Geometric Machine Learning

Apply affine coordinate changes to 3D points and displacement vectors using homogeneous coordinates.

Compute batched pairwise Euclidean distance matrices for point clouds in PyTorch.

Compute the eigendecomposition of a real symmetric matrix and verify that its eigenvectors form an orthonormal basis.

Compute the gradient of a scalar field with respect to spatial coordinates using the chain rule and verify it with PyTorch autograd.

Use a Jacobian to linearize a coordinate transformation near a point.

Derive reverse-mode gradients through an affine layer followed by a ReLU activation.

Implement gradient-descent and Adam parameter updates and diagnose an unstable learning rate.

Train a small multilayer perceptron in PyTorch using a loss function, backpropagation, and an optimizer.