A Demo Post with Math Equations

June 20, 2026 · demomathastro

Introduction

This post shows how KaTeX renders mathematical equations in your Astro blog. All standard $\LaTeX$ commands work out of the box.

Inline Math

You can write inline equations like $\pi \approx 3.14159$ or the policy gradient $\nabla_\theta J(\theta) = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a|s) Q^\pi(s,a)]$ .

Display Math with Numbering

Policy Gradient Theorem

The fundamental policy gradient theorem:

\nabla_\theta J(\theta) = \mathbb{E}_{\tau \sim \pi_\theta} \left[ \sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t | s_t) \, R(\tau) \right] \tag{1}

3D Diffusion Policy

Our 3D diffusion policy formulation:

\mathcal{L}_{\text{diff}} = \mathbb{E}_{\epsilon, t, \mathbf{a}_0} \left[ \| \epsilon - \epsilon_\theta(\mathbf{a}_t, t, \mathbf{s}_{\text{3D}}) \|^2 \right] \tag{2}

where $\mathbf{s}_{\text{3D}}$ is the 3D visual representation extracted from point cloud observations.

KL Divergence Regularization

For stable policy optimization:

D_{\text{KL}}(\pi_{\text{old}} \| \pi_{\text{new}}) = \mathbb{E}_{s \sim \rho^\pi} \left[ \sum_a \pi_{\text{old}}(a|s) \log \frac{\pi_{\text{old}}(a|s)}{\pi_{\text{new}}(a|s)} \right] \tag{3}

Referencing Equations

As shown in Equation $(1)$ , the policy gradient depends on the expected return. The diffusion loss in Equation $(2)$ uses 3D representations. KL regularization in Equation $(3)$ prevents policy collapse.

Matrix Notation

The linear system for our optimization:

\begin{bmatrix} K_{11} & K_{12} & \cdots & K_{1n} \\ K_{21} & K_{22} & \cdots & K_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ K_{n1} & K_{n2} & \cdots & K_{nn} \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix} \tag{4}

Code + Math

import torch

def diffusion_loss(model, x_0, condition):
    """Compute diffusion loss with 3D condition (Eq. 2)."""
    t = torch.randint(0, model.timesteps, (x_0.shape[0],))
    noise = torch.randn_like(x_0)
    x_t = model.q_sample(x_0, t, noise)
    pred_noise = model(x_t, t, condition)
    return torch.mean((noise - pred_noise) ** 2)

The loss function above implements Equation $(2)$ in practice.

This post demonstrates that KaTeX math rendering works seamlessly in the Astro blog setup.