Average-Case Reductions for $k$-XOR and Tensor PCA
Hi there, little explorer! Imagine you have a big puzzle, but some pieces are hidden, and some are tricky.
Scientists are like detectives trying to find the hidden pieces in two super-duper hard puzzles called "k-XOR" and "Tensor PCA." They're like treasure hunts!
This news says they found a special secret map! This map shows how to turn one hard puzzle into another, maybe a little easier one. It's like finding out that if you can solve one type of treasure hunt, you can also solve another!
This helps them understand which puzzles are the most tricky for our smart computers to solve. It's like learning which treasure chests need the strongest keys! Yay for clever detectives!
arXiv:2601.19016v2 Announce Type: replace-cross Abstract: We study the computational properties of two canonical planted average-case problems -- noisy planted $k$-XOR and Tensor PCA -- by formally unifying them into a family of planted problems parametrized by tensor order $k$, number of entries $m$, and noise level $\delta$. We build a wide range of poly-time average-case reductions within this family, across all regimes $m \in [1, n^k]$. In the denser $m \geq n^{k/2}$ regime, our reductions preserve proximity to the computational threshold, and, as a central application, reduce conjectured-hard $k$-XOR instances with $m \approx n^{k/2}$ to conjectured-hard instances of Tensor PCA. Additionally, we give new order-reducing maps at fixed densities (e.g., $5\to 4$ for $k$-XOR with $m \appro
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Abstract:We study the computational properties of two canonical planted average-case problems -- noisy planted $k$-XOR and Tensor PCA -- by formally unifying them into a family of planted problems parametrized by tensor order $k$, number of entries $m$, and noise level $\delta$. We build a wide range of poly-time average-case reductions within this family, across all regimes $m \in [1, n^k]$. In the denser $m \geq n^{k/2}$ regime, our reductions preserve proximity to the computational threshold, and, as a central application, reduce conjectured-hard $k$-XOR instances with $m \approx n^{k/2}$ to conjectured-hard instances of Tensor PCA. Additionally, we give new order-reducing maps at fixed densities (e.g., $5\to 4$ for $k$-XOR with $m \approx n^{k/2}$ entries and $7\to 4$ for Tensor PCA). In the sparser $m \leq n^{k/2}$ regime, we relate instances of different orders, reducing, for example, $7$-XOR with $m = n^{3.4}$ to the classical setting of $3$-XOR with $m = \widetilde\Theta(n^{1.4})$. Taken together, these results establish a hardness partial order in the space of planted tensor models.
Comments: 112 pages, 6 figures
Subjects:
Computational Complexity (cs.CC); Cryptography and Security (cs.CR); Probability (math.PR); Statistics Theory (math.ST)
Cite as: arXiv:2601.19016 [cs.CC]
(or arXiv:2601.19016v2 [cs.CC] for this version)
https://doi.org/10.48550/arXiv.2601.19016
arXiv-issued DOI via DataCite
Submission history
From: Alina Harbuzova [view email] [v1] Mon, 26 Jan 2026 23:05:54 UTC (175 KB) [v2] Thu, 2 Apr 2026 16:45:31 UTC (179 KB)
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