Tracking High-order Evolutions via Cascading Low-rank Fitting

• URL: http://arxiv.org/abs/2604.10980v1
• Date: Mon, 13 Apr 2026 04:39:49 GMT
• Title: Tracking High-order Evolutions via Cascading Low-rank Fitting
• Abstract summary: High-order dynamics has emerged as a promising frontier in generative modeling.<n>We introduce cascading low-rank fitting, an ordinary differential equation inspired method that approximates successive derivatives.<n>We prove that if the initial difference is linearly decomposable, the generic ranks of high-order derivatives are guaranteed to be monotonically non-increasing.
• Score: 10.036945747389058
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Diffusion models have become the de facto standard for modern visual generation, including well-established frameworks such as latent diffusion and flow matching. Recently, modeling high-order dynamics has emerged as a promising frontier in generative modeling. Rather than only learning the first-order velocity field that transports random noise to a target data distribution, these approaches simultaneously learn higher-order derivatives, such as acceleration and jerk, yielding a diverse family of higher-order diffusion variants. To represent higher-order derivatives, naive approaches instantiate separate neural networks for each order, which scales the parameter space linearly with the derivative order. To overcome this computational bottleneck, we introduce cascading low-rank fitting, an ordinary differential equation inspired method that approximates successive derivatives by applying a shared base function augmented with sequentially accumulated low-rank components. Theoretically, we analyze the rank dynamics of these successive matrix differences. We prove that if the initial difference is linearly decomposable, the generic ranks of high-order derivatives are guaranteed to be monotonically non-increasing. Conversely, we demonstrate that without this structural assumption, the General Leibniz Rule allows ranks to strictly increase. Furthermore, we establish that under specific conditions, the sequence of derivative ranks can be designed to form any arbitrary permutation. Finally, we present a straightforward algorithm to efficiently compute the proposed cascading low-rank fitting.

Related papers

• A Hyperfinite Framework for Score-Based Generative Modeling [0.8122270502556375]
  We develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis.<n>We show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics.<n>Under suitable assumptions, we derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives.
  arXiv  Detail & Related papers  (2026-08-03T18:51:44Z)
• Tensor-Train Joint Modeling for Few-Step Discrete Diffusion [25.442970567297326]
  We present a framework for explicit joint distribution modeling in discrete diffusion via tensor decomposition.<n>Our framework integrates into pretrained MDMs through lightweight fine-tuning, yielding substantial improvements in few-step generation at a fraction of the cost of training from scratch.
  arXiv  Detail & Related papers  (2026-07-04T09:32:12Z)
• Probability-Conserving Flow Guidance [49.03107678763765]
  Adaptive Manifold Guidance (AdaMaG) improves realism, reduces hallucinations, and controlled desaturation in high-guidance.<n>Across image generation benchmarks, AdaMaG improves realism, reduces hallucinations, and controlled desaturation in high-guidance.
  arXiv  Detail & Related papers  (2026-05-19T16:34:01Z)
• Hessian-vector products for tensor networks via recursive tangent-state propagation [1.5282767384702272]
  We introduce an analytical Hessian-vector product kernel designed for arbitrary compositions of linear maps.<n>Our second-order approach achieves up to a four-order-of-magnitude improvement in fidelity over naive Trotterization.
  arXiv  Detail & Related papers  (2026-04-22T09:38:08Z)
• Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation [56.361076943802594]
  CanonFlow achieves state-of-the-art performance on the challenging GEOM-DRUG dataset, and the advantage remains large in few-step generation.
  arXiv  Detail & Related papers  (2026-02-16T18:58:55Z)
• PRISM: Parallel Residual Iterative Sequence Model [52.26239951489612]
  We propose PRISM (Parallel Residual Iterative Sequence Model) to resolve this tension.<n>PRISM introduces a solver-inspired inductive bias that captures key structural properties of multi-step refinement in a parallelizable form.<n>We prove that this formulation achieves Rank-$L$ accumulation, structurally expanding the update manifold beyond the single-step Rank-$1$ bottleneck.
  arXiv  Detail & Related papers  (2026-02-11T12:39:41Z)
• Diffusion Models: A Mathematical Introduction [3.8673630752805437]
  We present a self-contained derivation of diffusion-based generative models.<n>We construct denoising diffusion probabilistic models from first principles.<n>Readers can both follow the theory and implement the corresponding algorithms in practice.
  arXiv  Detail & Related papers  (2025-11-13T16:20:52Z)
• Learning Optical Flow Field via Neural Ordinary Differential Equation [44.16275288019991]
  Recent works on optical flow estimation use neural networks to predict the flow field that maps positions of one image to positions of the other.<n>We introduce a novel approach for predicting the derivative of the flow using a continuous model, namely neural ordinary differential equations (ODE)
  arXiv  Detail & Related papers  (2025-06-03T18:30:14Z)
• Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows [40.884514919698596]
  Flow matching has emerged as a powerful framework for generative modeling.<n>We prove that higher-order flow matching preserves worst case optimality as a distribution estimator.
  arXiv  Detail & Related papers  (2025-03-12T05:07:07Z)
• On the Relation Between Linear Diffusion and Power Iteration [42.158089783398616]
  We study the generation process as a correlation machine'' We show that low frequencies emerge earlier in the generation process, where the denoising basis vectors are more aligned to the true data with a rate depending on their eigenvalues. This model allows us to show that the linear diffusion model converges in mean to the leading eigenvector of the underlying data, similarly to the prevalent power iteration method.
  arXiv  Detail & Related papers  (2024-10-16T07:33:12Z)
• Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-Based Decoding [84.3224556294803]
  Diffusion models excel at capturing the natural design spaces of images, molecules, DNA, RNA, and protein sequences. We aim to optimize downstream reward functions while preserving the naturalness of these design spaces. Our algorithm integrates soft value functions, which looks ahead to how intermediate noisy states lead to high rewards in the future.
  arXiv  Detail & Related papers  (2024-08-15T16:47:59Z)
• Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach [49.97755400231656]
  We show that a new accelerated DDPM sampler achieves accelerated performance for three broad distribution classes not considered before.<n>Our results show an improved dependency on the data dimension $d$ among accelerated DDPM type samplers.
  arXiv  Detail & Related papers  (2024-02-21T16:11:47Z)
• Linear convergence of forward-backward accelerated algorithms without knowledge of the modulus of strong convexity [14.0409219811182]
  We show that both Nesterov's accelerated gradient descent (NAG) and FISTA exhibit linear convergence for strongly convex functions. We emphasize the distinctive approach employed in crafting the Lyapunov function, which involves a dynamically adapting coefficient of kinetic energy.
  arXiv  Detail & Related papers  (2023-06-16T08:58:40Z)
• Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations [83.3201889218775]
  Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) when derived naively. However, the convergence properties of these methods are qualitatively different, even on simple bilinear games. We adopt a framework studied in fluid dynamics to design differential equation models for several saddle-point optimization methods.
  arXiv  Detail & Related papers  (2021-12-27T18:31:34Z)
• Estimating High Order Gradients of the Data Distribution by Denoising [81.24581325617552]
  First order derivative of a data density can be estimated efficiently by denoising score matching. We propose a method to directly estimate high order derivatives (scores) of a data density from samples.
  arXiv  Detail & Related papers  (2021-11-08T18:59:23Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.