mlsanityData to 7 Oct 2026

Continuous LWE is as Hard as LWE & Applications to Learning Gaussian Mixtures

arXiv 2204.02550 ↗First posted 6 Apr 2022cs.CR cs.LGcited by 26

Not on the map. Why?

Abstract (arXiv)

We show direct and conceptually simple reductions between the classical learning with errors (LWE) problem and its continuous analog, CLWE (Bruna, Regev, Song and Tang, STOC 2021). This allows us to bring to bear the powerful machinery of LWE-based cryptography to the applications of CLWE. For example, we obtain the hardness of CLWE under the classical worst-case hardness of the gap shortest vector problem. Previously, this was known only under quantum worst-case hardness of lattice problems. More broadly, with our reductions between the two problems, any future developments to LWE will also apply to CLWE and its downstream applications. As a concrete application, we show an improved hardness result for density estimation for mixtures of Gaussians. In this computational problem, given sample access to a mixture of Gaussians, the goal is to output a function that estimates the density function of the mixture. Under the (plausible and widely believed) exponential hardness of the classical LWE problem, we show that Gaussian mixture density estimation in ℝn with roughly log n Gaussian components given mathsfpoly(n) samples requires time quasi-polynomial in n. Under the (conservative) polynomial hardness of LWE, we show hardness of density estimation for nε Gaussians for any constant ε > 0, which improves on Bruna, Regev, Song and Tang (STOC 2021), who show hardness for at least √n Gaussians under polynomial (quantum) hardness assumptions. Our key technical tool is a reduction from classical LWE to LWE with k-sparse secrets where the multiplicative increase in the noise is only O(√k), independent of the ambient dimension n.

Key facts. Cited by 26 papers in the map (data through 7 Oct 2026). Builds on 8 of its 58 references that are in the map. First posted on arXiv 6 Apr 2022.

Builds on

8 of 58 references.

References · 8

  1. 2016Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixturescited by 114
  2. 2017Mixture Models, Robustness, and Sum of Squares Proofscited by 81
  3. 2017List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussianscited by 70
  4. 2020Continuous LWEcited by 37
  5. 2017On Learning Mixtures of Well-Separated Gaussianscited by 30
  6. 2014Tight bounds for learning a mixture of two gaussianscited by 29
  7. 2021On the Cryptographic Hardness of Learning Single Periodic Neuronscited by 20
  8. 2013Classical Hardness of Learning with Errorscited by 11

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 4; 2023: 5; 2024: 11; 2025: 4; 2026: 2.

26 papers.

Sort:

By area:

  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 440
  2. 2023Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginalscited by 31
  3. 2022Cryptographic Hardness of Learning Halfspaces with Massart Noisecited by 25
  4. 2022Hardness of Agnostically Learning Halfspaces from Worst-Case Lattice Problemscited by 16
  5. 2022Learning (Very) Simple Generative Models Is Hardcited by 9
  6. 2023Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noisecited by 9
  7. 2024Bypassing the Noisy Parity Barrier: Learning Higher-Order Markov Random Fields from Dynamicscited by 8
  8. 2024Improved Hardness Results for Learning Intersections of Halfspacescited by 6
  9. 2024Sum-of-squares lower bounds for Non-Gaussian Component Analysiscited by 6
  10. 2024Implicit High-Order Moment Tensor Estimation and Learning Latent Variable Modelscited by 5
  11. 2024Distribution-Specific Auditing For Subgroup Fairnesscited by 4
  12. 2024Computational-Statistical Gaps for Improper Learning in Sparse Linear Regressioncited by 4
14 more
  1. 2025Testing Noise Assumptions of Learning Algorithmscited by 4
  2. 2023SQ Lower Bounds for Random Sparse Planted Vector Problemcited by 3
  3. 2024Cryptographic Hardness of Score Estimationcited by 3
  4. 2025Quantum advantage for learning shallow neural networks with natural data distributionscited by 3
  5. 2025Low degree conjecture implies sharp computational thresholds in stochastic block modelcited by 2
  6. 2024Sparse Linear Regression and Lattice Problemscited by 1
  7. 2024On the Hardness of Learning One Hidden Layer Neural Networkscited by 1
  8. 2024CLUE-MARK: Watermarking Diffusion Models using CLWEcited by 1
  9. 2023SQ Lower Bounds for Learning Bounded Covariance GMMscited by 0
  10. 2023Polynomial Bounds for Learning Noisy Optical Physical Unclonable Functions and Connections to Learning With Errorscited by 0
  11. 2024Sample-Efficient Private Learning of Mixtures of Gaussianscited by 0
  12. 2025On Learning Parallel Pancakes with Mostly Uniform Weightscited by 0
  13. 2026Near-Optimal Cryptographic Hardness of Learning With Homogeneous Halfspaces Under Gaussian Marginalscited by 0
  14. 2026Implementing a White-Box Undetectable Backdoor for Random Fourier Featurescited by 0