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Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that \(\widetilde{O}(k/\varepsilon)\) iterations suffice to generate an \(\varepsilon\)-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

Diffusion and flow-based generative models produce strong images, yet their controllability remains largely endpoint-centric: users specify conditions and receive final outputs, while the intermediate generative dynamics remain hidden. Recent methods have begun to exploit generation order and process decomposition to improve sample quality, but still treat intermediate states as internal computation rather than objects for interaction. We propose Trajectory Forcing (TF), a trajectory-centric framework that makes the generation path explicit, semantic, and editable. TF organizes synthesis as a sequence of semantically structured stages, progressing from global layout to object-, part-, and detail-level representations. Each stage produces a decodable latent state that can be inspected, evaluated, and locally edited before the next stage begins. To instantiate this path, we derive coarse-to-fine teacher hierarchies by clustering pretrained visual representations such as DINOv2, and train a hierarchy-conditioned one-step flow-matching model at each level. We further introduce trajectory-aware metrics that measure structural consistency and local controllability beyond endpoint quality metrics such as FID. Experiments show that TF achieves competitive sample quality while exposing coherent intermediate states and supporting localized edits across semantic levels. By shifting the focus from final images to the generative path itself, TF opens a route toward controllable, trajectory-aware image synthesis.

Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FAPS), to our knowledge the first function-space posterior sampling framework that unifies stochastic-process regression and PDE inverse problems. Built on pretrained function-space flow-matching priors, FAPS enables likelihood-guided posterior inference from sparse and noisy observations, supports variable query discretizations, and avoids explicit prior-density evaluation. Its Langevin correction uses a low-rank covariance preconditioner to exploit dominant function-space correlations across discretizations. Across Gaussian and non-Gaussian stochastic-process regression benchmarks and diverse PDE inverse problems, FAPS produces coherent posterior samples with accurate uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better PDE noisy inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.

Recent generative video compression methods leverage powerful generative priors to achieve perceptually pleasing reconstructions. However, most existing approaches require additional training to adapt generative models to produce realistic reconstructions from compact representations. In this paper, we propose ZeroGVC, a zero-shot generative video compression framework that leverages pretrained autoregressive diffusion priors for low-delay video reconstruction. ZeroGVC encodes the first frame of each group of pictures (GOP) with an image codec and represents subsequent P-frames through Codebook-Guided Autoregressive Latent Compression. This design is motivated by our observation that the compression scheme of denoising diffusion codebook models is effective in few-step consistency sampling. By selecting compact combinations of reproducible codebook noise vectors, ZeroGVC steers the latent denoising trajectory toward the target P-frame while allowing the decoder to reproduce the same trajectory in only a few denoising steps. In addition, we design an optional bidirectional reference mode that mitigates error propagation by leveraging the next I-frame context without introducing any additional bitrate overhead. Extensive experiments on standard video compression benchmarks demonstrate that ZeroGVC achieves superior perceptual reconstruction quality at ultra-low bitrates without any additional training.

Path-based attribution methods such as Integrated Gradients (IG) are widely adopted for their strong axiomatic properties and effectiveness in attributing model predictions to input features by integrating gradients along a path from a baseline to the input. However, the choice of the attribution path largely affects the quality of explanations, and existing approaches rely on fixed or hand-crafted paths that often produce noisy or distorted attributions. To address this limitation, we propose Diffusion Integrated Gradients (DiffIG), a novel method that reformulates path generation as a conditional generative modeling problem. DiffIG first trains a diffusion model to learn a distribution over paths generated from a Stick-Breaking Process, then employs guided sampling to embed user guidance during the sampling procedure. We demonstrate that DiffIG quantitatively matches or outperforms existing path-based methods, achieving perceptually aligned explanations. This work introduces a new generative perspective for flexible, inference-time controllable Explainable Artificial Intelligence (XAI) methods.

UV parameterization is a fundamental step in 3D content creation, yet producing production-ready UV layouts remains challenging due to the gap between geometric distortion objectives and the stylistic preferences of professional artists. While classical methods optimize handcrafted energy functions, artist-authored UVs exhibit structural patterns such as straightened seams, axis-aligned islands, and flexible interior deformation, properties that are difficult to explicitly formulate. In this work, we present DreamUV, an end-to-end learning framework that formulates UV unwrapping as a generative Flow Matching problem. Rather than predicting a single optimal parameterization, DreamUV learns a mesh-conditioned transport process that maps noise samples to a distribution of artist-like UV layouts. To reflect real-world authoring practices, we introduce a boundary-aware training strategy that prioritizes seam geometry, and a Model-in-the-Loop Finetuning(MITL) scheme that explicitly accounts for discretization errors during sampling and stabilizes transport dynamics under heterogeneous supervision. We evaluate DreamUV on a large-scale dataset of professionally authored UV layouts. Experiments demonstrate that our method produces significantly straighter boundaries and tighter axis-aligned islands than both classical and learning-based baselines, while maintaining competitive distortion metrics. Qualitative results and a user study with professional artists further confirm that DreamUV generates UV layouts that are not only valid, but aligned with practical production requirements.

Adam is one of the most widely implemented and influential modern optimizers. Why is it effective across different optimization problems in practice? This question arguably lies at the center of the optimization community over the last decade and has motivated a substantial body of work aimed at understanding its convergence behavior. However, existing studies have mainly focused on the convergence rate of Adam in smooth nonconvex optimization, which unfortunately does not adequately capture practical settings, since many real-world problems are nonsmooth, such as those arising in training neural networks. Thus, these studies cannot fully explain the popularity and empirical success of Adam. Recently, an insightful and powerful framework called Online-to-Nonconvex Conversion has opened a new way to analyze Adam for nonsmooth nonconvex optimization. Unfortunately, prior works along this line share two common limitations. First, all of them ignore the important bias-correction term in the original Adam algorithm. Second and more importantly, many of them require extra operations that are not used in Adam, such as a clipping step. Therefore, the convergence guarantee for the original Adam method still remains unclear. In this work, we present the first finite-time analysis for the classical form of Adam, i.e., with the bias-correction step and without further algorithmic modifications, and prove that a randomly scaled learning rate ensures a convergence rate of \(1/T^{\frac{2}{13}}\) for nonsmooth nonconvex optimization. Moreover, our result provably applies to the modern heavy-tailed noise regime, which is closer to practice. Interestingly, our theory is established under the parameter choice \(β_1=β_2\), aligning with the recent empirical studies.

Pre-trained foundation models have demonstrated remarkable success in many domains, enabling a unified backbone to generalize across diverse downstream tasks. However, extending this paradigm to graph learning remains challenging due to the intrinsic mismatch between graph data and fixed architectural designs. In this work, we show that this limitation can be overcome via recurrent graph models. To achieve this, we conduct a systematic theoretical analysis, rigorously deriving step dependence as a necessary and sufficient condition for an adaptively convergent recurrent process. Building on this foundation, we propose AdaR, an Adaptive Recurrent graph model, empowering flexible test-time computing on various downstream tasks without changing model parameters. To enable adaptive inference, AdaR explicitly encodes normalized step information and representation-target relations into the recurrent updates. To ensure convergence of the recurrent process, AdaR employs gradient-based supervision signals that guide representation updates throughout the recurrence. Empirical results demonstrate that AdaR consistently outperforms strong baselines in both inductive and transductive settings.

We propose NullFlow, a principled framework for one-step generative image reconstruction. Our key idea is to confine the generative flow to a measurement-consistent subspace. Because the flow never leaves this subspace, NullFlow needs no separate data-fidelity corrections, unlike existing solvers. NullFlow samples in a single network evaluation by learning the flow's average velocity, avoiding the step-by-step integration of traditional flow matching methods. We prove that the average velocity of this constrained flow yields a training objective whose global minimizer is a one-step posterior sampler. We show on image inpainting that NullFlow matches state-of-the-art diffusion solvers while cutting inference from hundreds of network evaluations to one.

We show that replacing the standard MSE denoising loss in diffusion models with a nonlinear transformation induced by an f-divergence yields a simple robust training surrogate that empirically improves performance under data contamination, with small additional computational overhead. The theoretical foundation rests on a local divergence construction: under the Gaussian reverse-kernel structure of DDPM, each per-step likelihood ratio follows a lognormal distribution parameterized by a scalar mismatch, so the conditional f-divergence at each step reduces to a one-dimensional function of the denoising error. Summing these local divergences yields a training objective that unifies diffusion training as divergence induced weighted denoising, where the derivative of the induced divergence acts as a residual-space influence weight that controls the contribution of each sample. Bounded-influence divergences (Hellinger, negative exponential) suppress large error samples, with Hellinger yielding an explicit exponential weight, connecting the framework to robust M-estimation. Empirically, on CIFAR-10 under 30% contamination, NED reduces FID from 93.0 (KL) to 77.5, while also outperforming standard robust losses such as Huber and clipped MSE.

While longitudinal brain PET imaging is the gold standard for quantifying the spatiotemporal accumulation of Beta-amyloid, its widespread clinical utility is constrained by high operational costs and cumulative radiation risks. Recent deep generative models show promise in longitudinal image synthesis; however, they often fail to capture subtle pathological progression due to identity drift and a persistent bias toward trivially replicating baseline signal intensities rather than modeling temporal transition. To this end, we propose Delta-Diffusion, a novel progression-aware framework that redefines longitudinal PET synthesis as a conditional Poisson Diffusion Bridge (PDB) process. Unlike standard diffusion models that start from Gaussian noise, our PDB formulation is mathematically anchored to the subject's baseline PET, effectively transforming the generative task into a conditional distribution transition of the amyloid trajectory. To handle heteroscedastic nature of PET imaging, we introduce a physically-grounded Poisson perturbation within a Diffusion Transformer (DiT). This architecture uses adaptive scale-shift modulation to precisely calibrate the synthesis with the elapsed clinical interval and structural MRI context. A volume-of-interest balanced objective is designed to emphasize sparse, high-risk regions of amyloid accumulation. Validated on two cohorts with 542 subjects, Delta-Diffusion demonstrates superior performance in capturing longitudinal variations in amyloid deposition compared to state-of-the-art methods, offering a robust computational framework for tracking disease progression.

Few-step distillation for video diffusion models has attracted significant attention, driven by the urgent demand for efficient deployment in real-world scenarios. However, Distribution Matching Distillation (DMD), a leading paradigm, tends to degrade under limited NFE budgets, manifesting in video generation as layout instability, oversaturation, and broken motion dynamics. We trace this failure to a structural limitation: standard DMD is an intra-sample distribution-matching objective with coordinate-wise gradients, and thus imposes no explicit constraint on the relational geometry across batch elements or temporal frames, leaving the underlying copula largely unregulated. Combined with the mode-seeking tendency of its reverse-KL objective, this absence of relational guidance makes DMD prone to collapsing into local optima in the few-step regime. Motivated by this insight, we propose Copula-aware DMD (CoDMD), a lightweight relational regularizer that reuses score estimates already produced by the frozen teacher and the online fake model to construct pairwise relation matrices across samples and frames. These are matched through a supplementary distributional objective that requires no additional networks, datasets, or sampling trajectories. On the Wan-2.1-T2V model series at 1.3B & 14B scales, CoDMD distills 50-step teachers into 4-step students, achieving an approximate 25\(\times\) speed-up while attaining VBench scores of 84.46 & 84.87, outperforming prior trajectory-based (rCM 82.81 & 84.05) and distribution-based (DMD 83.38 & 83.81) methods.

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.

Characterizing the complete wall-pressure spectrum in turbulent wall-bounded flows requires simultaneous access to the viscous-scale high-wavenumber content and the outer-layer low-wavenumber content -- a requirement that neither short-domain direct numerical simulation (DNS) nor sparse experimental measurements alone can satisfy. We propose Patched Flow Matching (Patched FM), a generative framework that fuses these two complementary sources by learning a patch-local prior over inner-scaled wall-pressure statistics from short-domain DNS and assimilating sparse sensor measurements at inference time through training-free posterior sampling. The patch-additive decomposition of the flow matching vector field decouples the generative prior from the global domain size, enabling reconstruction on domains arbitrarily larger than the training configuration. By expressing the patch prior in inner-scaled coordinates, where high-wavenumber wall-pressure statistics are approximately Reynolds-number invariant, the framework extends to higher Reynolds numbers through hierarchical transfer learning with as few as \(500\) short-domain snapshots (\(2.5%\) of the base training data) at a fraction of the scratch-training cost. Applied to compressible channel-flow DNS at \(Re_τ= 180\), \(500\), and \(1000\), Patched FM reconstructs full-resolution wall-pressure fields on a domain four times larger than the training configuration (\(L_x^L = 16πδ\) versus \(L_x^S = 4πδ\)) from sensor coverage as low as \(0.25%\), recovering the low-wavenumber spectral content inaccessible to short-domain DNS with high fidelity in both streamwise and spanwise directions. Zero-shot generalization to unseen Reynolds numbers and ablation studies further confirm the role of inner scaling as a physical prerequisite for data-efficient Reynolds-number transfer.

Diffusion models excel at capturing multi-modal action distributions in robot imitation learning. However, in multi-task and long-horizon scenarios, monolithic architectures lack structural generalization capabilities, suffering from gradient conflicts between distinct semantic sub-stages. While pure data-driven Mixture-of-Experts (MoE) methods introduce labor division, they frequently trigger routing collapse, and instantiating full-scale experts causes parameter explosion and high expansion costs. To address these issues, we propose Concept-prior Routed Diffusion Experts (CoRDE), a structure-guided variational distillation framework. CoRDE extracts semantic distributions from a frozen concept encoder to guide the variational posterior responsibility via a learnable soft mapping matrix. This mechanism introduces an entropy-controlled responsibility inference process that encourages confident routing under reliable semantic predictions while preserving the stochastic diffusion term for behavioral diversity. To overcome parameter inflation, CoRDE employs a parameter-efficient expert pool using Low-Rank Adaptation (LoRA) on a shared frozen backbone. Theoretical analysis shows that the mixture score discrepancy is bounded by responsibility-weighted local expert errors, supporting high-fidelity generation under low-rank expert adaptation. Empirical evaluations confirm that, compared to existing baselines, CoRDE systematically reduces routing collapse, forming robust, semantically aligned expert allocations while achieving superior action quality and incremental learning efficiency.

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

Deep generative models reproduce the observational distribution of their training data, inheriting any spurious associations it contains. A common source is an unobserved confounder that shapes both an attribute the user wants to control at sampling time and an attribute expected to vary in response. Existing causal generative approaches resolve the resulting ambiguity by imposing structural assumptions strong enough to single out one interventional distribution; in image domains, such assumptions are rarely warranted, and the data is generally consistent with a set of distinct causal mechanisms -- a feasible region of interventional distributions. We propose CauVaDE (Causal Variational Deep Embedding), built on a canonical augmented SCM in which the unobserved confounder collapses, without loss of generality, into a discrete latent cluster of bounded support while continuous variation is absorbed into independent noises. We prove that this canonical class is dense, in both observational and interventional Wasserstein distance, in the class of augmented SCMs compatible with a given causal diagram, and instantiate it as a mixture variational autoencoder whose cluster variable plays the role of the canonical confounder. An entropy regularizer with weight \(γ\) on the cluster posterior then traces a family of candidate causal effects that fit the observational data to comparable likelihood while spanning the feasible region. Experiments on image data benchmarks show that CauVaDE produces diverse interventional samples and improves FID against an unconfounded reference.

We present AdaPrivate-TS, a differentially private contextual bandit algorithm that combines Thompson Sampling with batched zCDP composition. Our key insight is that differential privacy noise inflates the posterior covariance in a structured way: adding Gaussian noise \(N(0,σ^2 I)\) to \(b\) yields sampling covariance \(v^2 A^{-1} + σ^2 A^{-2}\), which Thompson Sampling interprets as increased uncertainty rather than pure corruption. Under event-level privacy (protecting individual interactions) with stochastic contexts, we prove that the privacy cost is only \(O(\sqrt{d}\,\log T/\sqrtρ)\), logarithmic in \(T\), because parallel composition amortizes noise across batches. Additionally, we explore privacy amplification via Poisson subsampling, which can reduce effective noise at stringent privacy budgets. Experiments on synthetic and real-world datasets demonstrate: (1) AdaPrivate-TS achieves 93-99% of non-private performance at \(\varepsilon \in [0.5, 5]\), outperforming UCB by 0.5-3.7% and up to 18% with tuned adaptive exploration at extreme \(\varepsilon\); (2) privacy amplification provides additional 2-5% gains at low \(\varepsilon\); (3) on MovieLens and Jester, AdaPrivate-TS achieves the best overall performance among event-level baselines, dominating at \(\varepsilon \geq 2\); (4) under DP-SVD private features, TS's advantage over UCB grows to +11%, confirming noise-as-uncertainty is not limited to reward privacy. We provide rigorous proofs for privacy guarantees under interactive zCDP composition and comprehensive evaluation including convergence curves, 12-seed CIs, and DP-SVD feature ablation.

This paper explores unsupervised disentangled representation learning from a functional perspective. We define latent concepts as factors that influence observations through locally orthogonal directions, formalized as an orthogonality constraint on the Jacobian of the generative mapping. We prove that this condition yields identifiability of general nonlinear generative models, without requiring statistical independence or causal assumptions, provided the latent domain admits all combinations of factor values. Experiments with orthogonality-regularized normalizing flows empirically confirm the theory, demonstrate reliable recovery of ground-truth factors, and shed light on the success of VAEs. These findings challenge the prevailing impossibility claims for unsupervised disentanglement and provide a principled alternative foundation.

We consider the parameter estimation problem in logistic regression with Gaussian design: the estimation of a fixed unknown parameter \(θ^*\in \mathbb{R}^d\) (\(\|θ^*\|_2\ge 1\)) from \(n\) i.i.d. samples \(\{(x_i,y_i)\}_{i=1}^n\), where \(x_i\sim N(0,I_d)\) and \(y_i|x_i \sim {\rm Bernoulli}(1/(1+\exp(-x_i^\top θ^*)))\). Our main aim is to characterize the finite-sample estimation performance and convergence behavior of gradient descent (GD) on the maximum likelihood objective (i.e., the logistic loss). Under small \(O(1)\) stepsize and \(0\) initialization, we show that GD linearly converges to a small neighborhood of \(θ^*\) achieving an \(\ell_2\) error of order \(O(\sqrt{\|θ^*\|_2^5d/n})\). This substantially goes beyond existing theoretical results that lack non-asymptotic estimation error rate and exhibit much slower parameter convergence. We also establish a faster local linear convergence to the same statistical error under a large \(Θ(\|θ^*\|_2)\) stepsize. The main technical component is to show that the gradient of the logistic loss satisfies a certain approximate invertibility condition (AIC). To that end, we uniformly control the deviation of the gradient from its population counterpart by covering and peeling arguments, and then show that the population GD is a contraction by a delicate analysis based on the eigenvalues of population Hessian matrices. Finally, we build upon the recent work Matsumoto and Mazumdar (2025) and devise a novel efficient estimator that attains a sharper rate in high dimensions. This indicates that the existing non-asymptotic guarantees exhibit sub-optimal dependence on \(\|θ^*\|_2\), and that in many regimes \(Θ(\sqrt{\|θ^*\|_2d/n})\) is the tight estimation error rate. Numerical examples are provided to corroborate our theoretical results.