Monthly Archives: March 2026

Problem: Let \(\boldsymbol{\Theta}\) be a smooth statistical manifold, and let \(D:\boldsymbol{\Theta}^2\to [0,\infty)\) be a smooth function. What does it mean for \((\boldsymbol{\Theta},D)\) to be a “divergence manifold“? Solution: The notion of a divergence manifold relaxes the axioms of a metric … Continue reading →

VAE $\textbf{Problem}$: Apply the PRAC-DTDT-ID workflow to a (vanilla) $\textit{autoencoder}$. $\textbf{Solution}$: Problem: to learn a lower-dimensional latent manifold representation of the input manifold (this is justified by the $\textit{submanifold hypothesis}$, namely that the data-generating distribution $p(\mathbf x)$ is essentially supported … Continue reading →

Problem: Give a broad sketch of the current state of the field of research in graph neural networks. Solution: Problem: Okay, so now explain what a graph neural network (GNN) actually is. Solution: A GNN is basically any neural network … Continue reading →

Problem: Consider a Landau-Ginzburg statistical field theory involving a single real scalar field \(\phi(\mathbf x)\) for \(\mathbf x\in\mathbf R^d\) governed by the canonically normalized free energy density: \[\mathcal F(\phi,\partial\phi/\partial\mathbf x,…)=\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2+\frac{\phi^2}{2\xi^2}+…\] Explain what the \(+…\) means, explain which terms have … Continue reading →

CNNs_Part_1 $\textbf{Problem}$: Write functions that take an arbitrary grayscale image and convolve them with a Sobel edge detection kernel. Apply both functions to a grayscale image of your choice. $\textbf{Solution}$: In [1]: import numpy as np import matplotlib.pyplot as plt sobel_horizontal_kernel … Continue reading →