Structure of solutions to continuous constraint satisfaction problems through the statistics of wedged and inscribed spheres

TL;DR

This work introduces a cost-free geometric framework for continuous constraint satisfaction problems by counting how many spheres can be uniquely inserted into the solution space. By defining wedged spheres of fixed radius and inscribed spheres of variable radius, the authors connect these counts to the topology of the feasible set, offering a complementary perspective to energy-based analyses. Applied to the spherical perceptron, the method reveals two topological regimes: one with a highly loopy, connected structure and another with a decomposition into simply connected components, with replica-symmetry-breaking transitions informing the complexity of the solution space. The approach provides insights into algorithmic performance and topology-driven optimization, and it can be extended to non-Euclidean configuration spaces and sublevel-set analyses.

Abstract

The study of random landscapes has long relied on counting stationary points: metastable states and the barriers between them. However, this method is useless for describing flat regions, common in constraint satisfaction problems. We introduce a characterization of flat regions by counting the number of spheres that can be uniquely inserted into them, either by wedging spheres of fixed radius or by inscribing spheres of variable radius. The ratio of these counts constrains the topology of the solution space. We apply this characterization to the spherical perceptron and show the existence of at least two topological regimes.

Figures (10)

• Figure 1: Geometric regimes of the spherical perceptron. The spherical perceptron can be interpreted as a random Lorentz gas on the $D$-sphere. Because the ambient space is curved, spherical obstacles can take qualitatively different geometries. Left: When the margin $\kappa$ is negative, the spherical obstacles have a have a positive radius of curvature and the problem is nonconvex. Center: When the margin $\kappa$ is zero, spherical obstacles are perfect hemispheres of the configuration space and have flat boundaries. Right: When the margin $\kappa$ is positive, spherical obstacles have a negative curvature and the problem is convex. In all three cases, the problem is analyzed in a limit where the obstacle boundaries are asymptotically flat.
• Figure 2: Wedging spheres into the set of solutions.Left: Spheres of fixed radius are uniquely specified in a $D$-dimensional configuration space by identifying $D$ points on their boundary. We define $\#_r$ as the number of wedged spheres with radius $r$. Right: In the problems we study, the number of wedged spheres of a given radius is equal to the number of points of intersection of $D$ constraint boundaries in a problem with a larger margin. We define $\#_0$ as the number wedged points.
• Figure 3: Inscribing spheres into the set of solutions. Spheres of maximal radius are uniquely specified in a $D$-dimensional configuration space by identifying $D+1$ points on their boundary. We define $\#_\text{insc}$ as the number of inscribed spheres.
• Figure 4: Wedged and inscribed spheres constrain solution topology.Left: The sets of wedged and inscribed spheres define a graph on the space of solutions whose internal vertices are centered on inscribed spheres and have degree $D+1$ and whose leaves are wedged points. The relative numbers of inscribed spheres and wedged points inform the topology of this graph and therefore of the solution space. Right: There are three ways for the topology of the graph to be modified as the margin is continuously varied. An existing leaf can split into $D$ new leaves at a point where a previously obscured decision boundary is uncovered. Alternatively, two leaves can meet and annihilate, forming an edge between the internal vertices they were previously connected to. Finally, a new isostatic void can appear where one was not previously, adding a vertex with $D+1$ leaves (not pictured).
• Figure 5: Approach of $\omega$-dependent annealed solution to its asymptotic value. The logarithm of the average number of wedged points in the spherical perceptron as a function of $\alpha$ for fixed $\kappa=-\frac{3}{2}$. The exact answer in black is directly compared to the answer obtained by fixing the parameter $\omega$ in \ref{['eq:fixed.size.det.2']} to fixed finite values. The answer quickly converges with increasing $\omega$.