Latest Posts
On Image Generation
Modern generative machine learning has achieved impressive empirical results, yet most existing approaches treat images merely as discrete pixel matrices. To build a mathematically rigorous generative paradigm, we must abandon the naive coordinate representation and view the image space through the lens of differential geometry and bundle theory.
Machine Learning on Stiefel Manifold
The Stiefel manifold is a useful geometric object in machine learning because there are applications where the target transformation must preserve the norm of input vectors.
Machine Learning on Spherical Manifold
There are a variety of application where an \(n\)-dimensional hypersphere is a natural domain for data. For example images from 360-degrees cameras or weather on the Earth. But basic machine learning algorithms are expected to take place on regular Euclidean space, therefore it is hard to preserve the spherical structure of the domen. In order to overcome this difficulty, we can introduce additional structure to basic optimization algorithms like Gradient Descent, to enforce them to work on sphere.