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Person# Harshit Gupta

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An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source r

In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often u

In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spl

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Shayan Aziznejad, Joaquim Gonçalves Garcia Barreto Campos, Harshit Gupta, Michaël Unser

We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output relation. To that end, we first establish a global bound for the Lipschitz constant of neural networks. Based on the obtained bound, we then formulate a variational problem for learning activation functions. Our variational problem is infinite-dimensional and is not computationally tractable. However, we prove that there always exists a solution that has continuous and piecewise-linear (linear-spline) activations. This reduces the original problem to a finite-dimensional minimization where an l(1) penalty on the parameters of the activations favors the learning of sparse nonlinearities. We numerically compare our scheme with standard ReLU network and its variations, PReLU and LeakyReLU and we empirically demonstrate the practical aspects of our framework.

Laurène Donati, Harshit Gupta, Michael Thompson McCann, Michaël Unser

We present CryoGAN, a new paradigm for single-particle cryo-electron microscopy (cryo-EM) reconstruction based on unsupervised deep adversarial learning. In single-particle cryo-EM, the structure of a biomolecule needs to be reconstructed from a large set of noisy tomographic projections with unknown orientations. Current reconstruction techniques are based on a marginalized maximum-likelihood formulation that requires calculations over the set of all possible poses for each projection image, a computationally demanding procedure. Our approach is to seek a 3D structure that has simulated projections that match the real data in a distributional sense, thereby sidestepping pose estimation or marginalization. We prove that, in an idealized mathematical model of cryo-EM, this approach results in recovery of the correct structure. Motivated by distribution matching, we propose CryoGAN, a specialized GAN that consists of a 3D structure, a cryo-EM physics simulator, and a discriminator neural network. During reconstruction, the 3D structure is optimized so that its projections obtained through the simulator resemble real data (to the discriminator). Simultaneously, the discriminator is trained to distinguish real projections from simulated projections. CryoGAN takes as input only real projection images and the distribution of the cryo-EM imaging parameters. It involves neither prior training nor an initial estimation of the 3D structure. CryoGAN currently achieves a 10.8 angstrom resolution on a realistic synthetic dataset. Preliminary results on experimental beta-galactosidase and 80S ribosome data demonstrate the ability of CryoGAN to exploit data statistics under standard experimental imaging conditions. We believe that this paradigm opens the door to a family of novel likelihood-free algorithms for cryo-EM reconstruction.

Harshit Gupta, Thanh-An Michel Pham, Michaël Unser, Fangshu Yang

Optical diffraction tomography is an effective tool to estimate the refractive indices of unknown objects. It proceeds by solving an ill-posed inverse problem for which the wave equation governs the scattering events. The solution has traditionally been derived by the minimization of an objective function in which the data-fidelity term encourages measurement consistency while the regularization term enforces prior constraints. In this work, we propose to train a convolutional neural network (CNN) as the projector in a projected-gradient-descent method. We iteratively produce high-quality estimates and ensure measurement consistency, thus keeping the best of CNN-based and regularization-based worlds. Our experiments on two-dimensional-simulated and real data show an improvement over other conventional or deep-learning-based methods. Furthermore, our trained CNN projector is general enough to accommodate various forward models for the handling of multiple-scattering events. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement