flashcards_differential_geometry_phase_1
What is a manifold?
A smooth collection of points in high-dimensional space that locally resembles Euclidean space.
What is the latent space in machine learning?
A low-dimensional space capturing the essence of observed data.
Why are manifolds important for generative models?
They help model high-dimensional data by leveraging its intrinsic low-dimensional structure.
What is geodesic distance?
The shortest path between two points on a manifold.
What is the manifold hypothesis?
High-dimensional data lies near a low-dimensional manifold within its observation space.
What is the tangent space?
The local Euclidean approximation of a manifold at a given point.
How is the tangent space derived?
Using the Jacobian matrix, which provides a linear approximation of a function.
What is a Riemannian metric?
A local measure of distance on a manifold, defined as G_x = J_x^T J_x.
How do you calculate geodesics?
By numerically solving for the shortest path using the Riemannian metric.
Why are geodesics unique in non-antipodal cases?
They result from the intersection of a manifold with a plane spanning the two points.
How do autoencoders approximate manifolds?
Through a decoder mapping latent space to data space and an encoder projecting data to latent space.
What is the autoencoder loss function?
Sum of squared reconstruction errors: Σ ||x - f(g(x))||^2.
How is geodesic length computed numerically?
By approximating the integral using discrete sums or optimization.
What makes geodesics computationally challenging?
The need for iterative optimization and accurate computation of metrics along the path.
What is an abstract density metric?
A metric defined inversely proportional to the data density, pulling geodesics toward dense regions.
