Showing 1–12 of 12 results for author: Azangulov, I

Masked Language Flow Models

Authors: Iskander Azangulov, Kianoosh Ashouritaklimi, Leo Zhang, Simon Vary, Patrick Rebeschini

Abstract: Masked Diffusion Models (MDMs) promise fast, parallel language generation, but their reverse transition factorises across token positions -- an approximation that breaks down in the few-step sampling regime where parallel generation ought to provide the greatest efficiency gains. Flow Language Models (FLMs) sidestep this limitation by learning a continuous flow that transports noise toward clean s… ▽ More Masked Diffusion Models (MDMs) promise fast, parallel language generation, but their reverse transition factorises across token positions -- an approximation that breaks down in the few-step sampling regime where parallel generation ought to provide the greatest efficiency gains. Flow Language Models (FLMs) sidestep this limitation by learning a continuous flow that transports noise toward clean sequences represented in Euclidean space, inducing a flow map that can be distilled for single-step generation. However, this makes complex tasks requiring multi-step reasoning problematic for FLMs, as FLMs are forced to decode every token during generation. To address this, we introduce Masked Language Flow Models (MLFMs), which incorporate masking into FLMs using a continuous stochastic interpolant to bridge partially masked and clean sequences. This design enables conditional generation via continuous flows and allows pretrained MDMs to be converted into MLFMs through a simple, lightweight adaptation. Leveraging this flexibility, we propose a novel sampler that alternates continuous denoising with the discrete unmasking of confident tokens to better support multi-step reasoning. We evaluate our approach on GSM8K and MT-Bench and find, for the first time, that flow-based language models can be scaled to solve downstream reasoning and instruction-following tasks. △ Less

Submitted 25 June, 2026; originally announced June 2026.

Comments: Preprint

Variance-Tilted Diffusion Models for Diverse Sampling

Authors: Iskander Azangulov, Leo Zhang, Kianoosh Ashouritaklimi

Abstract: Diffusion models are typically sampled independently, even when the downstream objective is to obtain a diverse set of candidates. We introduce a variance-weighted batch distribution that favours collections of samples with large empirical spread after a prescribed linear feature map. The target is specified explicitly, and the sampler is derived as the corresponding Doob $h$-transform of independ… ▽ More Diffusion models are typically sampled independently, even when the downstream objective is to obtain a diverse set of candidates. We introduce a variance-weighted batch distribution that favours collections of samples with large empirical spread after a prescribed linear feature map. The target is specified explicitly, and the sampler is derived as the corresponding Doob $h$-transform of independent diffusion dynamics. The resulting correction has a compact form: an interaction term that repels posterior denoised means, together with a curvature term that moves particles to the region of higher feature variance. This yields an interacting-particle sampler with a transparent probabilistic target rather than a heuristic repulsive drift. △ Less

Submitted 20 June, 2026; originally announced June 2026.

Comments: Accepted at SPIGM @ ICML workshop 2026

Exact Posterior Score Estimation for Solving Linear Inverse Problems

Authors: Abbas Mammadov, Ozgur Kara, Kaan Oktay, Iskander Azangulov, Adil Kaan Akan, Hyungjin Chung, James Matthew Rehg, Yee Whye Teh

Abstract: Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching correc… ▽ More Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers. △ Less

Submitted 15 June, 2026; originally announced June 2026.

Sampling from Flow Language Models via Marginal-Conditioned Bridges

Authors: Iskander Azangulov, Leo Zhang

Abstract: Flow Language Models (FLMs) are a recently introduced class of language models which adapt continuous flow matching for one-hot encoded token sequences. Their denoisers have a special structure absent from generic continuous diffusion models: each block of the denoising mean is a posterior marginal distribution over the clean token at that position. Standard DDPM-style samplers collapse these marg… ▽ More Flow Language Models (FLMs) are a recently introduced class of language models which adapt continuous flow matching for one-hot encoded token sequences. Their denoisers have a special structure absent from generic continuous diffusion models: each block of the denoising mean is a posterior marginal distribution over the clean token at that position. Standard DDPM-style samplers collapse these marginals to a single conditional-mean endpoint and bridge toward this simplex-valued point, which is generally not a valid one-hot sequence. We argue that the natural sampler for an FLM is instead posterior-predictive. At each reverse step, we sample a clean one-hot endpoint from the factorized posterior defined by the FLM token marginals, and then sample the next continuous state from the analytic Ornstein--Uhlenbeck bridge conditioned on that endpoint. The method is training-free, uses the same model evaluations as standard sampling, and gives a principled interface for token-level decoding controls such as temperature scaling and nucleus truncation. We show that, under exact posterior marginals, the endpoint approximation error is exactly the conditional multi-information among token positions. The induced one-step bridge kernel preserves all token-wise posterior-predictive marginals and loses only the residual cross-position dependence. Finally, we prove a Girsanov path-space comparison showing that the marginal-conditioned bridge has a no-larger denoising-error term than the frozen conditional-mean bridge, with strict improvement whenever intermediate coordinate-wise bridge observations reveal additional information about the clean token. Experiments with FLMs show that the sampler improves the quality--diversity tradeoff. Code is available at: github.com/imbirik/mcb. △ Less

Submitted 13 May, 2026; originally announced May 2026.

Parallel Sampling from Masked Diffusion Models via Conditional Independence Testing

Authors: Iskander Azangulov, Teodora Pandeva, Niranjani Prasad, Javier Zazo, Sushrut Karmalkar

Abstract: Masked diffusion models (MDMs) offer a compelling alternative to autoregressive models (ARMs) for discrete text generation because they enable parallel token sampling, rather than sequential, left-to-right generation. This means potentially much faster inference. However, effective parallel sampling faces two competing requirements: (i) simultaneously updated tokens must be conditionally independe… ▽ More Masked diffusion models (MDMs) offer a compelling alternative to autoregressive models (ARMs) for discrete text generation because they enable parallel token sampling, rather than sequential, left-to-right generation. This means potentially much faster inference. However, effective parallel sampling faces two competing requirements: (i) simultaneously updated tokens must be conditionally independent, and (ii) updates should prioritise high-confidence predictions. These goals conflict because high-confidence predictions often cluster and depend on each other, opportunities for parallel updates. We present PUNT, a model-agnostic sampler that reconciles this trade-off. Our method identifies token dependencies and removes lower-confidence tokens from conflicting groups. This produces sets of indices for unmasking that satisfy both independence and confidence criteria. Our approach ensures improved parallel unmasking through approximate conditional independence testing. Our experiments show that PUNT delivers a superior trade-off between accuracy and compute when compared to other strong training-free baselines, especially for generation of longer sequences. On the IFEval benchmark, it achieves up to 16\% higher accuracy over baseline methods, including sequential generation (one-by-one). These gains hold across different values of hyperparameters, mitigating the need for brittle hyperparameter tuning. Moreover, we observe that PUNT induces an emergent hierarchical generation strategy, where the model first establishes high-level paragraph structure before local refinement, suggesting a planning-like generation process that contributes to strong alignment performance. △ Less

Submitted 24 October, 2025; originally announced October 2025.

Adaptive Diffusion Guidance via Stochastic Optimal Control

Authors: Iskander Azangulov, Peter Potaptchik, Qinyu Li, Eddie Aamari, George Deligiannidis, Judith Rousseau

Abstract: Guidance is a cornerstone of modern diffusion models, playing a pivotal role in conditional generation and enhancing the quality of unconditional samples. However, current approaches to guidance scheduling--determining the appropriate guidance weight--are largely heuristic and lack a solid theoretical foundation. This work addresses these limitations on two fronts. First, we provide a theoretical… ▽ More Guidance is a cornerstone of modern diffusion models, playing a pivotal role in conditional generation and enhancing the quality of unconditional samples. However, current approaches to guidance scheduling--determining the appropriate guidance weight--are largely heuristic and lack a solid theoretical foundation. This work addresses these limitations on two fronts. First, we provide a theoretical formalization that precisely characterizes the relationship between guidance strength and classifier confidence. Second, building on this insight, we introduce a stochastic optimal control framework that casts guidance scheduling as an adaptive optimization problem. In this formulation, guidance strength is not fixed but dynamically selected based on time, the current sample, and the conditioning class, either independently or in combination. By solving the resulting control problem, we establish a principled foundation for more effective guidance in diffusion models. △ Less

Submitted 31 March, 2026; v1 submitted 25 May, 2025; originally announced May 2025.

Comments: AISTATS 2026

Linear Convergence of Diffusion Models Under the Manifold Hypothesis

Authors: Peter Potaptchik, Iskander Azangulov, George Deligiannidis

Abstract: Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into $D$-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in $D$ or polynomial (superline… ▽ More Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into $D$-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in $D$ or polynomial (superlinear) in $d$. The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension $d$. Moreover, we show that this linear dependency is sharp. △ Less

Submitted 23 April, 2025; v1 submitted 11 October, 2024; originally announced October 2024.

Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

Authors: Iskander Azangulov, George Deligiannidis, Judith Rousseau

Abstract: Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more applications in science and beyond. The \textit{manifold hypothesis} states that high-dimensional data often lie on lower-dimensional manifolds within the ambien… ▽ More Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more applications in science and beyond. The \textit{manifold hypothesis} states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of score learning. In terms of sampling complexity, we obtain rates independent of the ambient dimension w.r.t.\ the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes. △ Less

Submitted 6 August, 2026; v1 submitted 27 September, 2024; originally announced September 2024.

The GeometricKernels Package: Heat and Matérn Kernels for Geometric Learning on Manifolds, Meshes, and Graphs

Authors: Peter Mostowsky, Vincent Dutordoir, Iskander Azangulov, Noémie Jaquier, Michael John Hutchinson, Aditya Ravuri, Leonel Rozo, Alexander Terenin, Viacheslav Borovitskiy

Abstract: Kernels are a fundamental technical primitive in machine learning. In recent years, kernel-based methods such as Gaussian processes are becoming increasingly important in applications where quantifying uncertainty is of key interest. In settings that involve structured data defined on graphs, meshes, manifolds, or other related spaces, defining kernels with good uncertainty-quantification behavior… ▽ More Kernels are a fundamental technical primitive in machine learning. In recent years, kernel-based methods such as Gaussian processes are becoming increasingly important in applications where quantifying uncertainty is of key interest. In settings that involve structured data defined on graphs, meshes, manifolds, or other related spaces, defining kernels with good uncertainty-quantification behavior, and computing their value numerically, is less straightforward than in the Euclidean setting. To address this difficulty, we present GeometricKernels, a Python software package which implements the geometric analogs of classical Euclidean squared exponential - also known as heat - and Matérn kernels, which are widely-used in settings where uncertainty is of key interest. As a byproduct, we obtain the ability to compute Fourier-feature-type expansions, which are widely used in their own right, on a wide set of geometric spaces. Our implementation supports automatic differentiation in every major current framework simultaneously via a backend-agnostic design. In this companion paper to the package and its documentation, we outline the capabilities of the package and present an illustrated example of its interface. We also include a brief overview of the theory the package is built upon and provide some historic context in the appendix. △ Less

Submitted 2 March, 2026; v1 submitted 10 July, 2024; originally announced July 2024.

Journal ref: Journal of Machine Learning Research, 2025

Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces

Authors: Iskander Azangulov, Andrei Smolensky, Alexander Terenin, Viacheslav Borovitskiy

Abstract: Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most f… ▽ More Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners. △ Less

Submitted 12 September, 2024; v1 submitted 30 January, 2023; originally announced January 2023.

Journal ref: Journal of Machine Learning Research, 2024

Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case

Authors: Iskander Azangulov, Andrei Smolensky, Alexander Terenin, Viacheslav Borovitskiy

Abstract: Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most f… ▽ More Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners. △ Less

Submitted 27 February, 2026; v1 submitted 31 August, 2022; originally announced August 2022.

Comments: This version fixes two mathematical typos, in equations (58) and (65), where both sums should be taken only over the diagonal part $π^{(λ)}_{jj}$ and not over $π^{(λ)}_{jk}$ as had erroneously been written in the previous version. The proofs for both statements remain unchanged. We thank Nathaël Da Costa for making us aware of this pair of typos

Journal ref: Journal of Machine Learning Research, 2024

Matérn Gaussian Processes on Graphs

Authors: Viacheslav Borovitskiy, Iskander Azangulov, Alexander Terenin, Peter Mostowsky, Marc Peter Deisenroth, Nicolas Durrande

Abstract: Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic… ▽ More Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Matérn Gaussian processes - a widely-used model class in the Euclidean setting - to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Matérn Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks. △ Less

Submitted 15 May, 2026; v1 submitted 29 October, 2020; originally announced October 2020.

Journal ref: Artificial Intelligence and Statistics, 2021