Year: 2024
Type: preprint
Abstract: We consider sparse variants of the classical Learning Parities with random Noise (LPN) problem. Our main contribution is a new algorithmic framework that provides learning algorithms against low-noise for both Learning Sparse Parities (LSPN) problem and sparse LPN problem. Different from previous approaches for LSPN and sparse LPN, this framework has a simple structure and runs in polynomial space. Let $n$ be the dimension, $k$ denote the sparsity, and $η$ be the noise rate. As a fundamental problem in computational learning theory, Learning Sparse Parities with Noise (LSPN) assumes the hidden parity is $k$-sparse. While a simple enumeration algorithm takes ${n \choose k}=O(n/k)^k$ time, previously known results stills need ${n \choose k/2} = Ω(n/k)^{k/2}$ time for any noise rate $η$. Our framework provides a LSPN algorithm runs in time $O(η\cdot n/k)^k$ for any noise rate $η$, which improves the state-of-the-art of LSPN whenever $η\in ( k/n,\sqrt{k/n})$. The sparse LPN problem is closely related to the classical problem of refuting random $k$-CSP and has been widely used in cryptography as the hardness assumption. Different from the standard LPN, it samples random $k$-sparse vectors. Because the number of $k$-sparse vectors is ${n \choose k}n^{k/2}$. However, much less is known about learning algorithms for constant $k$ like 3 and $m
Source: arXiv (Cornell University)
Authors: Chen, Xue, Shu, Wenxuan, Zhou, Zhaienhe
Institution: University of Science and Technology of China
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G1997119393
OpenAlex ID: W4401201533