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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Generator Matching without Generators</title>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script>
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</head>
<body>
<div class="progress-bar" id="progressBar"></div>
<div class="slideshow-container">
<!-- SLIDE 3: The Setup -->
<div class="slide" data-slide="3">
<div class="slide-content">
<h2>Generator Matching: The Setup</h2>
<ul>
<li><strong>Goal:</strong> Train a model to generate samples from a target distribution (images, text, proteins, etc.)</li>
<li><strong>Approach:</strong> Transform an easy-to-sample "prior" into the target through many small steps</li>
</ul>
<div style="text-align: center; margin: 1.5rem 0;">
<img width="750" alt="Flow from prior to target" src="https://github.com/user-attachments/assets/1d156b8b-68aa-47dd-a39d-5f0e4e2b4675" />
</div>
<ul>
<li><strong>Time $t$:</strong> Goes from $t=0$ (prior) to $t=1$ (target)</li>
<li><strong>State $x_t$:</strong> The model state at time $t$
<ul>
<li>$x_0$ = sample from prior (e.g., random noise)</li>
<li>$x_1$ = sample from target (e.g., real image)</li>
</ul>
</li>
</ul>
</div>
</div>
<!-- SLIDE 4: Conditional Process -->
<div class="slide" data-slide="4">
<div class="slide-content">
<h2>The Conditional Process</h2>
<p style="font-size: 1.2rem; margin-bottom: 1.5rem;">GM requires that you can:</p>
<ol>
<li>Sample a latent $z$ from some distribution $p(z)$, which controls where the process ends at $t=1$</li>
<li>Conditioned on $z$, sample $x_{t+\Delta t}$ from $p(x_{t+\Delta t} \mid x_t, z, t)$ for small $\Delta t$</li>
</ol>
<p style="font-size: 1.2rem;">This means we can sample from $p(x_t \mid z, t)$ by first sampling $x_0$ from the prior, and then stepping through $p(x_{t+\Delta t} \mid x_t, z, t)$ for each step until $t$ is reached.</p>
<h3>What is $z$?</h3>
<ul>
<li>Often just holds a single training sample $x_1^z$</li>
<li>But $z$ is flexible—can include other information controlling the path</li>
</ul>
<div class="key-point">
<strong>Key constraint:</strong> When stepping through time repeatedly, at $t=1$ the state reaches $x_1^z$—the training sample associated with $z$.
</div>
</div>
</div>
<!-- SLIDE 5: Marginals and Posterior -->
<div class="slide" data-slide="5">
<div class="slide-content">
<h2>Marginals and Posterior</h2>
<h3>The Marginal Distribution $p(x_t \mid t)$</h3>
<ul>
<li>Sample $z$ from $p(z)$, then sample $p(x_t \mid z, t)$</li>
<li>With $n$ training samples of equal probability, one per $z$:</li>
</ul>
<p style="font-size: 1.2rem; margin: 1.5rem 0;">$$p(x_t \mid t) = \frac{1}{n} \sum_{i=1}^n p(x_t \mid z_i, t)$$</p>
<ul>
<li><strong>At $t=1$:</strong> This marginal equals the training data distribution</li>
</ul>
<h3>The Posterior $p(z \mid x_t, t)$</h3>
<ul>
<li>By Bayes: $p(z \mid x_t, t) \propto p(x_t \mid z, t) p(z)$</li>
<li>We won't compute it explicitly, but need to understand what it means</li>
</ul>
</div>
</div>
<!-- SLIDE 7: Interactive Posterior Visualization -->
<div class="slide" data-slide="7">
<div class="slide-content">
<h2>The Posterior $p(z \mid x_t, t)$</h2>
<div class="interactive-figure">
<h4>$t$ (horizontal); $x_t$ (vertical)</h4>
<canvas id="posteriorCanvas" width="600" height="450"></canvas>
<div class="controls">
<div style="border-left:1px solid var(--border-color); padding-left:10px; margin-left:10px;">
<label>N: <input type="number" id="inputN" value="40" min="1" max="100"></label>
<button id="btnResampleData" class="secondary">Resample Points</button>
</div>
</div>
<div class="caption">
<span style="color:#53df5a; font-weight:bold;">Green Dot</span> = current $(t, x_t)$ |
<span style="color:#a5a5a5; font-weight:bold;">White Lines</span> opacity = posterior $p(z|x_t,t)$
</div>
</div>
</div>
</div>
<!-- SLIDE 8: Resample-z Process Intro -->
<div class="slide" data-slide="8">
<div class="slide-content">
<h2>The Resample-$z$ Process</h2>
<p style="font-size: 1.2rem; margin-bottom: 1rem;">The conditional process samples $x_t$ by first sampling $z$ from $p(z)$, then sampling $x_t$ from $p(x_t \mid z, t)$ by stepping from $x_0$.
Consider a modified process that every now and again:</p>
<blockquote>
<ol>
<li>Samples fresh $z_{\text{new}}$ from the posterior $p(z \mid x_t, t)$</li>
<li>Takes steps with $p(x_{t+\Delta t} \mid x_t, z_{\text{new}}, t)$</li>
</ol>
</blockquote>
<p style="font-size: 1.2rem; margin: 1.5rem 0;">At points along the sample trajectory you forget old $z$, redraw from posterior, and continue stepping.</p>
<div class="key-point">
<strong>Critically:</strong> The marginal distribution $p(x_t \mid t)$ is the same for both the original conditional process and the resample-$z$ process.
</div>
<p style="font-size: 1.2rem; margin: 1.5rem 0;">Why?
Under the conditional process, sampling $z$ then $x_t$ has the joint distribution:
$$p(x_t \mid z, t) p(z) = p(x_t, z \mid t)$$
If you discard $z$ you're in the marginal distribution $p(x_t \mid t)$, and if you resample $z$ from the posterior you get:
$$p(z \mid x_t, t) p(x_t \mid t) = p(x_t, z \mid t)$$
After resampling $z$ the joint distribution is unchanged, and the step is the same, so the distribution at $p(x_{t+\Delta t}, z \mid t+\Delta t)$ will agree as well.
So since they both start from $p(x_0)$ at $t=0$, the marginals $p(x_t \mid t)$ will match for all time steps.
</p>
</div>
</div>
<!-- SLIDE 10: Interactive Resampling -->
<div class="slide" data-slide="10">
<div class="slide-content">
<h2>The Resample-$z$ Process</h2>
<div class="interactive-figure">
<h4>Watch paths change as resampling interval decreases</h4>
<canvas id="simCanvas" width="800" height="400"></canvas>
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<button id="btnAutoPlay" class="secondary">▶</button>
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<input type="range" id="stepSizeSlider" min="1" max="100" value="100">
</label>
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</div>
<div class="caption" id="simCaption">
<span style="color:var(--accent-color); font-weight:bold;">Red Path</span> → <span style="color:var(--primary-color); font-weight:bold">Blue Target</span> |
Click to set starting point
</div>
</div>
</div>
</div>
<!-- SLIDE 11: Infinite Switching Limit -->
<div class="slide" data-slide="11">
<div class="slide-content">
<h2>The limit of the Resample-$z$ Process</h2>
<p style="font-size: 1.2rem; margin: 1.5rem 0;">With our pure-drift flow matching example:</p>
<ul>
<li><strong>Without resampling:</strong> Straight lines from $x_0$ to $x_1$ (i.e. the conditional process)</li>
<li><strong>With slow resampling:</strong> Jagged, zig-zagging paths</li>
<li><strong>With fast resampling:</strong> Law of large numbers kicks in</li>
</ul>
<p style="font-size: 1.2rem; margin: 1.5rem 0;">As switching rate → ∞, the path converges to a <em>smooth posterior-weighted average</em> over all $z$ values.
$$ \bar{u}_t(x) = \sum_{k=1}^N p(z_k\mid x,t)\,u_t(x \mid z_k) $$
<strong>We can train a model to directly learn this average.</strong></p>
</div>
</div>
<!-- SLIDE 12: Interactive Averaged Velocity -->
<div class="slide" data-slide="12">
<div class="slide-content">
<h2>Posterior-Averaged Velocity</h2>
<div class="interactive-figure">
<h4>The red arrow shows $\bar{u}_t(x)$ pointing to the weighted average $x_1$</h4>
<canvas id="posteriorCanvas3" width="600" height="450"></canvas>
<div class="controls">
<div style="border-left:1px solid var(--border-color); padding-left:10px; margin-left:10px;">
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<button id="btnResampleData3" class="secondary">Resample Points</button>
</div>
</div>
<div class="caption">
<span style="color:#81c784; font-weight:bold;">Green Dot</span> = $(t, x_t)$ |
<span style="color:#e5eef2; font-weight:bold;">White Lines</span> = posterior weights |
<span style="color:var(--accent-color); font-weight:bold;">Red Arrow</span> = averaged velocity to $t=1$
</div>
</div>
</div>
</div>
<!-- SLIDE 11: Infinite Switching Limit -->
<div class="slide" data-slide="11">
<div class="slide-content">
<h2>Generator Matching:</h2>
<ul>
<li>The marginal matching and switching-rate-limit arguments hold for other kinds of conditional process.</li>
<li>Use the conditional process to generate training samples.</li>
<li>Use a Bregman divergence to train a model $f_\theta(t, x)$ that directly learns the average step.</li>
<li>Generate new samples by stepping from $x_0$ using the learned model.</li>
</ul>
</div>
</div>
<!-- SLIDE 15: Different State Spaces -->
<div class="slide" data-slide="15">
<div class="slide-content">
<h2>Different State Spaces</h2>
<div class="two-col">
<div>
<h3>Case I: Pure Drift (Flow)</h3>
<img width="400" src="assets/ex1_flow.svg" alt="Drift-only flow matching case"/>
<ul class="compact-list">
<li>Average the velocities</li>
</ul>
</div>
<div>
<h3>Case II: Drift + Diffusion</h3>
<img width="400" src="assets/ex2_driftdiffusion.svg" alt="Drift+Diffusion case"/>
<ul class="compact-list">
<li>Average the drift and the variance</li>
</ul>
</div>
</div>
<div class="two-col" style="margin-top: 0rem;">
<div>
<h3>Case III: Discrete States</h3>
<img width="400" src="assets/ex3_CTMC.svg" alt="Discrete state case"/>
<ul class="compact-list">
<li>Average the rates</li>
</ul>
</div>
<div>
<h3>Case IV: Jump Processes</h3>
<img width="400" src="assets/ex4_jumps.svg" alt="Jump process case"/>
<ul class="compact-list">
<li>Average event intensities but <em>mixture</em> of destinations.</li>
<li>i.e. a Gaussian jump in the conditional process is a mixture of Gaussians (one per $z$) in the posterior averaged step.</li>
</ul>
</div>
</div>
</div>
</div>
<div class="slide" data-slide="8">
<div class="slide-content">
<h2>Auxiliary Generator Matching with a time-dependent latent process</h2>
<ul>
<li>Consider a conditional process over $x_t,g_t$ with steps $p(x_{t+\Delta t}, g_{t+\Delta t} \mid x_t, g_t, z, t)$</li>
<li>Conditional process as before, still terminating at $x_1^z$ when $t=1$ (with no required constraint on $g_1$). Note: this process is not necessarily Markov in $x$ alone.</li>
<li>Induces a marginal $p(x_t \mid t)$, and a posterior $p(g_t, z \mid x_t, t)$.</li>
</ul>
<ul>
<li>Consider a posterior resampling process but now where $z$ and $g_t$ are both sometimes resampled from $p(g_t, z \mid x_t, t)$ as the process steps forward. As before, and by the same argument, the marginal $p(x_t \mid t)$ is unchanged by the resampling.</li>
</ul>
<div class="key-point">
Taking the limit as the resampling rate goes to infinity yields a process that has the same marginal $p(x_t \mid t)$ as the original conditional process. But now, since $g_t$ and $z$ have been marginalized out, the process is Markov in $x$ alone, and a model can be trained to take $x_t, t$ and predict the posterior-averaged step on $x$.
</div>
<ul>
<li>The details of step averaging depends on how $x_t$ and $g_t$ are related.</li>
<li>For example, in Edit Flows, $x_t = \phi(g_t)$ and $g_t$ is a sequence containing some "null" tokens, and $\phi$ removes the null tokens. This means a substitution-to-null in $g_t$ is a deletion in $x_t$, and a substitution from null to a non-null token in $g_t$ is an insertion in $x_t$. So the model takes in a sequence without null tokens, but predicts rates that include insertion and deletion operations. <!--The rates learned are the posterior-averaged rates over all $g_t$ such that $\phi(g_t) = x_t$.-->
</li>
</ul>
</div>
</div>
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