Stabilized Ellipsoid Embedding Problem

• The stabilized ellipsoid embedding problem is a framework combining convex geometry and symplectic topology to analyze embedding feasibility under prescribed spectral bounds and stabilization constraints.
• It identifies precise phase transitions and capacity thresholds—such as the n/d² ≈ 1/4 threshold in random point settings and supercritical scaling in symplectic embeddings.
• Proof techniques span random matrix theory, semidefinite programming, and holomorphic curve methods, offering insights into geometric obstructions and efficient algorithmic implementations.

The stabilized ellipsoid embedding problem encompasses both high-dimensional convex geometry and symplectic topology, addressing the existence, nonexistence, and phase transitions of embeddings between ellipsoidal domains under various stabilization regimes. Its rigorous characterization in the random convex setting and the symplectic category has elucidated sharp thresholds, universal phenomena, geometric obstructions, and algorithmic properties.

1. Definitions and Formulation

The stabilized ellipsoid embedding problem has two principal incarnations: fitting ellipsoids to random point clouds in convex geometry, and determining symplectic embeddings of domains augmented by additional Euclidean dimensions.

Convex geometry setting: Let $x_1, \ldots, x_n \subset \mathbb{R}^d$ be i.i.d. standard Gaussian vectors. The centered ellipsoid fitting problem seeks symmetric $S \succeq 0$ with prescribed spectral bounds, such that all points $\{x_\mu\}$ lie close to the boundary $\Sigma = \{x: x^\top S x = d\}$. For stabilization, spectrum constraints $[\lambda_-, \lambda_+]$ are imposed to avoid degeneracy; the loss function $\phi$ penalizes fitting error, and the feasible set is:

$$\Gamma(\phi, \lambda_-, \lambda_+, \epsilon) = \{ S: \operatorname{Spec}(S) \subset [\lambda_-, \lambda_+],\; \frac{1}{n}\sum_{\mu=1}^n \phi(\sqrt{d}[x_\mu^\top S x_\mu/d - 1]) \le \epsilon \}$$

Symplectic setting: In $\mathbb{R}^{4+2N}$ with the standard symplectic form, the ellipsoid $$E(a,b) = \{ \pi(\frac{x_1^2 + y_1^2}{a} + \frac{x_2^2 + y_2^2}{b}) \le 1 \}$$ is stabilized to $E(a,b) \times \mathbb{R}^{2N}$. The embedding problem seeks a symplectic embedding $$E(a,b) \times \mathbb{R}^{2N} \hookrightarrow M \times \mathbb{R}^{2N}$$, with the minimal scaling factor or capacity encapsulated in $E_M^N(a,b)$.

2. Phase Transitions and Capacity Thresholds

A precise phase transition in feasibility governs both settings.

Random point fitting: Set $\alpha = n/d^2$. Maillard–Bandeira established:

• If $\alpha < 1/4$, there exists a stabilized ellipsoid fit—i.e., spectra bounded and fitting error vanishing in probability as $d \to \infty$ for loss exponents $r \in [1, 4/3)$.
• If $\alpha > 1/4$, no such stabilized fit exists; any solution with bounded condition number incurs non-vanishing error (Maillard et al., 2023).

Symplectic stabilization: McDuff–Siegel proved for the stabilized embedding into the four-ball:

$$E_{B^4}^N(1,a) = \begin{cases} E_{B^4}(1,a), & 1 \le a \le \tau^4 \ \frac{3a}{a+1}, & a > \tau^4 \end{cases}$$

with $\tau^4 = (7 + 3\sqrt{5}) / 2$; the “supercritical” formula $3a/(a+1)$ is strictly below the volume bound $\sqrt{a}$ (Siegel, 17 Nov 2025).

3. Proof Strategies and Universality

Convex geometry: The existence and nonexistence proofs rely on random matrix theory, convex geometry (Gordon’s min-max theorem, Gaussian widths), and universality arguments. The transition at $n \approx d^2/4$ is established using a “Gaussian equivalent” model replacing point clouds by random GOE matrices:

• Infimum loss function is shown universal (ground-state universality proposition).
• Gaussian interpolation and Berry–Esseen bounds control approximation between ensembles.
• Gordon’s theorem quantifies feasibility in the cone of PSD matrices with bounded spectrum.

Symplectic topology: The proofs employ Gromov–Witten invariants, holomorphic curves, and symplectic folding constructions:

• Upper bounds: Hind’s symplectic folding shows existence of embeddings matching $3a/(a+1)$ (Siegel, 17 Nov 2025).
• Lower bounds: Holomorphic-curve techniques (Fredholm index computations, ECH capacities) provide sharp obstructions at “ghost staircase” and arithmetic-eccentricity points (McDuff, 2017, Cristofaro-Gardiner et al., 2017).

4. Higher Symplectic Capacities and Embedding Obstructions

The theory of higher symplectic capacities $c_k^{(m)}$ underpins recent advances:

• Capacities are monotonic, homogeneous under scaling, normalized on the ball, and invariant under stabilization.
• In dimension $4$, for opposite-parity eccentricities, the minimal scaling factor for embedding $$E(1,a)\times\mathbb{R}^{2N} \hookrightarrow E(1,b)\times\mathbb{R}^{2N}$$ is $\frac{2a}{a+b-1}$ (Cristofaro-Gardiner et al., 2021).
• Capacity computations rely on counting rational curves with prescribed tangency and cusp constraints, resulting in piecewise-linear and “staircase” phenomena in embedding functions.

These invariants sharply distinguish which symplectic embeddings are possible under stabilization, recover classical results, and resolve the rescaled polydisc limit conjecture for stabilized settings.

5. Algebraic and Tropical Tools: Curve Counting and Lattice Models

The classification and construction of obstructions depend on:

• Sesquicuspidal curves: Rational index-zero curves with prescribed cusp and tangency constraints encode obstructions. Existence in $\mathbb{CP}^2$ is linked to outer Fibonacci ratios or supercritical slopes (Siegel, 17 Nov 2025).
• Well-placed curves and toric models: Via blow-ups and cluster transformations, well-placed curves model embeddings and correspond to staircase breakpoints.
• Scattering diagrams and tropical vertex theorem: Tropical methods produce counts of rational affine lines, related by fundamental bijections to holomorphic curves in blowups, reconstructing the embedding capacity function and its breakpoints.

Such structures establish deep connections between algebraic, cluster, and symplectic topology frameworks governing stabilized embedding phenomena.

6. Algorithmic and Practical Implications

In convex geometric fitting, the stabilized ellipsoid problem reduces to semidefinite programming (SDP), efficiently solvable whenever feasible:

• Positive-regime spectral bounds allow for explicit initialization and warm-starting in optimization.
• The stabilization guarantees all principal axes are within fixed bounds, precluding highly degenerate ellipsoids.
• For $n < d^2/4$, nearly-exact fits with well-conditioned solutions are obtainable. For $n > d^2/4$, such algorithmic stabilization is impossible and any method incurs macroscopic fitting error (Maillard et al., 2023).

Symplectic setting embeds via explicit folding constructions when feasible, while obstructions rooted in holomorphic curve theory govern impossibility.

7. Generalizations, Open Problems, and Context

• Cluster-algebraic and mirror-symmetry perspectives encode mutation graphs and wall-crossing phenomena, reflecting the combinatorics of symplectic embeddings.
• Extensions to monotone del Pezzo surfaces yield analogous staircases and supercritical embedding formulas.
• Open questions remain on removing parity restrictions in high capacities, formulating combinatorial rules governing stabilized capacities, and the behavior of capacity sequences for non-integral and transition-region eccentricities.
• In dimensions $>4$, both the unstabilized and stabilized ellipsoid embedding problems become substantially more intricate, with current holomorphic-curve and obstruction techniques facing major technical barriers (Siegel, 17 Nov 2025, McDuff, 2017).

The stabilization phenomenon offers a unified view by which both geometric and topological phase transitions in embedding problems are governed by universal invariants and sharp threshold formulas.