Low-Degree Polynomial Methods

• Low-degree polynomial methods are techniques that use bounded-degree polynomials to approximate nonlinear functions, test algebraic structures, and simplify computational models.
• They enable efficient privacy-preserving neural inference, robust learning, and optimal property testing through approaches like quantization-aware least squares and Hermite expansions.
• Applications include deterministic approximate counting, spectral projection, and establishing complexity barriers, which yield practical insights for machine learning and algebraic computations.

A low-degree polynomial method refers to the broad spectrum of algorithmic, analytical, and complexity-theoretic techniques that utilize polynomials of bounded degree as core objects—whether as approximations to nonlinear functions, as algorithmic surrogates for hard-to-compute models, as tools for testing algebraic structure, or as complexity-theoretic barriers. The fundamental principle is to exploit the tractability and favorable computational properties of low-degree polynomials while ensuring sufficient expressivity for the intended task. This article surveys central concepts, modern methodologies, and key research areas united by low-degree polynomial methods, with a focus on their rigorous mathematical formulations and concrete algorithmic consequences.

1. Low-Degree Polynomial Approximation in Privacy-Preserving Neural Inference

Homomorphic encryption requires circuits of bounded multiplicative depth to remain efficient; thus, neural network inference in this setting replaces standard nonlinear activations with low-degree polynomial surrogates. Modern work achieves practical, non-interactive privacy-preserving models by fitting highly accurate, degree-2 (quadratic) polynomial approximations of the ReLU function over bounded intervals, yielding minimal multiplicative depth in the ciphertext circuit. The coefficients are computed via quantization-aware least squares, ensuring tight $L_\infty$-approximation within the activation range (e.g., $p_2(x) \approx 0.125\,x^2 + 0.499\,x$ on $[-2,2]$ with error $<10^{-3}$).

This approach is augmented with penalty-based training: the cross-entropy loss is regularized by a layer-wise "clip-range" term (see formula below) to prevent polynomial approximation error from compounding when pre-activations escape the fit interval, and all coefficients are quantization-aware. The structural optimization pipeline includes:

• Node fusing: merging chains of convolution, batch-normalization, and activation into single polynomial expressions, collapsing depth and rescaling overhead;
• Weight redistribution: rescaling and reparametrizing pooling and normalization to push all multiplicative scaling to one side, further reducing depth;
• Tower reuse: modulating modulus-chain sublevels and deferring rescaling—grouping "towers" of multiplicative operations—slashing RNS levels.

Data/parameter co-design is used: HW data layout for efficient ciphertext packing, slice-based clustering for convolutional weight sharing, and ensemble packing using unused ciphertext slots. On CIFAR-10/100 and ResNet variants, this yields plaintext-comparable accuracy to within 0.3 points and up to 4x speedup without bootstrapping (Chielle et al., 26 Sep 2025).

2. Low-Degree Polynomial Testing and Global Hypercontractivity

Low-degree polynomial testing—determining if a function $f: \mathbb F_q^n \to \mathbb F_q$ is degree-$d$, or $\delta$-far from such—is a foundational question in complexity theory, error correcting codes, and PCPs. The optimal family of testers, known as $t$-flat testers, query $f$ on a random $t$-dimensional affine subspace and accept iff $f|_A$ is degree-$d$. Matching lower and upper query complexity $O(q^d + 1/\delta)$ is achieved by showing that if $f$ is not degree-$d$, the probability that $f|_A$ violates degree-$d$ is controlled by expansion properties on the affine Grassmann graph and global hypercontractivity (Kaufman et al., 2022).

A central technical point is the structural analysis of "bad" subspaces: indicator functions of non-degree-$d$ flats with poor expansion reside (via Fourier analysis) primarily at level-1 in the harmonic decomposition, so their structure correlates with low-dimensional "zoom-in" subfamilies. This replaces earlier Ramsey-theoretic arguments and provides a polynomial dependence on $q$ rather than exponential/tower-type.

3. Low-Degree Model in Statistical Algorithms and Computational Barriers

Low-degree polynomial models are used as proxies for polynomial-time algorithms in high-dimensional statistics, detection, learning, and recovery tasks. For data $Y$ modeled as signal+noise, one defines $\mathrm{MMSE}_{\le D}$: the smallest achievable mean-squared error using any degree-$\le D$ polynomial function of the data. Tight upper and lower bounds are computed by explicit cumulant recursions (see formula for $\kappa_\alpha$) and Hermite expansions. In several planted recovery problems (e.g., submatrix, dense subgraph), degree-$\le D$ polynomials provably cannot achieve nontrivial correlation below known statistical thresholds, even when detection is algorithmically easy. This marks genuine computational-statistical gaps and implies the limitations of all polynomial-time estimators under conjectured hardness (Schramm et al., 2020, Carpentier et al., 11 Sep 2025).

Recent work introduces almost orthonormal polynomial bases—indexed by graph subtemplates, centered and variance-normalized—that span the relevant symmetries in latent-graph models. These bases allow the computation of exact degree-$D$ lower bounds on testing or estimation advantage in random graph models, precisely matching known computational thresholds in stochastic block models, hidden clique, and seriation (Carpentier et al., 11 Sep 2025).

4. Algorithmic Learning and Factorization with Low-Degree Polynomials

Learning and factorization tasks feature efficient algorithms powered by low-degree polynomial structure. Examples include:

• Attribute-efficient PTF learning: Sparse degree-$d$ polynomial threshold functions are learned with robustness to adversarial noise via Hermite basis expansion, sparsity certification in Chow vectors, and robust filtering based on sparse Frobenius norm certificates and degree-$2d$ polynomial outlier filters. This yields optimal $\epsilon^{O(d)}$-tolerance and $\tilde O(K^{4d}/\epsilon^2)$ sample complexity, achieved by iterative filtering and Hermite-based reconstruction (Zeng et al., 2023).
• Polynomial regression in low dimensions: Multi-index models where the target is a degree-$d$ polynomial of an unknown $r$-dimensional subspace can be learned in time $O_{r, d}(N n^2)$ with sample complexity $O_{r, d}(n\log^2(1/\epsilon)\,(\log n)^d)$ via trimmed PCA for subspace recovery and geodesic SGD methods for accurate estimation, exploiting Hermite expansions for gradient computations (Chen et al., 2020).
• Computing low-degree factors in lacunary and structured polynomials: For multivariate lacunary polynomials, Newton-Puiseux theory, Gap theorems, and combinatorial decompositions reduce the extraction of all degree-$d$ irreducible factors to polynomial-time algorithms relying on a small number of univariate and low-degree bivariate factorization steps, using projections of support polygons and Puiseux expansions (Grenet, 2014). For sparse and circuit-structured polynomials, deterministic reductions show that all constant-degree irreducible factors can be computed in time polynomial, quasipolynomial, or subexponential in input size, provided access to PIT and divisibility oracles for the class (Dutta et al., 26 Nov 2024).

5. Pseudorandomness and Derandomization via Low-Degree Techniques

Explicit pseudorandom generators (PRGs) for classes defined by low-degree polynomial threshold functions (PTFs) are constructed by leveraging low-degree moment matching and Hermite/Taylor expansions. By summing $L$ blocks of $R$-moment-matching bounded-independence Gaussian vectors, one exactly matches all degree-$2dR$ moments, so any Boolean combination of $k$ degree-$d$ PTFs cannot distinguish from true Gaussian vectors except with $O(\epsilon)$ error. Discretization via Box–Muller transformations and careful error control yields explicit, seed-efficient PRGs fooling all such functions in seed length $\mathrm{poly}(k,d,1/\epsilon)\cdot\log n$. These PRGs are critical in complexity theory, derandomizing learning, and property testing tasks (Yao et al., 15 Apr 2025).

Testing and learning low-degree polynomials, especially over finite fields, is central in PCPs, property testing, and coding. Near-optimal robust testers (e.g., the line-point low-degree test) achieve $O(d)$ query complexity and constant-factor soundness even in the high-error regime by new bivariate and bootstrapping techniques that combine explicit algebraic "pencil" constructions, Newton lifting, and expander-based arguments (Harsha et al., 2023).

6. Applications to Matrix Computation, Counting, and Approximation

Low-degree polynomial methods enable deterministic, efficient algorithms for problems previously believed computationally hard:

• Approximate counting for PTFs: Efficient deterministic algorithms approximate $\Pr_{x\sim \{\pm1\}^n}[p(x)\ge0]$ for degree-$d$ PTFs to additive $\pm \epsilon$ in time $O_{d,\epsilon}(1)\cdot\operatorname{poly}(n^d)$, using a new central limit theorem for Gaussian polynomials via Malliavin calculus and decompositions into eigenregular low-degree polynomials. This enables reduction from the Boolean cube to Gaussian space and finally to low-dimensional evaluation (De et al., 2013).
• Principal component projection: Spectral projectors can be computed without explicit eigen-decomposition by replacing rational matrix functions with carefully optimized low-degree polynomials, then applying Chebyshev approximations to sharp thresholds. Optimal linear and quadratic polynomial choices minimize approximation error, with empirical $10$–$50\times$ improvements in running time over prior methods (Farnham et al., 2019).
• Defining equations for algebraic varieties: In algebraic complexity, explicit low-degree ($n^3$) annihilating polynomials are constructed for the varieties of non-rigid matrices, small linear circuits, and tensor varieties, establishing Zariski-closure equations of polynomial degree and opening new avenues for derandomization/lower-bound proofs (Kumar et al., 2020).

7. Structure, Irreducibility, and Randomness in Low-Degree Polynomials

In algebraic settings, precise results are known for the divisibility of polynomials by nonnegative, low-degree multipliers, with exact formulas for quadratics and cubics and tight $O(d^2)$ general bounds. Constructive, finitely terminating membership criteria in real cones identify the least degree required. For random polynomials (including random matrices), the low-degree method combines algebraic integer counting and root delocalization bounds to upper bound the probability that a random polynomial has an irreducible factor of degree $\le d$. This two-step approach—delocalization and enumeration—yields exponentially small probabilities for low-degree factors in large random models, with broad applicability to random matrix theory and irreducibility (Kepka et al., 2012, O'Rourke et al., 2016).

Finally, in random algebraic geometry, it is shown that the zero set of a high-degree random Kostlan polynomial on $S^n$ can be $C^1$-approximated (and thus is isotopic) to the zero set of an explicit $O(\sqrt{d\log d})$-degree polynomial, with overwhelming probability. This is established via fine probabilistic tail bounds on spherical harmonic expansions and norm comparison inequalities, ensuring topological stability under low-degree approximation (Diatta et al., 2018).

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References

• Privacy-preserving ResNet models with depth-optimal low-degree polynomial activations (Chielle et al., 26 Sep 2025)
• Improved optimal low-degree tests via global hypercontractivity (Kaufman et al., 2022)
• Computational barriers to estimation with low-degree polynomials (Schramm et al., 2020)
• Low-degree lower bounds via almost orthonormal bases (Carpentier et al., 11 Sep 2025)
• Fast, deterministic approximate counting for low-degree polynomial threshold functions (De et al., 2013)
• Efficient low-degree factorization of sparse/lacunary polynomials (Grenet, 2014, Dutta et al., 26 Nov 2024)
• Robust, attribute-efficient learning of sparse low-degree PTFs (Zeng et al., 2023)
• Principal component projection with low-degree polynomials (Farnham et al., 2019)
• Robust and optimal low-degree tests in finite field settings (Harsha et al., 2023)
• Low-degree factors of random polynomials and random matrices (O'Rourke et al., 2016)
• Minimal degree divisors of polynomials with nonnegative coefficients (Kepka et al., 2012)
• Low-degree topological approximation of random polynomials (Diatta et al., 2018)