| Abstract: |
| Given a representation $V$ of a compact group $G$, the $\ell$-th moment is a tensor parametrizing the invariant polynomials of degree $\ell$. A classical theorem invariant theory implies that the
Invariant ring is finitely generated which implies that for $\ell$ sufficiently large closed orbits can be separated by the moments of order up to $\ell$. Unfortunately, the computational cost of computing the $\ell$-th moment grows exponentially in $\ell$. A problem that was originally motivated cryo-EM is to study representations for which moments of low degree separate (generic) orbits. In this talk we discuss the information determined by the second moment and describe a class of representations of the compact groups of classical type for which the third moment can separate generic orbits. While our original motivation for studying this problem was cryo-EM, we believe that understanding the structure of polynomial invariants has an important role in equivariant machine learning as well. |
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