Tuesday 29 Jun 2021: Symmetry Breaking and Bifurcation
Mike Field -
https://Universityofexeter.zoom.us/j/93410382638?pwd=U3kzMG5yUWwrRm1DcDlVZmQ5WVRwQT09 Meeting ID: 934 1038 2638 Password: 490214 15:30-16:30
We present two results in dynamics and bifurcation that were motivated by questions in machine learning (ML) related to the creation or annihilation of spurious minima (gradient descent dynamics).
A feature of non-convex optimization is the occurrence of “spurious minima” that appear (or disappear) through bifurcation not involving the global minima. A prototypical model is generic bifurcation on an absolutely irreducible representation admitting quadratic equivariants. For example, the standard representation of the symmetric group Sk or the external tensor product representation of Sk × Sd on the space of k × d matrices (row and column sums zero, k, d ≥ 3).
In 1984, Ihrig and Golubitsky proved that if there were quadratic equivariants then, under certain assumptions, axial solution branches could not be branches of sinks. Their assumption (H4), described as “the crucial hypothesis”, was that branches were axial (along axes of symmetry). Under rather general conditions, we show that if there are non-zero quadratic equivariants then, for generic bifurcation, all non- trivial solution branches are branches of hyperbolic saddles: there are no branches of sinks or sources. Our result applies, for example, when the quadratic equivariants are gradient and to examples where there are few or no axial solution branches.
Our second result is about forced breaking symmetry in generic bi- furcation on the standard representation of Sk, k > 2. We define a ‘minimal unfolding’ of the bifurcation to be a symmetry breaking per- turbation of the generic bifurcation that is stable (as a family) and has the minimum number of saddle-node bifurcations and the maximum number of branches of equilibria crossing the bifurcation point (all non- singular). In other words, minimal dynamic complexity. We indicate the explicit and relatively elementary construction of a minimal un- folding, the remarkable complexity that the Sk-equivariant bifurcation encodes and, time permitting, say how this relates to problems in ML and optimization.
The results described arose from collaborative research with Yossi Arjevani (NYU and Hebrew University of Jerusalem, from July).