Sum-of-Squares Hierarchy (SoS)

The Sum-of-Squares (SoS) hierarchy is a systematic framework of semidefinite programming relaxations for polynomial optimization and proof complexity, notable for its far-reaching impact across combinatorial optimization, theoretical computer science, quantum information theory, proof systems, and high-dimensional statistics. Structurally, the hierarchy enables one to "lift" nonlinear, intractable polynomial problems to a sequence of convex programs whose strength increases with the number of "rounds" or degree, often yielding the tightest known algorithmic relaxations for NP-hard problems and serving as a vehicle for both algorithm design and hardness proofs.

1. Mathematical and Proof-Theoretic Foundations

The SoS hierarchy originated in the work of Lasserre and Parrilo as an approach to global polynomial optimization. For a polynomial $f(x)$ minimized over a semialgebraic set defined by polynomial equalities/inequalities, the $r$-th level SoS relaxation asks: can $f(x) - \lambda$ be written as a sum of squares of polynomials of degree up to $2r$, perhaps after multiplication by the constraints' defining polynomials? Formally, for constraints $g_j(x) \geq 0$,

$\max \left\{ \lambda : f(x) - \lambda = \sum_{i} s_i(x) g_i(x),\ s_i \text{ SOS},\ \deg(s_i g_i) \leq 2r \right\}.$

Equivalently, the dual (moment) formulation seeks pseudoexpectation functionals on degree-$2r$ polynomials consistent with all given constraints. For Boolean or finite domains, this reduces to a hierarchy of semidefinite programs with size $n^{O(r)}$, which converge (under suitable conditions) to the exact optimum as $r$ increases.

This framework also embeds a proof system, where SoS certificates allow refutation of unsatisfiable systems by expressing $-1$ as a sum of squares modulo the constraints. The connections to the hierarchy's duality and proof complexity are detailed in works focusing on the degree/level tradeoffs needed for various combinatorial principles, such as the ordering principle, Max-Cut, and more (Barak et al., 2012 , Potechin, 2018 ).

2. Applications in Combinatorial Optimization

SoS hierarchies are central to the state of the art in multiple discrete optimization domains:

• Max Clique and Graph Densities: The SoS hierarchy provides increasingly tight semidefinite relaxations for the maximum clique problem, Max-Cut, and dense subgraph detection. The power of the hierarchy is seen in its connection with spectral methods: for example, Lovász's $\vartheta$ function can be realized at level-1 (SDP) and higher levels yield tighter bounds (Rajendran, 2022 ). For random graphs, higher-degree SoS relaxations can certify small clique numbers with high probability.
• Minimum Bisection: Guruswami and Sinop showed via spectral rounding that the SoS relaxation yields strong guarantees for bisection on graphs with low threshold-rank Laplacian, exploiting the graph’s eigenstructure to inform rounding algorithms.
• Constraint Satisfaction Problems (CSPs): For random CSPs, SoS lower bounds match the known limitations of polynomial-time algorithms, with explicit analysis for Max $K$-CSP, planted clique, and other domains. Pseudocalibration techniques allow constructing pseudoexpectations mimicking the distribution of satisfying assignments in the planted setting, yielding near-optimal integrality gaps (Rajendran, 2022 ).
• Tensor Completion and Machine Learning: In noisy tensor completion, the sixth-level SoS relaxation matches the best possible Rademacher complexity among polynomial-time norms and achieves recovery thresholds up to the known computational–statistical gap (Barak et al., 2015 ).

3. Hierarchy Structure, Convergence, and Computational Aspects

SoS relaxations are generally formulated as SDPs whose size scales exponentially in the degree/level $r$, but for fixed or moderate $r$ are solvable and yield strong approximations.

• Convergence Guarantees: For compact semialgebraic sets (Archimedean property), convergence to the global optimum is guaranteed (Putinar, Schmüdgen, and subsequent results). For key domains, quantitative rates are known: e.g., over the unit ball and simplex, the error decays as $O(1/r^2)$, which is optimal (Slot, 2021 , Fang et al., 2019 ).
• Polynomials on the Boolean Cube: For binary polynomial optimization, the rate of convergence is determined by the smallest root of Krawtchouk polynomials: for $r \approx t n$, the SoS bound is within $1/2 - \sqrt{t(1-t)}$ of the minimum for large $n$, and so exactness is achieved by about $n/2$ rounds (Slot et al., 2020 ).
• Spectral Hierarchies: Recent developments introduce spectral (eigenvalue-based) analogues to SoS with per-level computations as minimum eigenvalue problems, achieving $O(1/k)$ convergence and scaling to higher dimensions than SDP-based methods (Johnston et al., 2023 ).
• Computing with Probability Distributions: Continuous extensions of moment-SOS hierarchies permit direct optimization over probability distributions, not just finite supports, with well-defined convergence and statistical consistency under sampling (Tran et al., 20 Apr 2025 ).

4. Limitations and Lower Bounds

The power and limitations of SoS have been sharply delineated:

• Integrality Gaps and Lower Bounds: Explicit constructions achieve $\Omega(n)$-level SoS lower bounds for random CSPs, 3-XOR, and planted clique, matching the hardness of random instances and showing that substantial rounds are needed to refute such systems (Hopkins et al., 2022 ).
• Symmetry and Hardness: Highly symmetric problem instances allow reduction of SoS lower bounds to verifying univariate polynomial inequalities. This leads to optimal lower bounds for Max-Cut, Min-Knapsack, and general unconstrained binary polynomial optimization (Kurpisz et al., 2014 , Kurpisz et al., 2016 ).
• Comparison with Other Hierarchies: Some relaxation families—such as DSOS and SDSOS—are computationally more tractable but can fail to converge in cases where SoS succeeds (Josz, 2017 , Kurpisz et al., 2019 ). Over the Boolean cube, Schmüdgen-like versions of SoS, SDSOS, SONC, and SA are polynomially equivalent, but outside such domains, the relationships can be subtle and even incomparable.

5. Applications in Quantum Information and Beyond

The SoS hierarchy underpins much of quantum information’s convex relaxations:

• Separability and Quantum Proofs: SOS certificates resolve separability for the Best Separable State problem and approximate injective tensor norms, with tight $O((d/\ell)^2)$ convergence, generalizing previous results for quadratic cases (Fang et al., 2019 ). Duality with the DPS hierarchy enables transfer of convergence guarantees between quantum and polynomial settings.
• Perturbation Theory and Quantum Chemistry: Degree-4 SoS (2-RDM) captures quadratic/qubit systems but can fail for second-order perturbation in quartic (fermionic) Hamiltonians, where degree-6 (3-RDM, or specific "fragments" thereof) is necessary and sufficient (Hastings, 2022 ).

6. Methodological Innovations

• Polynomial Kernel Techniques: Improved convergence for sum-of-squares certificates on the sphere and other domains is achieved by selecting and analyzing polynomial kernels tailored to the harmonic structure of the problem (Fang et al., 2019 , Slot, 2021 ).
• Rademacher Complexity and Generalization: In tensor completion, the Rademacher complexity of SoS norm balls dictates generalization performance, with degree-6 SoS achieving the known information–computation frontier (Barak et al., 2015 ).
• Pseudocalibration and Indistinguishability: Modern integrality gap lower bound arguments use pseudocalibration, constructing pseudoexpectations that are indistinguishable from planted models by any low-degree test, thereby matching low-degree likelihood ratio barriers (Rajendran, 2022 , Rajendran, 2023 ).
• SOS for Probability Distributions: SOS hierarchies have been extended to optimize over probability measures directly, leading to tractable relaxations for functionals polynomial in these measures, with theoretical and practical consequences in metric geometry (e.g., Gromov-Wasserstein) and beyond (Tran et al., 20 Apr 2025 , Tran et al., 13 Feb 2025 ).

7. Future Directions and Open Questions

Areas for further research and development include:

• Optimizing over probability distributions with generalized polynomial objectives, beyond support on finite or discrete spaces.
• Efficient rounding and certification for rich pseudodistributions, such as those arising in high-entropy step (HES) SoS hierarchies, particularly for spin glass and high-dimensional inference problems (Sandhu et al., 25 Jan 2024 ).
• Explicit spectral hierarchies for broader classes of polynomial constraints, seeking alternatives to full SDP relaxations for high-dimensional or sparsely-interacting systems.
• Sharpness of SoS integrality gap lower bounds, especially for problems designed to "fool" SoS at linear or near-linear number of levels.

Summary Table: Major Features and Fact Patterns of the SoS Hierarchy

The Sum-of-Squares hierarchy integrates deep algebraic, analytic, and combinatorial tools, making it one of the most influential frameworks in modern discrete and convex optimization, quantum information, proof complexity, and high-dimensional data science. Its future developments are poised to further connect optimization, algorithms, and the geometry of computation.