The decomposition provides a theoretical basis for adapting likelihood-based LLM methods to flow matching, while distinguishing exact substitutions from controlled surrogates, while distinguishing exact substitutions from controlled surrogates.
Abstract
Flow matching enables likelihood-free training, yet alignment methods increasingly reuse conditional flow matching (CFM) losses as endpoint negative log-likelihoods (NLLs) and their old/new differences as log-likelihood ratios. We characterize when these substitutions are valid. For linear Gaussian paths, we exactly decompose endpoint NLL into entropy, a weighted CFM objective, an interior velocity--score residual, and a boundary residual. Thus CFM-only estimates and differences are exact only when the corresponding residuals cancel. At the off-policy population optimum, ordinary CFM is not generally a pointwise NLL estimator, whereas \(w_{\mathrm{sc}}(t)=(1-t)/t\) removes the interior residual; this positive result does not extend generally to training or on-policy alignment. On-policy log-ratios can remain biased even for identical endpoint laws or after surrogate optimization. Experiments across dimensions, distributions, and geometries support these conclusions and the mechanisms that make inexact ratios useful. **More broadly, the decomposition provides a theoretical basis for adapting likelihood-based LLM methods to flow matching, while distinguishing exact substitutions from controlled surrogates.**
Maximum likelihood (ML) estimation is a principled and statistically efficient approach for learning probabilistic models. However, for unnormalized models, ML estimation requires evaluating the partition function and differentiating through it, which may not always be tractable. Score matching provides a practically viable alternative that circumvents this obstacle by fitting the score in a way that eliminates dependence on the normalizing constant. We derive the generalized score matching objective on a convex subset of $\mathbb{R}^{d}$ constructively starting from Minimum Probability Flow (MPF) learning, and show how classical score matching as well as domain-adapted variants for non-negative data arise naturally within the proposed framework. We show that the resulting objective is a {\it proper local scoring rule} of second-order, which provides the theoretical guarantee that the true density is recovered when the objective is minimized. Furthermore, for a model belonging to the exponential family, we establish convexity of the objective together with consistency of the finite-sample estimator under standard regularity conditions. Our derivation sheds new light on the scope and applicability of generalized score matching in various problem settings. We compare generalized score matching-based estimators on constrained domains, where the partition function is analytically intractable. We provide experimental results on parameter estimation for model densities belonging to the exponential family defined over convex subsets of $\mathbb{R}^{d}$, and a generative modeling use-case to demonstrate broader applicability of the proposed generalized score matching framework.
Nishanth Shetty, Saisuchith Mahajan, C. Seelamantula· 0 citations
This work shows that the regression difficulty of Conditional Flow Matching varies systematically along the path, and proposes Difficulty-Calibrated Flow Matching, which derives the schedule from the model itself: a short pilot run with the linear path records the per-time loss, and the schedule is set to the quantile function of this difficulty profile.
The performance of Flow Matching largely depends on the quality of the coupling between the source and target distributions. However, independent coupling often leads to path crossings and local velocity ambiguity, while OT-based couplings typically incur high construction costs. To address this challenge, we propose Quantile AlignTree Flow Matching (QAT-FM), an efficient structured coupling strategy that constructs a hierarchical coupling between a Gaussian prior and the target data distribution via a quantile-aligned tree structure. QAT-FM constructs the coupling in $\mathcal{O}(Nd\log N)$ time and supports per-pair source sampling with $\mathcal{O}(d)$ complexity, enabling scalable training for large-scale high-dimensional generative tasks. Theoretically, we prove that the QAT coupling satisfies marginal consistency, induces non-crossing linear interpolation paths, and consistently improves path separation at intermediate times compared with independent coupling, thereby alleviating local velocity ambiguity. QAT-FM further extends naturally to conditional generation, enabling structured conditional coupling while preserving global Gaussian alignment. Experiments across diverse benchmark datasets demonstrate that QAT-FM achieves competitive generative performance while substantially reducing coupling construction cost.
Jun-Yi Lin, Meng-Yu Li, Jing-Xuan Hu et al.· 0 citations
This work proposes Missing-Data Flow Matching, which treats the missing coordinates of training samples as latent variables and averages the flow matching loss over the values they could take, and places the method alongside strong classical and deep imputation baselines on real tabular data.
Fairoz Nower Khan, Nabuat Zaman Nahim, Pei-Zhong Ju· arXiv.org· 0 citations
Bayesian model selection couples a discrete model indicator with a model-specific continuous parameter space. We introduce structured dimension-matched variational transdimensional inference (SM-VTI) for finite enumerable model spaces. A rooted construction graph expresses a model as a sequence of local stop/child decisions. Each typed edge compiles a declared scientific parent-child edit into an exact native-coordinate dimension-matching lifting; an edge-conditioned flow then learns the residual continuous transport. The resulting local policy and conditional flow define one direct joint variational distribution, without embedding every model in a saturated maximum-dimensional surrogate. We derive its exact path density and optimize the joint reverse-KL objective. On a controlled 15-model target, SM-VTI-Joint recovers terminal masses, local actions, and nonlinear conditional geometry. On a 128-model misspecified robust variable-selection problem, a 10-data-set nearly parameter-matched affine comparison with AVTI shows stronger early model-mass recovery and competitive final joint accuracy under the same target-evaluation budget.
Generative models like diffusion and flow matching excel in traffic imputation but suffer from high inference latency, as the learned vector fields typically induce curved generative trajectories requiring multi-step numerical integration. Furthermore, learning the transformation from non-informative priors introduces redundant computational overhead. To address these issues, we propose Low-Rank Prior-Induced Consistency Flow Matching (LOFT) for efficient and effective distribution modeling under highly sparse data. First, we construct a low-rank prior from sparse observations to recover inherent spatiotemporal correlations. Initializing the flow with this informative prior reduces the mapping complexity, allowing the model to focus on fine-grained variations. Second, to enable efficient inference by linearizing generative trajectories, we introduce an uncertainty-aware rectification mechanism. This mechanism resolves the gradient conflict between improving accuracy and trajectory linearization by dynamically arbitrating the optimization trade-off based on the training progress and data uncertainty. Experimental results demonstrate that LOFT surpasses state-of-the-art baselines using an NFE (Number of Function Evaluations) of 2, whereas competing methods typically require 20 to 50 NFE, achieving over a 10× improvement in inference efficiency. The code is available at https://github.com/maoxiaowei97/LOFT.
Xiaowei Mao, Tingrui Wu, Yawen Yang et al.· Proceedings of the 32nd ACM...· 0 citations
