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Average-Case Reductions for $k$-XOR and Tensor PCA

Guy Bresler, Alina Harbuzova

TL;DR

The paper develops a comprehensive framework of poly-time average-case reductions linking planted $k$-XOR and Tensor PCA across a continuum of densities and signal strengths. Central to the approach is the equation-combination primitive, along with discrete and Gaussian variants, that either densifies instances or reduces tensor order while preserving the planted structure and approximate output distributions. The authors establish a hardness partial order: hardness conjectures for one $(k,t)$ pair imply those for others, including reductions that connect canonical $k$-XOR to Tensor PCA and extend to $k$-sparse LWE. Beyond hardness, the work yields algorithmic consequences by transferring known strategies and gap phenomena across models, and introduces new normal-form variants (e.g., $k$-XOR$^{\mathrm{FULL}}$) that facilitate reductions toward Tensor PCA. These results illuminate a fundamental connection between average-case planted problems and bridge algorithmic techniques across discrete and Gaussian noise settings with implications for cryptography and high-dimensional statistics.

Abstract

We study two canonical planted average-case problems -- noisy $k\mathsf{\text{-}XOR}$ and Tensor PCA -- and relate their computational properties via poly-time average-case reductions. In fact, we consider a \emph{family of problems} that interpolates between $k\mathsf{\text{-}XOR}$ and Tensor PCA, allowing intermediate densities and signal levels. We introduce two \emph{densifying} reductions that increase the number of observed entries while controlling the decrease in signal, and, in particular, reduce any $k\mathsf{\text{-}XOR}$ instance at the computational threshold to Tensor PCA at the computational threshold. Additionally, we give new order-reducing maps (e.g., $5\to 4$ $k\mathsf{\text{-}XOR}$ and $7\to 4$ Tensor PCA) at fixed entry density.

Average-Case Reductions for $k$-XOR and Tensor PCA

TL;DR

The paper develops a comprehensive framework of poly-time average-case reductions linking planted -XOR and Tensor PCA across a continuum of densities and signal strengths. Central to the approach is the equation-combination primitive, along with discrete and Gaussian variants, that either densifies instances or reduces tensor order while preserving the planted structure and approximate output distributions. The authors establish a hardness partial order: hardness conjectures for one pair imply those for others, including reductions that connect canonical -XOR to Tensor PCA and extend to -sparse LWE. Beyond hardness, the work yields algorithmic consequences by transferring known strategies and gap phenomena across models, and introduces new normal-form variants (e.g., -XOR) that facilitate reductions toward Tensor PCA. These results illuminate a fundamental connection between average-case planted problems and bridge algorithmic techniques across discrete and Gaussian noise settings with implications for cryptography and high-dimensional statistics.

Abstract

We study two canonical planted average-case problems -- noisy and Tensor PCA -- and relate their computational properties via poly-time average-case reductions. In fact, we consider a \emph{family of problems} that interpolates between and Tensor PCA, allowing intermediate densities and signal levels. We introduce two \emph{densifying} reductions that increase the number of observed entries while controlling the decrease in signal, and, in particular, reduce any instance at the computational threshold to Tensor PCA at the computational threshold. Additionally, we give new order-reducing maps (e.g., and Tensor PCA) at fixed entry density.
Paper Structure (138 sections, 59 theorems, 407 equations, 7 figures, 23 algorithms)

This paper contains 138 sections, 59 theorems, 407 equations, 7 figures, 23 algorithms.

Table of Contents

  1. Introduction
  2. The two canonical problems.
  3. A new connection.
  4. Technique.
  5. A family of $k\mathsf{\text{-}XOR}$ problems
  6. Complexity Profiles
  7. Tasks.
  8. Computational Threshold.
  9. Information Theoretic Lower Bounds.
  10. Computational-Statistical Gap.
  11. Main Results: Hardness Partial Order
  12. Average-Case Reductions and Their Implications for Hardness
  13. Our Results
  14. Our Results: $k$-sparse LWE
  15. Main Reduction Ideas
  16. ...and 123 more sections

Key Result

Theorem 1.1

Fix constants $k' \geq 2, \varepsilon \in (0,1)$, and $\eta > 0$. For some sufficiently large $k = k(k',\varepsilon)$, if there is no polynomial-time algorithm solving detection (resp. recovery) for $k\mathsf{\text{-}XOR}^{}(n, m, \delta^\mathsf{xor})$ with then there is no polynomial-time algorithm solving detection (resp. recovery) for $k'\mathsf{\text{-}TensorPCA}(n, \delta^{\mathsf{tpca}})$ a

Figures (7)

  • Figure 1: Thm. \ref{['thm:decrease_k_intro']}: blue arrows represent reductions to odd $k'$, red -- to even.
  • Figure 2: Example Reductions of Thm. \ref{['thm:densifying_general']}: red arrow exemplifies reductions to sparsities $t' \in \left( 2t/(1-t), 1 \right)$ and blue arrows the reductions to Tensor PCA ($t'=1$).
  • Figure 3: Reduction of Cor. \ref{['cor:densifying_gen']}: conjectured hardness in the blue region implies hardness for the red region.
  • Figure 4: Example Reductions of Thm. \ref{['thm:specific_intro']}: red arrow exemplifies reductions to sparsities $t' \in \left( 2t/(1-t), 1 \right)$ and blue arrows the reductions to Tensor PCA ($t'=1$).
  • Figure 5: Conjectured hardness implication order "$\Rightarrow$" in the space of parameters $\left( k,t \right)$.
  • ...and 2 more figures

Theorems & Definitions (119)

  • Theorem 1.1: Corollary of Thm. \ref{['thm:densifying_general']}
  • Theorem 1.2: Corollary of Thm. \ref{['thm:decrease_k_intro']}
  • Definition 1.1: Family of $k\mathsf{\text{-}XOR}, k\mathsf{\text{-}Gauss}$ Models
  • Remark 1.1: Polynomial Scale
  • Proposition 1.1: Computational Equivalence of $k\mathsf{\text{-}XOR}$ and $k\mathsf{\text{-}Gauss}$
  • Proposition 1.2: Computational Equivalence of $k\mathsf{\text{-}Gauss}$ and Canonical Tensor PCA
  • Definition 1.2: Density Parameter $t$
  • Remark 1.2: Choice of Null Distribution
  • Definition 1.3: $k\mathsf{\text{-}XOR}$ Family, Reparametrized
  • Definition 1.4: Hardness Partial Order $\Rightarrow$
  • ...and 109 more