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Abstract Deep Learning (DL) has recently enabled unprecedented advances in one of the grand challenges in computational biology: the half-century-old problem of protein structure prediction. In this paper we discuss recent advances, limitations, and future perspectives of DL on five broad areas: protein structure prediction, protein function prediction, genome engineering, systems biology and data integration, and phylogenetic inference. We discuss each application area and cover the main bottlenecks of DL approaches, such as training data, problem scope, and the ability to leverage existing DL architectures in new contexts. To conclude, we provide a summary of the subject-specific and general challenges for DL across the biosciences. 

DeepTensor is a computationally efficient framework for low-rank decomposition of matrices and tensors using deep generative networks. We decompose a tensor as the product of low-rank tensor factors where each low-rank tensor is generated by a deep network (DN) that is trained in a self-supervised manner to minimize the mean-square approximation error. Our key observation is that the implicit regularization inherent in DNs enables them to capture nonlinear signal structures that are out of the reach of classical linear methods like the singular value decomposition (SVD) and principal components analysis (PCA). We demonstrate that the performance of DeepTensor is robust to a wide range of distributions and a computationally efficient drop-in replacement for the SVD, PCA, nonnegative matrix factorization (NMF), and similar decompositions by exploring a range of real-world applications, including hyperspectral image denoising, 3D MRI tomography, and image classification. 

We take a random matrix theory approach to random sketching and show an asymptotic first-order equivalence of the regularized sketched pseudoinverse of a positive semidefinite matrix to a certain evaluation of the resolvent of the same matrix. We focus on real-valued regularization and extend previous results on an asymptotic equivalence of random matrices to the real setting, providing a precise characterization of the equivalence even under negative regularization, including a precise characterization of the smallest nonzero eigenvalue of the sketched matrix. We then further characterize the second-order equivalence of the sketched pseudoinverse. We also apply our results to the analysis of the sketch-and-project method and to sketched ridge regression. Last, we prove that these results generalize to asymptotically free sketching matrices, obtaining the resulting equivalence for orthogonal sketching matrices and comparing our results to several common sketches used in practice.