Near Optimal Algorithms for Noisy $k$-XOR under Low-Degree Heuristic

Abstract: Noisy $k$-XOR is a basic average-case inference problem in which one observes random noisy $k$-ary parity constraints and seeks to recover, or more weakly, detect, a hidden Boolean assignment. A central question is to characterize the tradeoff among sample complexity, noise level, and running time. We give a recovery algorithm, and hence also a detection algorithm, for noisy $k$-XOR in the high-noise regime. For every parameter $D$, our algorithm runs in time $n<sup>{D+O(1)}$ and succeeds whenever $$m \ge C_k \frac{n^{{k/2}}{D^{{\,k/2-1}δ^2},}}$$ where $C_k$ is an explicit constant depending only on $k$, and $δ$ is the noise bias. Our result matches the best previously known time--sample tradeoff for detection, while simultaneously yielding recovery guarantees. In addition, the dependence on the noise bias $δ$ is optimal up to constant factors, matching the information-theoretic scaling. We also prove matching low-degree lower bounds. In particular, we show that the degree-$D$ low-degree likelihood ratio has bounded $L<sup>2$-norm below the same threshold, up to the same factor $D<sup>{k/2-1}$. Under the low-degree heuristic, this implies that our algorithm is near-optimal over a broad range of parameters. Our approach combines a refined second-moment analysis with color coding and dynamic programming for structured hypergraph embedding statistics. These techniques may be of independent interest for other average-case inference problems.

• The paper introduces an explicit algorithm for noisy k-XOR recovery that eliminates previous sample complexity overhead by matching the optimal noise bias threshold.
• It leverages hypergraph statistics, color coding, and dynamic programming to achieve a precise tradeoff between runtime and sample complexity.
• A corresponding low-degree lower bound confirms that any n^(O(D))-time algorithm requires Ω(n^(k/2)/(D^(k/2-1)δ^2)) samples, establishing the algorithm’s optimality.

Near-Optimal Algorithms for Noisy $k$-XOR under the Low-Degree Heuristic

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Problem Statement and Context

The paper addresses the noisy $k$-XOR problem, equivalently known as sparse Learning Parity with Noise (LPN). Here, one observes $m$ random $k$-XOR constraints—parity checks over exactly $k$ variables—each presenting a noisy label: $z_\alpha = x_\alpha \xi_\alpha$, where the hidden Boolean vector $x \in \{\pm1\}^n$ is to be recovered, and $\xi_\alpha$ is a Rademacher noise with mean $\delta$. The central computational task is either recovery (estimating $x$) or detection (distinguishing a planted assignment from pure noise), in a parameterized regime (number of samples $k$0, arity $k$1, noise bias $k$2).

Noisy $k$3-XOR is foundational in average-case complexity, learning theory, and cryptography (e.g., as a hardness assumption for various primitives). Prior work established the statistical threshold for detection at $k$4, but efficient recovery algorithms matching this were unattainable in the high-noise regime—there was a persistent overhead in sample complexity (especially in $k$5).

The paper studies this gap exhaustively, seeking to resolve the optimal tradeoff between sample complexity, running time, and noise bias for recovery (and thus, detection), within the low-degree paradigm.

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Main Results

1. Algorithmic Upper Bound

The authors develop an explicit algorithm for noisy $k$6-XOR that, for any arity $k$7 and parameter $k$8, achieves runtime $k$9 and succeeds at recovery whenever

$m$0

where $m$1 is a constant depending only on $m$2. This eliminates the prior polylogarithmic factors and, crucially, is information-theoretically optimal in $m$3, matching the lower bound implied by channel capacity considerations.

Unlike existing algorithms (e.g., Basu et al.), this method closes the sample complexity gap in $m$4, unifying detection and recovery thresholds. The result also produces a tight computational-sample tradeoff: as $m$5 grows, runtime increases while the required number of samples decreases, interpolating all the way down to the statistical threshold.

2. Matching Low-Degree Lower Bound

Theoretical guarantees are established by analyzing the low-degree likelihood ratio, following the framework where the existence of low-degree distinguishing polynomials correlates with the existence of efficient algorithms. The paper proves that:

$m$6

is required for any degree-$m$7 polynomial (i.e., any $m$8-time algorithm) to distinguish or recover, thus conjecturally establishing optimality of the presented algorithm. The combinatorial enumeration of even $m$9-uniform hypergraphs is essential for the explicit form of these bounds.

3. Algorithmic Construction

The algorithm leverages:

• Hypergraph statistics: Aggregated sums over structured $k$0-uniform hypergraph patterns with specific regularity and connectivity constraints. Character sums are evaluated over families of degree-2 regular cycles and paths.
• Color coding and dynamic programming: Efficient enumeration of bounded-width hypergraph embeddings by introducing random coloring and DP recursion, with variance reduction via averaging—a blueprint adapted from recent advances in average-case inference (notably tensor PCA and planted CSPs).
• Clean-up (Rounding) Step: After a weak recovery step (recovering $k$1 up to a fixed error), a majority-vote postprocessing with fresh samples resolves all bits, exploiting independence to eliminate residual bias.

This modularity enables direct control over the runtime-sample tradeoff via the hypergraph gadget’s "width" ($k$2).

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Numerical Guarantees and Tightness

The paper’s main theorems give precise, explicit constants and asymptotics: for arity $k$3 and every $k$4 (a multiple of $k$5) in a specified range,

Algorithmic Guarantee:

• Runtime: $k$6
• Sample Complexity: $k$7
• Success Probability: $k$8, approaching $k$9 as $k$0 or $k$1 grows

Low-Degree Barrier:

• No $k$2-time algorithm can succeed for $k$3

The dependence on $k$4 is strictly quadratic, which is provably optimal, while the scaling in $k$5 and $k$6 matches the precise combinatorial threshold predicted by the low-degree method.

Contrasts with Prior Work

The results strictly improve over Basu et al. (who incur an $k$7 or worse overhead), and show that earlier search-to-decision reductions for $k$8-XOR are suboptimal except in regimes where computational hardness is unconstrained.

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Technical Contributions

Hypergraph-Based Mean-Variance Analysis

The detection statistic is a sum over hypergraph patterns with intricate regularity properties, selected so as to maximize signal under the planted model and minimize variance under the null model. Second-moment and combinatorial analyses guarantee separation in all relevant parameter regimes.

Color Coding and DP for Efficient Computation

The adoption of color coding (Alon et al.) transforms the inherently exponential pattern-matching task into a polynomial-time DP on bounded width forests, underpinning the runtime guarantees. Multiple colorings are averaged to mitigate additional variance without loss of bias.

Sample Model Equivalence and Statistical Reductions

Careful analysis shows that Bernoulli, fixed-with and fixed-without replacement versions of the model are asymptotically equivalent in the parameter regime, up to negligible additive differences. All technical results are formulated in the most convenient model for each argument.

Information-Computational Barrier under Low-Degree Heuristic

Matching upper and lower bounds in the low-degree framework provide strong evidence that the computational barrier found is tight, at least for constant or slowly growing noise rates. The combinatorial enumeration of even degree-$k$9 hypergraphs is addressed explicitly.

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Theoretical and Practical Implications

• Theoretically, the work precisely locates the onset of computational hardness for noisy $z_\alpha = x_\alpha \xi_\alpha$0-XOR, and specifies the optimal tradeoff attainable by any polynomial-time (or sub-exponential time) algorithm, aligning detection and recovery thresholds.
• Practically, these algorithms have potential application for cryptanalysis in cryptographic primitives based on sparse LPN assumptions, as they close possible gaps between decision and search hardness at high noise rates—the regime exploited in cryptography.
• The general technique—pattern statistics + color coding for average-case inference—serves as a model for advanced algorithmic design in high-dimensional, noisy combinatorial problems.

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Future Directions

The analysis reveals that in the low-noise regime ($z_\alpha = x_\alpha \xi_\alpha$1), the standard low-degree heuristic may fail to capture all tractability phenomena, as certain algorithmic reductions (cf. Chen et al.) go beyond low-degree predictions. A precise conjectural framework interpolating constant- and vanishing-noise complexity remains open.

Further, extensions to more general planted CSPs, variants with additional combinatorial structure, and observation models with more general noise or sampling may admit similar hypergraph-based reductions and yield new algorithmic frontiers.

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Conclusion

This work settles the optimal sample--runtime--noise tradeoff for recovery in the noisy $z_\alpha = x_\alpha \xi_\alpha$2-XOR problem, establishing that low-degree polynomial algorithms achieve the finest possible threshold predicted by current computational heuristics. The synthesis of hypergraph statistics, color coding, and low-degree analysis provides a robust framework expandable to a broad class of average-case high-dimensional inference problems.