Inductive Moment Matching

Introduction

Generative models for continuous domains have enabled numerous applications in images (Rombach et al., 2022; Saharia et al., 2022; Esser et al., 2024), videos (Ho et al., 2022a; Blattmann et al., 2023; OpenAI, 2024), and audio (Chen et al., 2020; Kong et al., 2020; Liu et al., 2023), yet achieving high-fidelity outputs, efficient inference, and stable training remains a core challenge — a trilemma that continues to motivate research in this domain. Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020b), one of the leading techniques, require many inference steps for high-quality results, while step-reduction methods, such as diffusion distillation (Yin et al., 2024; Sauer et al., 2025; Zhou et al., 2024; Luo et al., 2024a) and Consistency Models (Song et al., 2023; Geng et al., 2024; Lu & Song, 2024; Kim et al., 2023), often risk training collapse without careful tuning and regularization (such as pre-generating data-noise pair and early stopping).

To address the aforementioned trilemma, we introduce Inductive Moment Matching (IMM), a stable, single-stage training procedure that learns generative models from scratch for single- or multi-step inference. IMM operates on the time-dependent marginal distributions of stochastic interpolants (Albergo et al., 2023) — continuous-time stochastic processes that connect two arbitrary probability density functions (data at $t=0$ and prior at $t=1$). By learning a (stochastic or deterministic) mapping from any marginal at time $t$ to any marginal at time $s<t$, it can naturally support one- or multi-step generation (Figure 2).

IMM models can be trained efficiently from mathematical induction. For time $s<r<t$, we form two distributions at $s$ by running a one-step IMM from samples at $r$ and $t$. We then minimize their divergence, enforcing that the distributions at $s$ are independent of the starting time-steps. This construction by induction guarantees convergence to the data distribution. To help with training stability, we model IMM based on certain stochastic interpolants and optimize the objective with stable sample-based divergence estimators such as moment matching (Gretton et al., 2012). Notably, we prove that Consistency Models (CMs) are a single-particle, first-moment matching special case of IMM, which partially explains the training instability of CMs.

On ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$, IMM surpasses diffusion models and achieves 1.99 FID with only 8 inference steps using standard transformer architectures. On CIFAR-10, IMM similarly achieves state-of-the-art of 1.98 FID with 2-step generation for a model trained from scratch.

Preliminaries

When $\gamma_{t}\equiv 0$ and ${\bm{I}}_{t}({\mathbf{x}},\bm{\epsilon})=\alpha_{t}{\mathbf{x}}+\sigma_{t}\bm{\epsilon}$ for $\alpha_{t},\sigma_{t}$ defined in FM, the intermediate variable ${\mathbf{x}}_{t}=\alpha_{t}{\mathbf{x}}+\sigma_{t}\bm{\epsilon}$ becomes a deterministic interpolation and its interpolant velocity ${\mathbf{v}}_{t}=\alpha_{t}^{\prime}{\mathbf{x}}+\sigma_{t}^{\prime}\bm{\epsilon}$ reduces to FM velocity. Thus, its training and inference both reduce to that of FM. When $\bm{\epsilon}\sim{\mathcal{N}}(0,{I})$, stochastic interpolants reduce to $v$-prediction diffusion.

2 Maximum Mean Discrepancy

Inductive Moment Matching

We introduce Inductive Moment Matching (IMM), a method that trains a model of both high quality and sampling efficiency in a single stage. To do so, we assume a time-augmented interpolation between data (distribution at $t=0$) and prior (distribution at $t=1$) and propose learning an implicit one-step model (i.e. a one-step sampler) that transforms the distribution at time $t$ to the distribution at time $s$ for any $s<t$ (Section 3.1). The model enables direct one-step sampling from $t=1$ to $s=0$ and few-step sampling via recursive application from any $t$ to any $r<t$ and then to any $s<r$ until $s=0$; this allows us to learn the model from its own samples via bootstrapping (Section 3.2).

Given data ${\mathbf{x}}\sim q({\mathbf{x}})$ and prior $\bm{\epsilon}\sim p(\bm{\epsilon})$, the time-augmented interpolation ${\mathbf{x}}_{t}$ defined in Albergo et al. (2023) follows ${\mathbf{x}}_{t}\sim q_{t}({\mathbf{x}}_{t}|{\mathbf{x}},\bm{\epsilon})$. This implies a marginal interpolating distribution

We learn a model distribution implicitly defined by a one-step sampler that transforms $q_{t}({\mathbf{x}}_{t})$ into $q_{s}({\mathbf{x}}_{s})$ for some $s\leq t$. This can be done via a special class of interpolants, which preserves the marginal distribution $q_{s}({\mathbf{x}}_{s})$ while interpolating between ${\mathbf{x}}$ and ${\mathbf{x}}_{t}$. We term these marginal-preserving interpolants among a class of generalized interpolants. Formally, we define ${\mathbf{x}}_{s}$ as a generalized interpolant between ${\mathbf{x}}$ and ${\mathbf{x}}_{t}$ if, for all $s\in[0,t]$, its distribution follows

and satisfies constraints ${\bm{I}}_{t|t}({\mathbf{x}},{\mathbf{x}}_{t})={\mathbf{x}}_{t}$, ${\bm{I}}_{0|t}({\mathbf{x}},{\mathbf{x}}_{t})={\mathbf{x}}$, $\gamma_{t|t}=\gamma_{0|t}=0$, and $q_{t|1}({\mathbf{x}}_{t}|{\mathbf{x}},\bm{\epsilon})\equiv q_{t}({\mathbf{x}}_{t}|{\mathbf{x}},\bm{\epsilon})$. When $t=1$, it reduces to regular stochastic interpolants. Next, we define marginal-preserving interpolants.

A generalized interpolant ${\mathbf{x}}_{s}$ is marginal-preserving if for all $t\in$ and for all $s\in[0,t]$, the following equality holds:

That is, this class of interpolants has the same marginal at $s$ regardless of $t$. For all $t\in$, we define our noisy model distribution at $s\in[0,t]$ as

where the interpolant is marginal preserving and $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})$ is our clean model distribution implicitly parameterized as a one-step sampler. This definition also enables multistep sampling. To produce a clean sample ${\mathbf{x}}$ given ${\mathbf{x}}_{t}\sim q_{t}({\mathbf{x}}_{t})$ in two steps via an intermediate $s$: (1) we sample $\hat{{\mathbf{x}}}\sim p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})$ followed by $\hat{{\mathbf{x}}}_{s}\sim q_{s|t}({\mathbf{x}}_{s}|\hat{{\mathbf{x}}},{\mathbf{x}}_{t})$ and (2) if the marginal of $\hat{{\mathbf{x}}}_{s}$ matches $q_{s}({\mathbf{x}}_{s})$, we can obtain ${\mathbf{x}}$ by ${\mathbf{x}}\sim p_{0|s}^{\theta}({\mathbf{x}}|\hat{{\mathbf{x}}}_{s})$. We are therefore motivated to minimize divergence between Eq. (4) and (6) using the objective below.

Naïve objective. As one can easily draw samples from the model, it can be naïvely learned by directly minimizing

with time distribution $p(s,t)$ and a sample-based divergence metric $D(\cdot,\cdot)$ such as MMD or GAN (Goodfellow et al., 2020). If an interpolant ${\mathbf{x}}_{s}$ is marginal-preserving, then the minimum loss is 0 (see Lemma 3). One might also notice the similarity between right-hand sides of Eq. (4) and (6). However, $q_{s}({\mathbf{x}}_{s})=p_{s|t}^{\theta}({\mathbf{x}}_{s})$ does not necessarily imply $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})=q_{t}({\mathbf{x}}|{\mathbf{x}}_{t})$. In fact, the minimizer $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})$ is not unique and, under mild assumptions, a deterministic minimizer exists (see Section 4).

2 Learning via Inductive Bootstrapping

While sound, the naïve objective in Eq. (7) is difficult to optimize in practice because when $t$ is far from $s$, the input distribution $q_{t}({\mathbf{x}}_{t})$ can be far from the target $q_{s}({\mathbf{x}}_{s})$. Fortunately, our interpolant construction implies that the model definition in Eq. (6) satisfies boundary condition $q_{s}({\mathbf{x}}_{s})=p_{s|s}^{\theta}({\mathbf{x}}_{s})$ regardless of $\theta$ (see Lemma 4), which indicates that $p_{s|t}^{\theta}({\mathbf{x}}_{s})\approx q_{s}({\mathbf{x}}_{s})$ when $t$ is close to $s$. Furthermore, the interpolant enforces $p_{s|t}^{\theta}({\mathbf{x}}_{s})\approx p_{s|r}^{\theta}({\mathbf{x}}_{s})$ for any $r<t$ close to $t$ as long as the model is continuous around $t$. Therefore, we can construct an inductive learning algorithm for $p_{s|t}^{\theta}({\mathbf{x}}_{s})$ by using samples from $p_{s|r}^{\theta}({\mathbf{x}}_{s})$.

For better analysis, we define a sequence number $n$ for parameter $\theta_{n}$ and function $r(s,t)$ where $s\leq r(s,t)<t$ such that $p_{s|t}^{\theta_{n}}({\mathbf{x}}_{s})$ learns to match $p_{s|r}^{\theta_{n-1}}({\mathbf{x}}_{s})$.Note that $n$ is different from optimization steps. Advancing from $n-1$ to $n$ can take arbitrary number of optimization steps. We omit $r$’s arguments when context is clear and let $r(s,t)$ be a finite decrement from $t$ but truncated at $s\leq t$ (see Appendix B.3 for well-conditioned $r(s,t)$).

General objective. With marginal-preserving interpolants and mapping $r(s,t)$, we learn $\theta_{n}$ in the following objective:

where $w(s,t)$ is a weighting function. We choose MMD as our objective due to its superior optimization stability and show that this objective learns the correct data distribution.

Assuming $r(s,t)$ is well-conditioned, the interpolant is marginal-preserving, and $\theta_{n}^{*}$ is a minimizer of Eq. (8) for each $n$ with infinite data and network capacity, for all $t\in$, $s\in[0,t]$,

In other words, $\theta_{n}$ eventually learns the target distribution $q_{s}({\mathbf{x}}_{s})$ by parameterizing a one-step sampler $p_{s|t}^{\theta_{n}}({\mathbf{x}}|{\mathbf{x}}_{t})$.

Simplified Formulation and Practice

We present algorithmic and practical decisions below.

Despite theoretical soundness, it remains unclear how to empirically choose a marginal-preserving interpolant. First, we present a sufficient condition for marginal preservation.

Given $s,t\in$, $s\leq t$, an interpolant ${\mathbf{x}}_{s}\sim q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})$ is self-consistent if for all $r\in[s,t]$, the following holds:

In other words, ${\mathbf{x}}_{s}$ has the same distribution if one (1) directly samples it by interpolating ${\mathbf{x}}$ and ${\mathbf{x}}_{t}$ and (2) first samples any ${\mathbf{x}}_{r}$ (given ${\mathbf{x}}$ and ${\mathbf{x}}_{t}$) and then samples ${\mathbf{x}}_{s}$ (given ${\mathbf{x}}$ and ${\mathbf{x}}_{r}$). Furthermore, self-consistency implies marginal preservation (Lemma 5).

DDIM interpolant. Denoising Diffusion Implicit Models (Song et al., 2020a) was introduced as a fast ODE sampler for diffusion models, defined as

(Informal) If $\gamma_{s|t}\equiv 0$ and ${\bm{I}}_{s|t}({\mathbf{x}},{\mathbf{x}}_{t})$ satisfies mild assumptions, there exists a deterministic $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})$ that attains 0 loss for Eq. (7).

See Appendix B.6 for formal statement and proof. This allows us to define $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})=\delta({\mathbf{x}}-{\bm{g}}_{\theta}({\mathbf{x}}_{t},s,t))$ for a neural network ${\bm{g}}_{\theta}({\mathbf{x}}_{t},s,t)$ with parameter $\theta$ by default.

Eliminating stochasticity. We use DDIM interpolant, deterministic model, and prior $p(\bm{\epsilon})={\mathcal{N}}(0,\sigma_{d}^{2}{I})$ where $\sigma_{d}$ is the data standard deviation (Lu & Song, 2024). As a result, one can draw ${\mathbf{x}}_{s}$ from model via ${\mathbf{x}}_{s}={\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})\vcentcolon=\operatorname{DDIM}({\mathbf{x}}_{t},{\bm{g}}_{\theta}({\mathbf{x}}_{t},s,t),s,t)$ where ${\mathbf{x}}_{t}\sim q_{t}({\mathbf{x}}_{t})$.

Re-using ${\mathbf{x}}_{t}$ for ${\mathbf{x}}_{r}$. Inspecting Eq. (8) and (6), one requires ${\mathbf{x}}_{r}\sim q_{r}({\mathbf{x}}_{r})$ to generate samples from the target distribution. Instead of sampling ${\mathbf{x}}_{r}$ given a new $({\mathbf{x}},\bm{\epsilon})$ pair, we can reduce variance by reusing ${\mathbf{x}}_{t}$ and ${\mathbf{x}}$ such that ${\mathbf{x}}_{r}=\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},r,t)$. This is justified because ${\mathbf{x}}_{r}$ derived from ${\mathbf{x}}_{t}$ preserves the marginal distribution $q_{r}({\mathbf{x}}_{r})$ (see Appendix C.2).

Stop gradient. We set $n$ to optimization step number, i.e. advancing from $n-1$ to $n$ is a single optimizer step where $\theta_{n}$ is initialized from $\theta_{n-1}$. Equivalently, we can omit $n$ from $\theta_{n}$ and write $\theta_{n-1}$ as the stop-gradient parameter $\theta^{-}$.

Simplified objective. Let ${\mathbf{x}}_{t},{\mathbf{x}}_{t}^{\prime}$ be i.i.d. random variables from $q_{t}({\mathbf{x}}_{t})$ and ${\mathbf{x}}_{r},{\mathbf{x}}_{r}^{\prime}$ are variables obtained by reusing ${\mathbf{x}}_{t},{\mathbf{x}}_{t}^{\prime}$ respectively, the training objective can be derived from the MMD definition in Eq. (1) (see Appendix C.3) as

where ${\mathbf{y}}_{s,t}={\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})$, ${\mathbf{y}}_{s,t}^{\prime}={\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t}^{\prime})$, ${{\mathbf{y}}}_{s,r}={\bm{f}}_{s,r}^{\theta^{-}}({\mathbf{x}}_{r})$, ${{\mathbf{y}}}_{s,r}^{\prime}={\bm{f}}_{s,r}^{\theta^{-}}({\mathbf{x}}_{r}^{\prime})$, $k(\cdot,\cdot)$ is a kernel function, and $w(s,t)$ is a prior weighting function.

An empirical estimate of the above objective uses $M$ particle samples to approximate each distribution indexed by $t$. In practice, we divide a batch of model output with size $B$ into $B/M$ groups within which share the same $(s,t)$ sample, and the objective is approximated by instantiating $B/M$ number of $M\times M$ matrices. Note that the number of model passes does not change with respect to $M$ (see Appendix C.4). A $M=2$ version is visualized in Figure 3 and a simplified training algorithm is shown in Algorithm 1. A full training algorithm is shown in Appendix D.

2 Other Implementation Choices

We defer detailed analysis of each decision to Appendix C.

Flow trajectories. We investigate the two most used flow trajectories (Nichol & Dhariwal, 2021; Lipman et al., 2022),

Cosine. $\alpha_{t}=\cos(\frac{1}{2}\pi t)$, $\sigma_{t}=\sin(\frac{1}{2}\pi t)$.

Network ${\bm{g}}_{\theta}({\mathbf{x}}_{t},s,t)$. We set ${\bm{g}}_{\theta}({\mathbf{x}}_{t},s,t)=c_{\text{skip}}(t){\mathbf{x}}_{t}+c_{\text{out}}(t){\bm{G}}_{\theta}(c_{\text{in}}(t){\mathbf{x}}_{t},c_{\text{noise}}(s),c_{\text{noise}}(t))$ with a neural network ${\bm{G}}_{\theta}$, following EDM (Karras et al., 2022). For all choices we let $c_{\text{in}}(t)=1/\sqrt{\alpha_{t}^{2}+\sigma_{t}^{2}}/\sigma_{d}$ (Lu & Song, 2024). Listed below are valid choices for other coefficients.

Identity. $c_{\text{skip}}(t)=0$, $c_{\text{out}}(t)=1$.

Simple-EDM (Lu & Song, 2024). $c_{\text{skip}}(t)=\alpha_{t}/(\alpha_{t}^{2}+\sigma_{t}^{2})$, $c_{\text{out}}(t)=-\sigma_{d}\sigma_{t}/\sqrt{\alpha_{t}^{2}+\sigma_{t}^{2}}$.

Euler-FM. $c_{\text{skip}}(t)=1$, $c_{\text{out}}(t)=-t\sigma_{d}$. This is specific to OT-FM schedule.

We show in Appendix C.5 that ${\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})$ similarly follows the EDM parameterization of the form ${\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})=c_{\text{skip}}(s,t){\mathbf{x}}_{t}+c_{\text{out}}(s,t){\bm{G}}_{\theta}(c_{\text{in}}(t){\mathbf{x}}_{t},c_{\text{noise}}(s),c_{\text{noise}}(t))$.

Noise conditioning $c_{\text{noise}}(\cdot)$. We choose $c_{\text{noise}}(t)=ct$ for some constant $c\geq 1$. We find our model convergence relatively insensitive to $c$ but recommend using larger $c$, e.g. $1000$ (Song et al., 2020b; Peebles & Xie, 2023), because it enables sufficient distinction between nearby $r$ and $t$.

Mapping function $r(s,t)$. We find that $r(s,t)$ via constant decrement in $\eta_{t}=\sigma_{t}/\alpha_{t}$ works well where the decrement is chosen in the form of $(\eta_{\text{max}}-\eta_{\text{min}})/2^{k}$ for some appropriate $k$ (details in Appendix C.7).

Weighting $w(s,t)$ and distribution $p(s,t)$. We follow VDM (Kingma et al., 2021; Kingma & Gao, 2024) and define $p(t)={\mathcal{U}}(\epsilon,T)$ and $p(s|t)={\mathcal{U}}(\epsilon,t)$ for constants $\epsilon,T\in$. Similarly, weighting is defined as

3 Sampling

Restart sampling. Similar to Xu et al. (2023); Song et al. (2023), one can introduce stochasticity during sampling by re-noising a sample to a higher noise-level before sampling again. For example, a two-step restart sampler from ${\mathbf{x}}_{t}$ requires $s\in(0,t)$ for drawing sample $\hat{{\mathbf{x}}}={\bm{f}}_{0,s}^{\theta}({\mathbf{x}}_{s})$ where ${\mathbf{x}}_{s}\sim q_{s}({\mathbf{x}}_{s}|{\bm{f}}_{0,t}^{\theta}({\mathbf{x}}_{t})).$

Classifier-free guidance. Given a data-label pair $({\mathbf{x}},{\mathbf{c}})$, during inference time, classifier-free guidance (Ho & Salimans, 2022) with weight $w$ replaces conditional model output ${\bm{G}}_{\theta}({\mathbf{x}}_{t},s,t,{\mathbf{c}})$ by a reweighted model output via

where $\varnothing$ denotes the null-token indicating unconditional output. Similarly, we define our guided model as ${\bm{f}}_{s,t,w}^{\theta}({\mathbf{x}}_{t})=c_{\text{skip}}(s,t){\mathbf{x}}_{t}+c_{\text{out}}(s,t){\bm{G}}_{\theta}^{w}({\mathbf{x}}_{t},s,t,{\mathbf{c}})$ where ${\bm{G}}_{\theta}^{w}({\mathbf{x}}_{t},s,t,{\mathbf{c}})$ is as defined in Eq. (14) and we drop $c_{\text{in}}(\cdot)$ and $c_{\text{noise}}(\cdot)$ for notational simplicity. We justify this decision in Appendix E. Similar to diffusion models, ${\mathbf{c}}$ is randomly dropped with probability $p$ during training without special practices.

We present pushforward sampling in Algorithm 2 and detail both samplers in Appendix F.

Connection with Prior Works

Our work is closely connected with many prior works. Detailed analysis is found in Appendix G.

We show in the following Lemma that CM objective with ${\mathcal{L}}_{2}$ distance is a single-particle estimate of IMM objective with energy kernel.

This single-particle estimate ignores the repulsion force imposed by $k(\cdot,\cdot)$. Energy kernel also only matches the first moment, ignoring all higher moments. These decisions can be significant contributors to training instability and performance degradation of CMs.

Improved CMs (Song & Dhariwal, 2023) propose pseudo-huber loss as $d(\cdot,\cdot)$ which we justify in the Lemma below.

Negative pseudo-huber loss $k_{c}(x,y)=c-\sqrt{\norm{x-y}^{2}+c^{2}}$ for $c>0$ is a conditionally positive definite kernel that matches all moments of $x$ and $y$ where weights on higher moments depend on $c$.

From a moment-matching perspective, the improved performance is explained by the loss matching all moments of the distributions. In addition to pseudo-huber loss, many other kernels (Laplace, RBF, etc.) are all valid choices in the design space.

We also extend IMM loss to the differential limit by taking $r(s,t)\rightarrow t$, the result of which subsumes the continuous-time CM (Lu & Song, 2024) as a single-particle estimate (Appendix H). We leave experiments for this to future work.

Generative Moment Matching Network. GMMN (Li et al., 2015) directly applies MMD to train a generator ${\bm{G}}_{\theta}({\mathbf{z}})$ where ${\mathbf{z}}\sim{\mathcal{N}}(0,{I})$ to match the data distribution. It is a special case of IMM in that when $t=1$ and $r(s,t)\equiv s=0$ our loss reduces to naïve GMMN objective.

Related Works

Diffusion, Flow Matching, and stochastic interpolants. Diffusion models (Sohl-Dickstein et al., 2015; Song et al., 2020b; Ho et al., 2020; Kingma et al., 2021) and Flow Matching (Lipman et al., 2022; Liu et al., 2022) are widely used generative frameworks that learn a score or velocity field of a noising process from data into a simple prior. They have been scaled successfully for text-to-image (Rombach et al., 2022; Saharia et al., 2022; Podell et al., 2023; Chen et al., 2023; Esser et al., 2024) and text-to-video (Ho et al., 2022a; Blattmann et al., 2023; OpenAI, 2024) tasks. Stochastic interpolants (Albergo et al., 2023; Albergo & Vanden-Eijnden, 2022) extend these ideas by explicitly defining a stochastic path between data and prior, then matching its velocity to facilitate distribution transfer. While IMM builds on top of the interpolant construction, it directly learns one-step mappings between any intermediate marginal distributions.

Diffusion distillation. To resolve diffusion models’ sampling inefficiency, recent methods (Salimans & Ho, 2022; Meng et al., 2023; Yin et al., 2024; Zhou et al., 2024; Luo et al., 2024a; Heek et al., 2024) focus on distilling one-step or few-step models from pre-trained diffusion models. Some approaches (Yin et al., 2024; Zhou et al., 2024) propose jointly optimizing two networks but the training requires careful tuning in practice and can lead to mode collapse (Yin et al., 2024). Another recent work (Salimans et al., 2024) explicitly matches the first moment of the data distribution available from pre-trained diffusion models. In contrast, our method implicitly matches all moments using MMD and can be trained from scratch with a single model.

Few-step generative models from scratch. Early one-step generative models primarily relied on GANs (Goodfellow et al., 2020; Karras et al., 2020; Brock, 2018) and MMD (Li et al., 2015, 2017) (or their combination) but scaling adversarial training remains challenging. Recent independent classes of few-step models, e.g. Consistency Models (CMs) (Song et al., 2023; Song & Dhariwal, 2023; Lu & Song, 2024), Consistency Trajectory Models (CTMs) (Kim et al., 2023; Heek et al., 2024) and Shortcut Models (SMs) (Frans et al., 2024) still face training instability and require specialized components (Lu & Song, 2024) (e.g., JVP for flash attention) or other special practices (e.g., high weight decay for SMs, combined LPIPS (Zhang et al., 2018) and GAN losses for CTMs, and special training schedules (Geng et al., 2024)) to remain stable. In contrast, our method trains stably with a single loss and achieves strong performance without special training practices.

Family Method FID$(\downarrow)$ Steps $(\downarrow)$ #Params GAN BigGAN (Brock, 2018) 6.95 1 112M GigaGAN (Kang et al., 2023) 3.45 1 569M StyleGAN-XL (Karras et al., 2020) 2.30 1 166M Masked & AR VQGAN (Esser et al., 2021) 26.52 1024 227M MaskGIT (Chang et al., 2022) 6.18 8 227M MAR (Li et al., 2024) 1.98 100 400M VAR-$d20$ (Tian et al., 2024a) 2.57 10 600M VAR-$d30$ (Tian et al., 2024a) 1.92 10 2B Diffusion & Flow ADM (Dhariwal & Nichol, 2021) 10.94 250 554M CDM (Ho et al., 2022b) 4.88 8100 - SimDiff (Hoogeboom et al., 2023) 2.77 512 2B LDM-4-G (Rombach et al., 2022) 3.60 250 400M U-DiT-L (Tian et al., 2024b) 3.37 250 916M U-ViT-H (Bao et al., 2023) 2.29 50 501M DiT-XL/2 ($w=1.0$) (Peebles & Xie, 2023) 9.62 250 675M DiT-XL/2 ($w=1.25$) (Peebles & Xie, 2023) 3.22 250 675M DiT-XL/2 ($w=1.5$) (Peebles & Xie, 2023) 2.27 250 675M SiT-XL/2 ($w=1.0$) (Ma et al., 2024) 9.35 250 675M SiT-XL/2 ($w=1.5$) (Ma et al., 2024) 2.15 250 675M Few-Step from Scratch iCT (Song et al., 2023) 34.24 1 675M 20.3 2 675M Shortcut (Frans et al., 2024) 10.60 1 675M 7.80 4 675M 3.80 128 675M IMM (ours) (XL/2, $w=1.25$) 7.77 1 675M 5.33 2 675M 3.66 4 675M 2.77 8 675M IMM (ours) (XL/2, $w=1.5$) 8.05 1 675M 3.99 2 675M 2.51 4 675M 1.99 8 675M

Experiments

We evaluate IMM’s empirical performance (Section 7.1), training stability (Section 7.2), sampling choices (Section 7.3), scaling behavior (Section 7.4) and ablate our practical decisions (Section 7.5).

We present FID (Heusel et al., 2017) results for unconditional CIFAR-10 and class-conditional ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$ in Table 1 and 2. For CIFAR-10, we separate baselines into diffusion and flow models, distillation models, and few-step models from scratch. IMM belongs to the last category in which it achieves state-of-the-art performance of 1.98 using pushforward sampler. For ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$, we use the popular DiT (Peebles & Xie, 2023) architecture because of its scalability, and compare it with GANs, masked and autoregressive models, diffusion and flow models, and few-step models trained from scratch.

We observe decreasing FID with more steps and IMM achieves 1.99 FID with 8 steps (with $w=1.5$), surpassing DiT and SiT (Ma et al., 2024) using the same architecture except for trivially injecting time $s$ (see Appendix I). Notably, we also achieve better 8-step FID than the 10-step VAR (Tian et al., 2024a) of comparable size. At 16 steps, IMM also achieves 1.90 FID outperforming VAR’s 2B variant (see Appendix I.4). However, different from VAR, IMM grants flexibility of variable number of inference steps and the large improvement in FID from 1 to 8 steps additionally demonstrates IMM’s efficient inferece-time scaling capability. Lastly, we similarly surpass Shortcut models’ (Frans et al., 2024) best performance with only 8 steps. We defer inference details to Section 7.3 and Appendix I.2.

2 IMM Training is Stable

We show that IMM is stable and achieves reasonable performance across a range of parameterization choices.

Positional vs. Fourier embedding. A known issue for CMs (Song et al., 2023) is its training instability when using Fourier embedding with scale $16$, which forces reliance on positional embeddings for stability. We find that IMM does not face this problem (see Figure 4). For Fourier embedding we use the standard NCSN++ (Song et al., 2020b) architecture and set embedding scale to $16$; for positional embeddings, we adopt DDPM++ (Song et al., 2020b). Both embedding types converge reliably, and we include samples from the Fourier embedding model in Figure 4.

Particle number. Particle number $M$ for estimating MMD is an important parameter for empirical success (Gretton et al., 2012; Li et al., 2015), where the estimate is more accurate with larger $M$. In our case, naïvely increasing $M$ can slow down convergence because we have a fixed batch size $B$ in which the samples are grouped into $B/M$ groups of $M$ where each group shares the same $t$. The larger $M$ means that fewer $t$’s are sampled. On the other hand, using extremely small numbers of particles, e.g. $M=2$, leads to training instability and performance degradation, especially on a large scale with DiT architectures. We find that there exists a sweet spot where a few particles effectively help with training stability while further increasing $M$ slows down convergence (see Figure 5). We see that in ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$, training collapses when $M=1$ (which is CM) and $M=2$, and achieves lowest FID under the same computation budget with $M=4$. We hypothesize $M<4$ does not allow sufficient mixing between particles and larger $M$ means fewer $t$’s are sampled for each step, thus slowing convergence. A general rule of thumb is to use a large enough $M$ for stability, but not too large for slowed convergence.

Noise embedding $c_{\text{noise}}(\cdot)$. We plot in Figure 9 the log absolute mean difference of $t$ and $r(s,t)$ in the positional embedding space. Increasing $c$ increases distinguishability of nearby distributions. We also observe similar convergence on ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$ across different $c$, demonstrating the insensitivity of our framework w.r.t. noise function.

3 Sampling

We investigate different sampling settings for best performance. One-step sampling is performed by simple pushforward from $T$ to $\epsilon$ (concrete values in Appendix I.2). On CIFAR-10 we use 2 steps and set intermediate time $t_{1}$ such that $\eta_{t_{1}}=1.4$, a choice we find to work well empirically. On ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$ we go beyond 2 steps and, for simplicity, investigate (1) uniform decrement in $t$ and (2) EDM (Karras et al., 2024) schedule (detailed in Appendix I.2). We plot FID of all sampler settings in Figure 6 with guidance weight $w=1.5$. We find pushforward samplers with uniform schedule to work the best on ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$ and use this as our default setting for multi-step generation. Additionally, we concede that pushforward combined with restart samplers can achieve superior results. We leave such experiments to future works.

4 Scaling Behavior

Similar to diffusion models, IMM scale with training and inference compute as well as model size on ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$. We plot in Figure 7 FID vs. training and inference compute in GFLOPs and we find strong correlation between compute used and performance. We further visualize samples in Figure 8 with increasing model size, i.e. DiT-S, DiT-B, DiT-L, DiT-XL, and increasing inference steps, i.e. 1, 2, 4, 8 steps. The sample quality increases along both axes as larger transformers with more inference steps capture more complex distributions. This also explains that more compute can sometimes yield different visual content from the same initial noise as shown in the visual results.

5 Ablation Studies

All ablation studies are done with DDPM++ architecture for CIFAR-10 and DiT-B for ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$. FID comparisons use 2-step samplers by default.

Flow schedules and parameterization. We investigate all combinations of network parameterization and flow schedules: Simple-EDM + cosine (sEDM/cos), Simple-EDM + OT-FM (sEDM/FM), Euler-FM + OT-FM (eFM), Identity + cosine (id/cos), Identity + OT-FM (id/FM). Identity parameterization consistently fall behind other types of parameterization, which all show similar performance across datasets (see Table 3). We see that on smaller scale (CIFAR-10), sEDM/FM works the best but on larger scale (ImageNet-$256{\mkern-1.0mu\times\mkern-1.0mu}256$), eFM works the best, indicating that OT-FM schedule and Euler paramaterization may be more scalable than other choices.

Mapping function $r(s,t)$. Our choices for ablation are (1) constant decrement in $\eta_{t}$, (2) constant decrement in $t$, (3) constant decrement in $\lambda_{t}=\log(\alpha_{t}^{2}/\sigma_{t}^{2})$, (4) constant increment in $1/\eta_{t}$ (see Appendix C.6). For fair comparison, we choose the decrement gap so that the minimum $t-r(s,t)$ is $\approx 10^{-3}$ and use the same network parameterization. FID progression in Figure 11 show that (1) consistently outperforms other choices. We additionally ablate the mapping gap using $M=4$ in (1). The constant decrement is in the form of $(\eta_{\text{max}}-\eta_{\text{min}})/2^{k}$ for an appropriately chosen $k$. We show in Figure 10 that the performance is relatively stable across $k\in\{11,12,13\}$ but experiences instability for $k=14$. This suggests that, for a given particle number, there exists a largest $k$ for stable optimization.

Conclusion

We present Inductive Moment Matching, a framework that learns a few-step generative model from scratch. It trains by first leveraging self-consistent interpolants to interpolate between data and prior and then matching all moments of its own distribution interpolated to be closer to that of data. Our method guarantees convergence in distribution and generalizes many prior works. Our method also achieves state-of-the-art performance across benchmarks while achieving orders of magnitude faster inference. We hope it provides a new perspective on training few-step models from scratch and inspire a new generation of generative models.

Impact Statement

This paper advances research in diffusion models and generative AI, which enable new creative possibilities and democratize content creation but also raise important considerations. Potential benefits include expanding artistic expression, assisting content creators, and generating synthetic data for research. However, we acknowledge challenges around potential misuse for deepfakes, copyright concerns, and impacts on creative industries. While our work aims to advance technical capabilities, we encourage ongoing discussion about responsible development and deployment of these technologies.

Acknowledgement

We thank Wanqiao Xu, Bokui Shen, Connor Lin, and Samrath Sinha for additional technical discussions and helpful suggestions.

References

Appendix A Background: Properties of Stochastic Interpolants

We note some relevant properties of stochastic interpolants for our exposition.

Boundary satisfaction. For an interpolant distribution $q_{t}({\mathbf{x}}_{t}|{\mathbf{x}},\bm{\epsilon})$ defined in Albergo et al. (2023), and the marginal $q_{t}({\mathbf{x}}_{t})$ as defined in Eq. (2), we can check that $q_{1}({\mathbf{x}}_{1})=p({\mathbf{x}}_{1})$ and $q_{0}({\mathbf{x}}_{0})=q({\mathbf{x}}_{0})$ so that ${\mathbf{x}}_{1}=\bm{\epsilon}$ and ${\mathbf{x}}_{0}={\mathbf{x}}$.

Joint distribution. The joint distribution of ${\mathbf{x}}$ and ${\mathbf{x}}_{t}$ is written as

in which case ${\mathbf{x}}_{1}=\bm{\epsilon}$.

Appendix B Theorems and Derivations

Assuming marginal-preserving interpolant and metric $D(\cdot,\cdot)$, a minimizer $\theta^{*}$ of Eq. (7) exists, i.e. $p_{s|t}^{\theta^{*}}({\mathbf{x}}|{\mathbf{x}}_{t})=q_{t}({\mathbf{x}}|{\mathbf{x}}_{t})$, and the minimum is 0.

We directly substitute $q_{t}({\mathbf{x}}|{\mathbf{x}}_{t})$ into the objective to check. First,

where $(a)$ is due to definition of marginal preservation. So the objective becomes

In general, the minimizer $q_{t}({\mathbf{x}}|{\mathbf{x}}_{t})$ exists. However, this does not show that the minimizer is unique. In fact, the minimizer is not unique in general because a deterministic minimizer can also exist under certain assumptions on the interpolant (see Appendix B.6). ∎

Failure Case without Marginal Preservation. We additionally show that marginal-preservation property of the interpolant $q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})$ is important for the naïve objective in Eq. (7) to attain 0 loss (Lemma 3). Consider the failure case below where the constructed interpolant is a generalized interpolant but not necessarily marginal-preserving. Then we show that there exists a $t$ such that $p_{s|t}^{\theta}({\mathbf{x}}_{s})$ can never reach $q_{s}({\mathbf{x}}_{s})$ regardless of $\theta$.

Let $q({\mathbf{x}})=\delta({\mathbf{x}})$, $p(\bm{\epsilon})=\delta(\bm{\epsilon}-1)$, and suppose an interpolant ${\bm{I}}_{s|t}({\mathbf{x}},{\mathbf{x}}_{t})=(1-\frac{s}{t}){\mathbf{x}}+\frac{s}{t}{\mathbf{x}}_{t}$ and $\gamma_{s|t}=\frac{s}{t}\sqrt{1-\frac{s}{t}}\mathmybb{1}(t<1)$, then $D(q_{s}({\mathbf{x}}_{s}),p_{s|t}^{\theta}({\mathbf{x}}_{s}))>0$ for all $0<s<t<1$ regardless of the learned distribution $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})$ given any metric $D(\cdot,\cdot)$.

This example first implies the learning target

is a delta distribution. However, we show that if we select any $t<1$ and $0<s<t$, $p_{s|t}^{\theta}({\mathbf{x}}_{s})$ can never be a delta distribution.

Now, we show the model distribution has non-zero variance under these choices of $t$ and $s$. Expectations are over $p_{s|t}^{\theta}({\mathbf{x}}_{s})$ or conditional interpolant $q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})$ for all equations below.

B.2 Boundary Satisfaction of Model Distribution

The operator output $p_{s|t}^{\theta}({\mathbf{x}}_{s})$ satisfies boundary condition.

For all $s\in$ and all $\theta$, the following boundary condition holds.

B.3 Definition of Well-Conditioned r​(s,t)r(s,t)

For simplicity, the mapping function $r(s,t)$ is well-conditioned if

where $\Delta(t)\geq\epsilon>0$ is a positive function such that $r(s,t)$ is increasing for $t\geq s+c_{0}(s)$ where $c_{0}(s)$ is the largest $t$ that is mapped to $s$. Formally, $c_{0}(s)=\sup\{t:r(s,t)=s\}$. For $t\geq s+c_{0}(s)$, the inverse w.r.t. $t$ exists, i.e. $r^{-1}(s,\cdot)$ and $r^{-1}(s,r(s,t))=t$. All practical implementations follow this general form, and are detailed in Appendix C.6.

B.4 Main Theorem

We prove by induction on sequence number $n$. First, $r(s,t)$ is well-conditioned by following the definition in Eq. (46). Furthermore, for notational convenience, we let $r_{n}^{-1}(s,\cdot)\vcentcolon=r^{-1}(s,r^{-1}(s,r^{-1}(s,\dots)))$ be $n$ nested application of $r^{-1}(s,\cdot)$ on the second argument. Additionally, $r_{0}^{-1}(s,t)=t$.

Base case: $n=1$. Given any $s\geq 0$, $r(s,u)=s$ for all $s<u\leq c_{0}(s)$, implying

for $u\leq c_{0}(s)$ where $(a)$ is implied by Lemma 4 and $(b)$ is implied by Lemma 3.

Inductive assumption: $n-1$. We assume $p_{s|u}^{\theta_{n-1}^{*}}({\mathbf{x}}_{s})=q_{s}({\mathbf{x}}_{s})$ for all $s\leq u\leq r_{n-2}^{-1}(s,c_{0}(s))$.

We inspect the target distribution $p_{s|r(s,u)}^{\theta_{n-1}^{*}}({\mathbf{x}}_{s})$ in Eq. (8) if optimized on $s\leq u\leq r_{n-1}^{-1}(s,c_{0}(s))$. On this interval, we can apply $r(s,\cdot)$ to the inequality and get $s=r(s,s)\leq r(s,u)\leq r(s,r_{n-1}^{-1}(s,c_{0}(s)))=r_{n-2}^{-1}(s,c_{0}(s))$ since $r(s,\cdot)$ is increasing. And by inductive assumption $p_{s|r(s,u)}^{\theta_{n-1}^{*}}({\mathbf{x}}_{s})=q_{s}({\mathbf{x}}_{s})$ for $s\leq r(s,u)\leq r_{n-2}^{-1}(s,c_{0}(s))$, this implies minimizing

on $s\leq u\leq r_{n-1}^{-1}(s,c_{0}(s))$ is equivalent to minimizing

for $s\leq u\leq r_{n-1}^{-1}(s,c_{0}(s))$. Lemma 3 implies that its minimum achieves $p_{s|u}^{\theta_{n}^{*}}({\mathbf{x}}_{s})=q_{s}({\mathbf{x}}_{s})$.

B.5 Self-Consistency Implies Marginal Preservation

Without assuming marginal preservation, it is important to define the marginal distribution of ${\mathbf{x}}_{s}$ under generalized interpolants $q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})$ as

and we show that with self-consistent interpolants, this distribution is invariant of $t$, i.e. $q_{s|t}({\mathbf{x}}_{s})=q_{s}({\mathbf{x}}_{s})$.

If the interpolant $q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})$ is self-consistent, the marginal distribution $q_{s|t}({\mathbf{x}}_{s})$ as defined in Eq. (49) satisfies $q_{s}({\mathbf{x}}_{s})=q_{s|t}({\mathbf{x}}_{s})$ for all $t\in[s,1]$.

where $(a)$ uses definition of self-consistent interpolants and $(b)$ uses definition of our generalized interpolant. ∎

We show in Appendix C.1 that DDIM is an example self-consistent interpolant. Furthermore, DDPM posteior (Ho et al., 2020; Kingma et al., 2021) is also self-consistent (see Lemma 6).

B.6 Existence of Deterministic Minimizer

We present the formal statement for the deterministic minimizer.

which pushes forward the measure $q_{t}({\mathbf{x}}_{t})$ to $q_{s}({\mathbf{x}}_{s})$. We define:

Then, since $\gamma_{s|t}\equiv 0$, $q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})=\delta({\mathbf{x}}_{s}-{\bm{I}}_{s|t}({\mathbf{x}},{\mathbf{x}}_{t}))$ where ${\mathbf{x}}\sim\delta({\mathbf{x}}-h_{s|t}({\mathbf{x}}_{t}))$. Therefore,

whose marginal follows $q_{s}({\mathbf{x}}_{s})$ due to it being the result of PF-ODE trajectories starting from $q_{t}({\mathbf{x}}_{t})$. ∎

Concretely, DDIM interpolant satisfies all of the deterministic assumption, the regularity condition, and the invertibility assumption because it is a linear function of ${\mathbf{x}}$ and ${\mathbf{x}}_{t}$. Therefore, any diffusion or FM schedule with DDIM interpolant will enjoy a deterministic minimizer $p_{s|t}^{\theta}({\mathbf{x}}|{\mathbf{x}}_{t})$.

Appendix C Analysis of Simplified Parameterization

We check that DDIM interpolant is self-consistent. By definition, $q_{s|t}({\mathbf{x}}_{s}|{\mathbf{x}},{\mathbf{x}}_{t})=\delta({\mathbf{x}}_{s}-\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},t,s))$. We check that for all $s\leq r\leq t$,

Therefore, $\delta({\mathbf{x}}_{s}-\operatorname{DDIM}(\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},t,r),{\mathbf{x}},r,s))=\delta({\mathbf{x}}_{s}-\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},t,s))$. So DDIM is self-consistent.

It also implies a Gaussian forward process $q_{t}({\mathbf{x}}_{t}|{\mathbf{x}})={\mathcal{N}}(\alpha_{t}{\mathbf{x}},\sigma_{t}^{2}\sigma_{d}^{2}{I})$ as in diffusion models. By definition,

so that ${\mathbf{x}}_{t}$ is a deterministic transform given ${\mathbf{x}}$ and $\bm{\epsilon}$, i.e., ${\mathbf{x}}_{t}=\operatorname{DDIM}(\bm{\epsilon},{\mathbf{x}},t,1)=\alpha_{t}{\mathbf{x}}+\sigma_{t}\bm{\epsilon}$, which implies $q_{t}({\mathbf{x}}_{t}|{\mathbf{x}})={\mathcal{N}}(\alpha_{t}{\mathbf{x}},\sigma_{t}^{2}\sigma_{d}^{2}{I})$.

C.2 Reusing 𝐱t{\mathbf{x}}_{t} for 𝐱r{\mathbf{x}}_{r}

We propose that instead of sampling ${\mathbf{x}}_{r}$ via forward flow $\alpha_{r}{\mathbf{x}}+\sigma_{r}\bm{\epsilon}$ we reuse ${\mathbf{x}}_{t}$ such that ${\mathbf{x}}_{r}=\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},r,t)$ to reduce variance. In fact, for any self-consistent interpolant, one can reuse ${\mathbf{x}}_{t}$ via ${\mathbf{x}}_{r}\sim q_{r|t}({\mathbf{x}}_{r}|{\mathbf{x}},{\mathbf{x}}_{t})$ and ${\mathbf{x}}_{r}$ will follow $q_{r}({\mathbf{x}}_{r})$ marginally. We check

where $(a)$ is due to Lemma 5. We can see that sampling ${\mathbf{x}},{\mathbf{x}}_{t}$ first then ${\mathbf{x}}_{r}\sim q_{r|t}({\mathbf{x}}_{r}|{\mathbf{x}},{\mathbf{x}}_{t})$ respects the marginal distribution $q_{r}({\mathbf{x}}_{r})$.

C.3 Simplified Objective

We derive our simplified objective. Given MMD defined in Eq. (1), we write our objective as

where $\langle\cdot,\cdot\rangle$ is in RKHS, $(a)$ is due to the correlation between ${\mathbf{x}}_{r}$ and ${\mathbf{x}}_{t}$ by re-using ${\mathbf{x}}_{t}$.

C.4 Empirical Estimation

As proposed in Gretton et al. (2012), MMD is typically estimated with V-statistics by instantiating a matrix of size $M\times M$ such that a batch of $B$ ${\mathbf{x}}$ samples, $\{x^{(i)}\}_{i=1}^{B}$, is separated into groups of $M$ (assume $B$ is divisible by $M$) particles $\{x^{(i,j)}\}_{i=1,j=1}^{B/M,M}$ where each group share a $(s^{i},r^{i},t^{i})$ sample. The Monte Carlo estimate becomes

Computational efficiency. First we note that regardless of $M$, we require only 2 model forward passes - one with and one without stop gradient, since the model takes in all $B$ instances together within the batch and produce outputs for the entire batch. For the calculation of our loss, although the need for $M$ particles may imply inefficient computation, the cost of this matrix computation is negligible in practice compared to the complexity of model forward pass. Suppose a forward pass for a single instance is ${\mathcal{O}}(K)$, then the total computation for computation loss for a batch of $B$ instances is ${\mathcal{O}}(BK)+{\mathcal{O}}(BM)$. Deep neural networks often has $K\gg M$, so ${\mathcal{O}}(BK)$ dominates the computation.

C.5 Simplified Parameterization

We derive ${\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})$ for each parameterization, which now generally follows the form

Identity. This is simply DDIM with $x$-prediction network.

When noise schedule is cosine, ${\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})=\cos(\frac{1}{2}\pi(t-s)){\mathbf{x}}_{t}-\sin(\frac{1}{2}\pi(t-s))\sigma_{d}{\bm{G}}_{\theta}\left(\frac{{\mathbf{x}}_{t}}{\sigma_{d}\sqrt{\alpha_{t}^{2}+\sigma_{t}^{2}}},c_{\text{noise}}(s),c_{\text{noise}}(t)\right)$. And similar to Lu & Song (2024), we can show that predicting ${\mathbf{x}}_{s}=\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},s,t)$ with ${\mathcal{L}}_{2}$ loss is equivalent to $v$-prediction with cosine schedule.

This reduces to $v$-target if cosine schedule is used, and it deviates from $v$-target if FM schedule is used instead.

This results in Euler ODE from ${\mathbf{x}}_{t}$ to ${\mathbf{x}}_{s}$. We also show that the network output reduces to $v$-prediction if matched with ${\mathbf{x}}_{s}=\operatorname{DDIM}({\mathbf{x}}_{t},{\mathbf{x}},s,t)$. To see this,

which is $v$-target under OT-FM schedule. This parameterization naturally allows zero-SNR sampling and satisfies boundary condition at $s=0$, similar to Simple-EDM above. This is not true for Identity parametrization using ${\bm{G}}_{\theta}$ as it satisfies boundary condition only at $s>0$.

C.6 Mapping Function r​(s,t)r(s,t)

We discuss below the concrete choices for $r(s,t)$. We use a constant decrement $\epsilon>0$ in different spaces.

Constant decrement in $\eta(t)\vcentcolon=\eta_{t}=\sigma_{t}/\alpha_{t}$. This is the choice that we find to work better than other choices in practice. First, let its inverse be $\eta^{-1}(\cdot)$,

We choose $\epsilon=(\eta_{\text{max}}-\eta_{\text{min}})/2^{k}$ for some $k$. We generally choose $\eta_{\text{max}}\approx 160$ and $\eta_{\text{min}}\approx 0$. We find $k=\{10,\dots,15\}$ works well enough depending on datasets.

Constant decrement in $\lambda(t)\vcentcolon=\text{log-SNR}_{t}=2\log(\alpha_{t}/\sigma_{t})$.

Let its inverse be $\lambda^{-1}(\cdot)$, then

We choose $\epsilon=(\lambda_{\text{max}}-\lambda_{\text{min}})/2^{k}$. This choice comes close to the first choice, but we refrain from this because $r(s,t)$ becomes close to $t$ both when $t\approx 0$ and $t\approx 1$ instead of just $t\approx 1$. This gives more chances for training instability than the first choice.

We choose $\epsilon=(1/\eta(t)_{\text{min}}-1/\eta(t)_{\text{max}})/2^{k}$.

C.7 Time Distribution p​(s,t)p(s,t)

In all cases we choose $p(t)={\mathcal{U}}(\epsilon,T)$ and $p(s|t)={\mathcal{U}}(\epsilon,t)$ for some $\epsilon\geq 0$ and $T\leq 1$. The decision for time distribution is coupled with $r(s,t)$. We list the constraints on $p(s,t)$ for each $r(s,t)$ choice below.

Constant decrement in $\eta(t)$. We need to choose $T<1$ because, for example, assuming OT-FM schedule, $\eta_{t}=t/(1-t)$, one can observe that constant decrement in $\eta_{t}$ when $t\approx 1$ results in $r(s,t)$ that is too close to $t$ due to $\eta_{t}$’s exploding gradient around $1$. We need to define $T<1$ such that $r(s,T)$ is not too close to $T$ for $s$ reasonably far away. With $\eta_{\text{max}}\approx 160$, we can choose $T=0.994$ for OT-FM and $T=0.996$ for VP-diffusion.

Constant decrement in $t$. No constraints needed. $T=1$, $\epsilon=0$.

Constant decrement in $\lambda_{t}$. One can similarly observe exploding gradient causing $r(s,t)$ to be too close to $t$ at both $t\approx 0$ and $t\approx 1$, so we can choose $\epsilon>0$, e.g. $0.001$, in addition to choosing $T=0.994$ for OT-FM and $T=0.996$ for VP-diffusion.

Constant increment in $1/\eta_{t}$. This experience exploding gradient for $t\approx 0$, so we require $\epsilon>0$, e.g. $0.005$. And $T=1$.

C.8 Kernel Function

whose magnitude can vary a lot depending on how far $x$ is from $y$.

C.9 Weighting Function w​(s,t)w(s,t)

To review VDM (Kingma et al., 2021), the negative ELBO loss for diffusion model is

where $\bm{\epsilon}_{\theta}$ is the noise-prediction network and $\lambda_{t}=\text{log-SNR}_{t}$. The weighted-ELBO loss proposed in Kingma & Gao (2024) introduces an additional weighting function $w(t)$ monotonically increasing in $t$ (monotonically decreasing in $\text{log-SNR}_{t}$) understood as a form of data augmentation. Specifically, they use sigmoid as the function such that the weighted ELBO is written as

where $\sigma(\cdot)$ is sigmoid function.

The $\alpha_{t}$ is tailored towards the Simple-EDM and Euler-FM parameterization as we have shown in Appendix C.5 that the networks $\sigma_{d}{\bm{G}}_{\theta}$ amounts to $v$-prediction in cosine and OT-FM schedules. Notice that ELBO diffusion loss matches $\bm{\epsilon}$ instead of ${\mathbf{v}}$. Inspecting the gradient of Laplace kernel, we have (again, for simplicity we let $\hat{{\bm{G}}}_{\theta}({\mathbf{x}}_{t},s,t)={\bm{G}}_{\theta}(\frac{{\mathbf{x}}_{t}}{\sigma_{d}\sqrt{\alpha_{t}^{2}+\sigma_{t}^{2}}},c_{\text{noise}}(s),c_{\text{noise}}(t))$)

for some constant $\hat{{\bm{G}}}_{\text{target}}$. We can see that gradient $\partialderivative{\theta}{\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})$ is guided by vector $\hat{{\bm{G}}}_{\theta}({\mathbf{x}}_{t},s,t)-\hat{{\bm{G}}}_{\text{target}}$. Assuming $\hat{{\bm{G}}}_{\theta}({\mathbf{x}}_{t},s,t)$ is $v$-prediction, as is the case for Simple-EDM parameterization with cosine schedule and Euler-FM parameterization with OT-FM schedule, we can reparameterize $v$- to $\epsilon$-prediction with $\bm{\epsilon}_{\theta}$ as the new parameterization. We omit arguments to network for simplicity.

We show below that for both cases $\bm{\epsilon}_{\theta}-\bm{\epsilon}_{\text{target}}=\alpha_{t}(\hat{{\bm{G}}}_{\theta}-\hat{{\bm{G}}}_{\text{target}})$ for some constants $\bm{\epsilon}_{\text{target}}$ and $\hat{{\bm{G}}}_{\text{target}}$. For Simple-EDM, we know $x$-prediction from $v$-prediction parameterization (Salimans & Ho, 2022), ${\mathbf{x}}_{\theta}=\alpha_{t}{\mathbf{x}}_{t}-\sigma_{t}\hat{{\bm{G}}}_{\theta}$, and we also know $x$-prediction from $\epsilon$-prediction, ${\mathbf{x}}_{\theta}=({\mathbf{x}}_{t}-\sigma_{t}\bm{\epsilon}_{\theta})/\alpha_{t}$. We have

For Euler-FM, we know $x$-prediction from $v$-prediction parameterization, ${\mathbf{x}}_{\theta}={\mathbf{x}}_{t}-t\hat{{\bm{G}}}_{\theta}$ and we also know $x$-prediction from $\epsilon$-prediction, ${\mathbf{x}}_{\theta}=({\mathbf{x}}_{t}-t\bm{\epsilon}_{\theta})/(1-t)$. We have

In both cases, $(\hat{{\bm{G}}}_{\theta}({\mathbf{x}}_{t},s,t)-\hat{{\bm{G}}}_{\text{target}})$ can be rewritten to $(\bm{\epsilon}_{\theta}({\mathbf{x}}_{t},s,t)-\bm{\epsilon}_{\text{target}})$ by multiplying a factor $\alpha_{t}$, and the guidance vector now matches that of the ELBO-diffusion loss. Therefore, we are motivated to incorporate $\alpha_{t}$ into $w(s,t)$ as proposed.

The exponent $a$ for $\alpha_{t}^{a}$ takes a value of either 1 or 2. We explain the reason for each decision here. When $a=1$, we guide the gradient $\partialderivative{\theta}{\bm{f}}_{s,t}^{\theta}({\mathbf{x}}_{t})$, with score difference $(\bm{\epsilon}_{\theta}({\mathbf{x}}_{t},s,t)-\bm{\epsilon}_{\text{target}})$. To motivate $a=2$, we first note that the weighted gradient

and as shown above that $\epsilon$-prediction is parameterized as

Multiplying an additional $\alpha_{t}$ to $\partialderivative{\theta}\hat{{\bm{G}}}_{\theta}({\mathbf{x}}_{t},s,t)$ therefore implicitly reparameterizes our model into an $\epsilon$-prediction model. The case of $a=2$ therefore implicitly reparameterizes our model into an $\epsilon$-prediction model guided by the score difference $(\bm{\epsilon}_{\theta}({\mathbf{x}}_{t},s,t)-\bm{\epsilon}_{\text{target}})$. Empirically, $\alpha_{t}^{2}$ additionally downweights loss for larger $t$ compared to $\alpha_{t}$, allowing the model to train on smaller time-steps more effectively.

Lastly, the division of $\alpha_{t}^{2}+\sigma_{t}^{2}$ is inspired by the increased weighting for middle time-steps (Esser et al., 2024) for Flow Matching training. This is purely an empirical decision.

Appendix D Training Algorithm

We present the training algorithm in Algorithm 3.

Appendix E Classifier-Free Guidance

We refer readers to Appendix C.5 for analysis of each parameterization. Most notably, the network ${\bm{G}}_{\theta}$ in both (1) Simple-EDM with cosine diffusion schedule and (2) Euler-FM with OT-FM schedule are equivalent to $v$-prediction parameterization in diffusion (Salimans & Ho, 2022) and FM (Lipman et al., 2022). When conditioned on label ${\mathbf{c}}$ during sampling, it is customary to use classifier-free guidance to reweight this $v$-prediction network via