We propose a variant of the classical conditional gradient method (CGM) for sparse inverse problems with differentiable measurement models. Such models arise in many practical problems including superresolution, time-series modeling, and matrix completion. Our algorithm combines nonconvex and convex optimization techniques: we propose global conditional gradient steps alternating with nonconvex local search exploiting the differentiable measurement model. This hybridization gives the theoretical global optimality guarantees and stopping conditions of convex optimization along with the performance and modeling flexibility associated with nonconvex optimization. Our experiments demonstrate that our technique achieves state-of-the-art results in several applications.

We propose a variant of the classical conditional gradient method (CGM) for sparse inverse problems with differentiable measurement models. Such models arise in many practical problems including superresolution, time-series modeling, and matrix completion. Our algorithm combines nonconvex and convex optimization techniques: we propose global conditional gradient steps alternating with nonconvex local search exploiting the differentiable measurement model. This hybridization gives the theoretical global optimality guarantees and stopping conditions of convex optimization along with the performance and modeling flexibility associated with nonconvex optimization. Our experiments demonstrate that our technique achieves state-of-the-art results in several applications.

Single-molecule localization super-resolution microscopy (SMLM) techniques like STORM and PALM have transformed cellular microscopy by substantially increasing spatial resolution. In this paper we introduce a new algorithm for a critical part of the SMLM process: estimating the number and locations of the fluorophores in a single frame. Our algorithm can analyze a 20000-frame experimental 3D SMLM dataset in about one second -substantially faster than real-time and existing algorithms. Our approach is straightforward but very different from existing algorithms: we train a neural network to minimize the Bayes' risk under a generative model for single SMLM frames. The neural network maps a frame directly to a collection of fluorophore locations, which we compare to the ground truth using a novel loss function. While training the neural network takes several hours, it only has to be done once for a given experimental setup. After training, localizing fluorophores in new images is extremely fast -orders of magnitude faster than existing algorithms. Faster recovery opens the door to real-time calibration and accelerated acquisition, and future work could tackle more complicated optical systems and more realistic simulators.

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