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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). 

Although graph convolutional networks (GCNs) that extend the convolution operation from images to graphs have led to competitive performance, the existing GCNs are still difficult to handle a variety of applications, especially cheminformatics problems. Recently multiple GCNs are applied to chemical compound structures which are represented by the hydrogen-depleted molecular graphs of different size. GCNs built for a binary adjacency matrix that reflects the connectivity among nodes in a graph do not account for the edge consistency in multiple molecular graphs, that is, chemical bonds (edges) in different molecular graphs can be similar due to the similar enthalpy and interatomic distance. In this paper, we propose a variant of GCN where a molecular graph is first decomposed into multiple views of the graph, each comprising a specific type of edges. In each view, an edge consistency constraint is enforced so that similar edges in different graphs can receive similar attention weights when passing information. Similarly to prior work, we prove that in each layer, our method corresponds to a spectral filter derived by the first order Chebyshev approximation of graph Laplacian. Extensive experiments demonstrate the substantial advantages of the proposed technique in quantitative structure-activity relationship prediction.