Constructing the Training Target

Constructing the Training Target

Conditional Probability Path

A conditional probability path is a set of distribution \(\mathbf{p}_t(x | z)\) over \(\mathbb{R}^d\) such that

\[ \begin{aligned} p_0(\cdot | z) = p_{init}, \quad p_1(\cdot | z) = \delta_z \quad \forall z \in \mathbb{R}^d \end{aligned} \tag{1}\] In other words, a conditional probability path interpolates between the initial distribution \(p_{init}\) and a single data point from \(p_{data}\).

Marginal Probability Path

Every conditional probability path induces a marginal probability path \(p_t(x)\). We obtain it by first sampling \(z \sim p_{data}\) and then sampling \(x \sim p_t(\cdot | z)\):

\[ \begin{aligned} &z \sim p_{data}, \quad x \sim p_t(\cdot | z) \\ p_t(x) &= \int p_t(x, z) dz \\ &= \int p_t(x | z) p_{data}(z) dz \end{aligned} \]

We can verify that \(p_t(x)\) interpolates between \(p_{init}\) and \(p_{data}\).

At \(t=0\): The conditional distribution equals the initial distribution for all \(z\), so \(p_{init}(x)\) factors out of the integral:

\[ \begin{aligned} p_0(x) &= \int p_0(x | z) p_{data}(z) dz \\ &= \int p_{init}(x) p_{data}(z) dz && \text{(since } p_0(\cdot|z) = p_{init} \text{)} \\ &= p_{init}(x) \int p_{data}(z) dz && \text{(factor out } p_{init}(x) \text{)} \\ &= p_{init}(x) && \text{(probability integrates to 1)} \end{aligned} \]

At \(t=1\): The conditional distribution collapses to a point mass at \(z\). To evaluate this integral, we use the sifting property of the Dirac delta:

\[ \int f(z) \delta_z(x) \, dz = f(x) \]

To understand this, think of \(\delta_z(x)\) as the limit of increasingly narrow, tall functions that integrate to 1 — for example, a Gaussian centered at \(z\) with vanishing variance:

\[ \delta_{z,\epsilon}(x) = \frac{1}{\epsilon\sqrt{2\pi}} e^{-(x-z)^2/(2\epsilon^2)} \xrightarrow{\epsilon \to 0} \delta_z(x) \]

As \(\epsilon \to 0\), all the “mass” concentrates at \(x = z\). So when we integrate \(f(z)\delta_{z,\epsilon}(x)\) over \(z\), only the value near \(z = x\) contributes, giving \(f(x)\). Applying this:

\[ \begin{aligned} p_1(x) &= \int p_1(x | z) p_{data}(z) dz \\ &= \int \delta_z(x) p_{data}(z) dz && \text{(since } p_1(\cdot|z) = \delta_z \text{)} \\ &= p_{data}(x) && \text{(sifting property)} \end{aligned} \]

Example: Gaussian probability paths

A Gaussian probability path is a family of Gaussian distributions that are parameterized by a time-dependent mean and covariance matrix. The conditional Gaussian probability path is defined as:

\[ \mathbf{p}_t(\cdot | z) = \mathcal{N}(\alpha_t z, \beta_t^2 I_d) \]

where \(\alpha_t\) and \(\beta_t\) are noise schedulers, and they are continously, monotonically differentiable functions with the following conditions:

\[ \begin{aligned} \alpha_0 &= \beta_1 = 0 \\ \beta_0 &= \alpha_1 = 1 \end{aligned} \]

which implies

\[ \begin{aligned} \mathbf{p}_0(\cdot | z) &= \mathcal{N}(0, I_d) \\ \mathbf{p}_1(\cdot | z) &= \lim_{\beta \to 0} \mathcal{N}(z, \beta^2 I_d) = \delta_z \end{aligned} \tag{2}\]

As the variance vanishes, the Gaussian concentrates all its mass at the mean \(z\), converging (in distribution) to the Dirac delta \(\delta_z\).

Therefore, this Gaussian probability path satisfies (1) and is a valid conditional probability path.

Intuitively, given a data point \(z\) at time \(t=1\), we gradually add noise to it for lower t till \(t=0\).