Metric Rarity and the Rise of Symmetry in Invariant Optimization

Abstract

In the non-convex landscapes of $G$-invariant optimization problems -- spanning particle physics, neural networks, and algebraic models -- symmetric critical points defy statistical intuition. While generic configurations dominate the space, empirical evidence reveals two striking regimes: (I) symmetry as the norm, with asymmetric points vanishingly rare, and (II) a profound energetic ordering, where global minima exhibit disproportionately higher symmetry. This paper unveils a unified geometric mechanism rooted in algebraic quotients. We reframe the problem on the quotient space $Y = X \sslash G$, where the physical domain is the real image $L = π(X(\mathbb{R}))$ -- a metrically rare subset within the ambient quotient $Y(\mathbb{R})$. Theorems quantify this rarity: for the symmetric group $S_n$, the volume of the real image $L$ decays exponentially as $e^{-Cn \log n}$. Regime I emerges from an ``empty interior'': critical points evade the smooth locus of $L$, settling on boundaries tied to non-trivial stabilizers. Regime II invokes the ``Active Constraint'' -- a global gradient channeling minima to sharp corners of $L$, manifesting as funnel topographies in physical systems that intercept the descent at crystalline structures with high symmetry.

Figures (11)

• Figure 1: Regime II symmetry emergence: (a) Random $S_n$ invariant polynomials. (b) Physical potential (LJ13).
• Figure 2: Visualization of the real image of $\pi: X(\mathbb{R}) \to Y(\mathbb{R})$ for $n=3$. The scatter plot displays the real image $L$ of the quotient map $\pi: \mathbb{R}^3 \to \mathbb{R}^3$ defined by elementary symmetric polynomials, given by $\pi(x_1,x_2,x_3) = (e_1, e_2, e_3) = (x_1 + x_2 + x_3, x_1 x_2 + x_2 x_3 + x_3 x_1, x_1 x_2 x_3)$. The geometry reveals a hierarchy of singularities corresponding to the stabilizer subgroups discussed in the text: White (Ambient Space, $Y(\mathbb{R})$): The full quotient space represents all real coefficients of monic cubic polynomials. Yellow (Smooth Region, $L^\circ$): Points with trivial stabilizers ($x_i \neq x_j$) form the 3D volume, representing coefficients of polynomials with three distinct real roots. Green (Boundary, $\partial L$): Points with $S_2$ stabilizers (two equal coordinates) form a 2D smooth boundary fold, representing coefficients of polynomials with 2 distinct real roots (one double root). Red (Deep Singularity): Points with full $S_3$ symmetry ($x_1=x_2=x_3$) collapse onto a 1D "cusp" or edge, representing coefficients of polynomials with a triple real root.
• Figure 3: Topographical evolution of the real image $L$. The disconnectivity graphs (visualized here as merge trees of energy sublevel sets) for $LJ_{13}$ (left) and $LJ_{55}$ (right) exhibit a unified "funnel" structure. All minima are shown for $LJ_{13}$, while for $LJ_{55}$ only branches leading to the 900 lowest-energy minima are displayed. Numbers adjacent to nodes indicate how many minima the node represents, and selected branches are labeled by the energetic rank of their minima. In the $LJ_{55}$ panel, the black inset is a zoom-in of the branch marked by $i$. Reprinted with permission from J. P. K. Doye, M. A. Miller, and D. J. Wales, J. Chem. Phys.111, 8417 (1999). Copyright 1999 AIP Publishing.
• Figure 4: Robust boundary preference when invariance is imposed via Reynolds symmetrization across multiple $(n,d)$ and coercive strengths. In each panel, energetic extremes exhibit fewer distinct coordinate values than the bulk, indicating higher symmetry near the boundary strata.
• Figure 5: Boundary preference under the quotient construction $f=P\circ \pi$ across unbounded and $y$-space bounded runs. With the notable exception of \ref{['fig:exception_quotient_n2']}, the $y$-space bounded examples consistently show lower distinct values near the low-energy end (left tail). This persistence indicates that the symmetry preference is not generally an artifact of $\tilde{f}$ being unbounded on $Y(\mathbb{R})$.

Theorems & Definitions (50)

• Definition 2.1: The $G$-Variety Setup
• Definition 2.2: Stabilizer Subgroup
• Definition 2.3: Fixed Point Subspace
• Definition 2.4: Invariant Functions
• Definition 2.5: The Quotient Space
• Theorem 2.6: Principle of Symmetric Criticality Palais1979
• Remark 2.7: Heuristic Counting Baseline
• Theorem 3.1: Surjectivity of the Quotient Differential
• proof
• Theorem 3.2: The Boundary of the Real Image and Stabilizer Parity