Generator Matching Framework

• Generator Matching Framework is a unified, modality-agnostic approach that leverages Markov process generators to interpolate between simple priors and target data distributions.
• It subsumes and generalizes methods such as diffusion models, flow matching, and discrete jump processes, enabling robust learning across continuous and discrete state spaces.
• The framework offers stable optimization with explicit convergence guarantees and supports practical extensions in energy systems, irregular time series, and latent space matching.

The Generator Matching Framework is a unified, modality-agnostic approach for generative modeling built upon the infinitesimal generator formalism for Markov processes. It subsumes and generalizes flow matching, diffusion models, discrete diffusion and jump processes, enabling principled learning of generative samplers across both continuous and discrete state spaces. The framework operates by matching the generator (which encodes the infinitesimal evolution of a Markov process) to a conditional or marginal “target generator,” guaranteeing that the learned process transports a simple prior distribution to the data distribution along an interpolating probability path. This paradigm enables construction and learning of highly expressive generative models including pure flows, diffusions, hybrid jump processes, and superpositions thereof.

1. Mathematical Foundations and General Formulation

Generator Matching is grounded in the theory of continuous-time Markov processes. For a state space $S$ (e.g., $\mathbb{R}^d$, discrete sets, or multimodal mixtures), a Markov process $X_t$ is characterized by its transition kernel and infinitesimal generator $\mathcal{L}_t$:

$$\mathcal{L}_t f(x) = \lim_{h\to 0} \frac{\mathbb{E}[f(X_{t+h}) | X_t = x] - f(x)}{h}$$

Given a path of distributions $\{ p_t \}_{t \in [0,1]}$ interpolating between a simple prior $p_0$ and the data $p_1$, the Generator Matching objective is to find a parameterization $\mathcal{L}_t^\theta$ whose marginals track $p_t$ over time. The loss functional is

$$L_{\mathrm{GM}}(\theta) = \mathbb{E}_{t \sim U[0,1], x \sim p_t} [ D(F_t(x), F_t^\theta(x)) ]$$

where $F_t$ and $F_t^\theta$ are vector-valued generator parameters (e.g., drift, diffusion, jump rates), and $D$ is a chosen Bregman divergence (e.g., squared-error, KL for jump processes) (Holderrieth et al., 2024, Patel et al., 2024). Marginal generators are typically learned via conditional generators associated to each training sample $z$, with the marginal determined by averaging over conditional paths.

The framework is rigorously justified for arbitrary choices of time-reweighting and state/time-dependent parameterizations; positive weights and measures over time leave the optimal solution unchanged, which enables practical stabilization strategies (Billera et al., 20 Nov 2025).

2. Subsumed Generative Modeling Paradigms

Generator Matching unifies several classical and modern generative models as special cases:

• Diffusion Models: Here, the generator is second-order and matches the drift and diffusion of a stochastic differential equation (SDE), e.g.,

$$\mathcal{L}_t^{\mathrm{diff}} f(x) = \nabla f(x)^T f_t x + \tfrac{1}{2} g_t^2 \Delta f(x)$$

The learning objective recovers score-based or denoising training procedures (Patel et al., 2024).

• Flow Matching: The generator reduces to first-order drift ODEs:

$$\mathcal{L}_t^{\mathrm{flow}} f(x) = \nabla f(x)^T u_t(x)$$

Matching the conditional or marginal velocity field yields transport from prior to data. This encompasses various deterministic flows and rectified flows (Patel et al., 2024, Huang et al., 2024).

• Discrete Diffusion and Discrete Flow: The generator is a time-dependent CTMC rate matrix, and the loss typically uses a KL-form Bregman divergence. Generator matching yields non-asymptotic error bounds and enables exact sampling via uniformization (Wan et al., 26 Sep 2025).
• Jump Processes and Superpositions: Generator Matching naturally extends to arbitrary jump kernels ($\mathcal{L}_t f(x) = \int [f(y) - f(x)] Q_t(dy|x)$), or convex superpositions of flows, diffusions and jumps, preserving the marginal distribution along the probability path (Holderrieth et al., 2024, Jahn et al., 29 May 2025).

3. Loss Functions, Conditional Paths, and Training Algorithms

Losses in Generator Matching are built upon Bregman divergences between the true and parametric generator fields. For continuous models, squared-error is used; for discrete (jump) processes, KL or cross-entropy is typical. The most general form is

$$L_{\mathrm{GM}}(\theta) = \mathbb{E}_{t \sim \mathcal{D}, x \sim p_t}[w(t) D_{t,x}(F_t(x), F_t^\theta(x))]$$

where both the divergence and parameterization may explicitly depend on $t$ and $x$, with all positive weights yielding equivalent optima (Billera et al., 20 Nov 2025). For conditional modeling, a conditional path $p_t(\cdot|z)$ and its generator $F_t^z(x)$ are sampled, and the loss averaged over data samples (Holderrieth et al., 2024).

Sampling from learned Generator Matching models involves simulating the appropriate ODE, SDE, or CTMC using learned generator parameters, optionally combining flow, diffusion, and jumps as prescribed by the superposition principle (Holderrieth et al., 2024).

4. Practical Extensions: Energy-Based, Trajectory, and Latent Matching

Recent works extend Generator Matching to energy-based settings, irregular time series, and latent-space matching:

• Energy-Based Generator Matching (EGM): Enables training of Markov-process samplers using only energy oracles, via self-normalized importance sampling and bootstrapped variance reduction. This supports general continuous, discrete, and mixed state spaces (Woo et al., 26 May 2025).
• Trajectory Generator Matching: Handles irregularly sampled time series by matching infinitesimal generators (both diffusion and jumps) via closed-form losses, with analytical treatment of jump kernels via scaled Gaussians and Kullback–Leibler minimization (Jahn et al., 29 May 2025). This approach is simulation-free for the generator.
• Latent and Feature Matching: For high-dimensional generative tasks, approaches such as manifold or moment matching encode data and generated distributions into lower-dimensional embeddings, using MMD or metric-learning losses to stabilize and enhance generative diversity (Liao et al., 2021, Dai et al., 2021, Gao et al., 2023).

5. Computational and Theoretical Advantages

Generator Matching presents several advantages over adversarial and standard generative approaches:

• Linear and convex optimization properties often yield stable and rapid convergence (notably in linear convex LP settings for energy-economic problems (Azad et al., 2024)).
• Non-saddlepoint decoupled losses (as in GRAM networks) lead to robust training unaffected by critic-generator imbalance (Srivastava et al., 2018).
• Exact sampling via uniformization for discrete flows removes truncation and discretization errors (Wan et al., 26 Sep 2025).
• Theoretical results show that matching the generator field in expectation directly controls the pathwise and marginal KL/TV errors through Girsanov-type identities for SDEs and CTMCs, yielding non-asymptotic convergence bounds (Wan et al., 26 Sep 2025, Patel et al., 2024).

6. Applications and Empirical Performance

Generator Matching has demonstrated efficacy across domains:

• Image and Multimodal Generation: Unifies Flows, Diffusions, Discrete Diffusion—superpositions with jumps demonstrably improve sample diversity (Holderrieth et al., 2024).
• One-Step Distillation: Flow Generator Matching enables reduction of multi-step ODE samplers to single-step generators, maintaining sample quality while reducing inference cost (e.g., unconditional CIFAR-10 FID=3.08, state-of-the-art for one-step flow-matching-based models) (Huang et al., 2024).
• Energy Systems Design: Linear convex generator-storage matching frameworks optimize dispatch and capacity for generator-storage pairs in energy systems, facilitating economic feasibility studies (Azad et al., 2024).
• Irregular Time-Series Sampling: Trajectory generator matching captures both diffusive and jump behavior in stochastic time-series observed at arbitrary intervals (Jahn et al., 29 May 2025).
• Physics-Informed PDE Solving: Physics-informed latent-space generator-encoder matching yields superior generalization and training stability for SDEs (Gao et al., 2023).
• Neutrino Physics: Generator-matching in NUISANCE enables systematic comparison and tuning of cross-section MC generators using external reweighting and likelihood-based fits (Stowell et al., 2016).

7. Design Space Expansion and Future Directions

Generator Matching drastically expands the generative modeling design space, enabling construction of samplers for arbitrary probability paths via flows, diffusions, jumps, or mixtures. It accommodates arbitrary state/time-dependent parameterizations, supports energy-only training, and enables superpositions for enhanced multimodal and stochastic representation (Holderrieth et al., 2024, Woo et al., 26 May 2025). The unified framework provides rigorous non-asymptotic error guarantees, robust theoretical foundations for loss weighting and optimization strategies, and principled extensions for modality-agnostic, simulation-free learning.

References:

• (Holderrieth et al., 2024) Generator Matching: Generative modeling with arbitrary Markov processes
• (Patel et al., 2024) Exploring Diffusion and Flow Matching Under Generator Matching
• (Huang et al., 2024) Flow Generator Matching
• (Wan et al., 26 Sep 2025) Error Analysis of Discrete Flow with Generator Matching
• (Woo et al., 26 May 2025) Energy-based generator matching: A neural sampler for general state space
• (Jahn et al., 29 May 2025) Trajectory Generator Matching for Time Series
• (Billera et al., 20 Nov 2025) Time dependent loss reweighting for flow matching and diffusion models is theoretically justified
• (Gao et al., 2023) Physics-Informed Generator-Encoder Adversarial Networks with Latent Space Matching for Stochastic Differential Equations
• (Liao et al., 2021) Scenario Generation for Cooling, Heating, and Power Loads Using Generative Moment Matching Networks
• (Dai et al., 2021) Manifold Matching via Deep Metric Learning for Generative Modeling
• (Stowell et al., 2016) NUISANCE: a neutrino cross-section generator tuning and comparison framework
• (Azad et al., 2024) A general framework for supporting economic feasibility of generator and storage energy systems through capacity and dispatch optimization
• (Srivastava et al., 2018) Generative Ratio Matching Networks