Grunnleggende konsepter
The author presents results on the uniform Ck approximation of G-invariant and antisymmetric functions using polynomials, highlighting the independence of embedding dimensions from regularity and accuracy.
Sammendrag
The content discusses the uniform Ck approximation of G-invariant and antisymmetric functions using polynomials. It explores the independence of embedding dimensions from regularity and accuracy, providing insights into deep learning challenges.
Key points include:
- Results for approximating G-invariant functions by polynomials.
- Embedding dimension independence from regularity and accuracy.
- Theoretical understanding in deep learning with symmetries.
- Applications in science and technology.
- Counterexample to exact representation theorems for antisymmetric functions.
- Theorems on Ck approximations for symmetric and antisymmetric functions.
- Representations of totally symmetric polynomials with finite generators.
The study contributes to understanding neural networks' behavior with symmetries, offering practical implications for various applications.
Statistikk
For any subgroup G of the symmetric group Sn on n symbols, we present results for the uniform Ck approximation of G-invariant functions by G-invariant polynomials.
We show that a similar procedure allows us to obtain a uniform Ck approximation of antisymmetric functions as a sum of K terms, where each term is a product of smooth totally symmetric function and smooth antisymmetric homogeneous polynomial.