Continuous LWE
This work resolves open problems in computational complexity for learning mixtures of Gaussians without separability assumptions and addresses computational hardness questions in robust machine learning, though it is incremental in extending LWE to a continuous setting.
The authors introduced a continuous analogue of the Learning with Errors (LWE) problem called CLWE, providing a polynomial-time quantum reduction from worst-case lattice problems to CLWE, which establishes similar hardness guarantees as LWE.
We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to those of LWE. Alternatively, our result can also be seen as opening new avenues of (quantum) attacks on lattice problems. Our work resolves an open problem regarding the computational complexity of learning mixtures of Gaussians without separability assumptions (Diakonikolas 2016, Moitra 2018). As an additional motivation, (a slight variant of) CLWE was considered in the context of robust machine learning (Diakonikolas et al.~FOCS 2017), where hardness in the statistical query (SQ) model was shown; our work addresses the open question regarding its computational hardness (Bubeck et al.~ICML 2019).