Study Links k‑XOR and Tensor PCA via Average‑Case Reductions

Global: Study Links k‑XOR and Tensor PCA via Average‑Case Reductions

Researchers Guy Bresler and Alina Harbuzova submitted a paper on 26 January 2026 that proposes new average‑case reductions connecting the noisy k‑XOR problem with Tensor Principal Component Analysis (Tensor PCA). The work, posted on the arXiv preprint server, aims to relate the computational properties of these two canonical planted problems and to clarify their respective thresholds for efficient algorithms.

Background

Both k‑XOR, a Boolean constraint satisfaction problem, and Tensor PCA, a high‑dimensional signal recovery task, serve as benchmark challenges in computational complexity, cryptography, and statistical inference. Researchers study them to understand where polynomial‑time algorithms succeed or fail under random noise.

Key Contributions

The authors introduce a family of intermediate problems that smoothly interpolate between k‑XOR and Tensor PCA, allowing variations in observation density and signal strength. This framework enables systematic comparison of the two problems across a broader parameter space.

Densifying Reductions

According to the paper, two novel “densifying” reductions increase the number of observed entries while carefully controlling the accompanying loss of signal. These reductions can transform any k‑XOR instance positioned at its computational threshold into a Tensor PCA instance that lies at the corresponding threshold, thereby establishing a direct computational equivalence.

Order‑Reducing Maps

The study also presents new order‑reducing mappings, such as a 5‑to‑4 reduction for k‑XOR and a 7‑to‑4 reduction for Tensor PCA, that preserve entry density while lowering problem order. The authors note that these maps provide additional tools for analyzing problem hardness at fixed densities.

Implications for Complexity Theory

By linking the thresholds of k‑XOR and Tensor PCA, the reductions suggest that breakthroughs—or limitations—in algorithms for one problem may translate to the other. The authors propose that future work could leverage this connection to explore tighter hardness results or to design algorithms that operate near these thresholds.

Publication Details

The paper is classified under Computational Complexity (cs.CC), Cryptography and Security (cs.CR), Probability (math.PR), and Statistics Theory (math.ST) on arXiv. It carries the identifier arXiv:2601.19016 and is accessible via DOI https://doi.org/10.48550/arXiv.2601.19016. As a preprint, it is available under an open‑access license.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.