, when players only observe their realized payoffs.

cs.GT↗

arXiv2023

Unbalanced Diffusion Schrödinger Bridge

Schrödinger bridges (SBs) provide an elegant framework for modeling the temporal evolution of populations in physical, chemical, or biological systems. Such natural processes are commonly subject to changes in population size over time due to the emergence of new species or birth and death events. However, existing neural parameterizations of SBs such as diffusion Schrödinger bridges (DSBs) are restricted to settings in which the endpoints of the stochastic process are both probability measures and assume conservation of mass constraints. To address this limitation, we introduce unbalanced DSBs which model the temporal evolution of marginals with arbitrary finite mass. This is achieved by deriving the time reversal of stochastic differential equations with killing and birth terms. We present two novel algorithmic schemes that comprise a scalable objective function for training unbalanced DSBs and provide a theoretical analysis alongside challenging applications on predicting heterogeneous molecular single-cell responses to various cancer drugs and simulating the emergence and spread of new viral variants.

cs.LG↗

arXiv2023

The Schrödinger Bridge between Gaussian Measures has a Closed Form

The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks to recover an optimal map, or more generally a coupling, to morph a Gaussian into another. It has been well studied and applied to a wide variety of tasks. Here we focus on the dynamic formulation of OT, also known as the Schrödinger bridge (SB) problem, which has recently seen a surge of interest in machine learning due to its connections with diffusion-based generative models. In contrast to the static setting, much less is known about the dynamic setting, even for Gaussian distributions. In this paper, we provide closed-form expressions for SBs between Gaussian measures. In contrast to the static Gaussian OT problem, which can be simply reduced to studying convex programs, our framework for solving SBs requires significantly more involved tools such as Riemannian geometry and generator theory. Notably, we establish that the solutions of SBs between Gaussian measures are themselves Gaussian processes with explicit mean and covariance kernels, and thus are readily amenable for many downstream applications such as generative modeling or interpolation. To demonstrate the utility, we devise a new method for modeling the evolution of single-cell genomics data and report significantly improved numerical stability compared to existing SB-based approaches.

cs.LG↗

arXiv2022

Riemannian stochastic approximation algorithms

We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport, but their behavior is much less understood compared to the Euclidean case because of the lack of a global linear structure on the manifold. We overcome this difficulty by introducing a suitable Fermi coordinate frame which allows us to map the asymptotic behavior of the Riemannian Robbins-Monro (RRM) algorithms under study to that of an associated deterministic dynamical system. In so doing, we provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes, despite the significant complications that arise due to the curvature and topology of the underlying manifold. We showcase the flexibility of the proposed framework by applying it to a range of retraction-based variants of the popular optimistic / extra-gradient methods for solving minimization problems and games, and we provide a unified treatment for their convergence.

math.OC↗

arXiv2022

Continuous-time Analysis for Variational Inequalities: An Overview and Desiderata

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.

stat.ML↗

arXiv2021

The limits of min-max optimization algorithms: convergence to spurious non-critical sets

Compared to ordinary function minimization problems, min-max optimization algorithms encounter far greater challenges because of the existence of periodic cycles and similar phenomena. Even though some of these behaviors can be overcome in the convex-concave regime, the general case is considerably more difficult. On that account, we take an in-depth look at a comprehensive class of state-of-the art algorithms and prevalent heuristics in non-convex / non-concave problems, and we establish the following general results: a) generically, the algorithms' limit points are contained in the ICT sets of a common, mean-field system; b) the attractors of this system also attract the algorithms in question with arbitrarily high probability; and c) all algorithms avoid the system's unstable sets with probability 1. On the surface, this provides a highly optimistic outlook for min-max algorithms; however, we show that there exist spurious attractors that do not contain any stationary points of the problem under study. In this regard, our work suggests that existing min-max algorithms may be subject to inescapable convergence failures. We complement our theoretical analysis by illustrating such attractors in simple, two-dimensional, almost bilinear problems.

math.OC↗