Recent years have seen the rapid growth of new approaches to optical imaging, with an emphasis on extracting threedimensional (3D) information from what is normally a twodimensional (2D) image capture. Perhaps most importantly, the rise of computational imaging enables both new physical layouts of optical components and new algorithms to be implemented. This paper concerns the convergence of two advances: the development of a transparent focal stack imaging system using graphene photodetector arrays, and the rapid expansion of the capabilities of machine learning including the development of powerful neural networks. This paper demonstrates 3D tracking of pointlike objects withmore »
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Abstract 
Free, publiclyaccessible full text available November 1, 2022

This paper discusses algorithms for phase retrieval where the measurements follow independent Poisson distributions. We developed an optimization problem based on maximum likelihood estimation (MLE) for the Poisson model and applied Wirtinger flow algorithm to solve it. Simulation results with a random Gaussian sensing matrix and Poisson measurement noise demonstrated that the Wirtinger flow algorithm based on the Poisson model produced higher quality reconstructions than when algorithms derived from Gaussian noise models (Wirtinger flow, Gerchberg Saxton) are applied to such data, with significantly improved computational efficiency.

We propose a randomized algorithm with quadratic convergence rate for convex optimization problems with a selfconcordant, composite, strongly convex objective function. Our method is based on performing an approximate Newton step using a random projection of the Hessian. Our first contribution is to show that, at each iteration, the embedding dimension (or sketch size) can be as small as the effective dimension of the Hessian matrix. Leveraging this novel fundamental result, we design an algorithm with a sketch size proportional to the effective dimension and which exhibits a quadratic rate of convergence. This result dramatically improves on the classical linearquadraticmore »

We develop a convex analytic approach to analyze finite width twolayer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rankone data. We also characterize the classification decision regions in terms of a kernel matrix and minimum `1norm solutions. This is in contrast to Neural Tangent Kernel which is unable tomore »

Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although posttraining quantization is wellstudied, training optimal quantized NNs involves combinatorial nonconvex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in twolayer neural networks from the recent literature, semidefinite lifting, and Grothendieck’s identity. Surprisingly, we show that certain quantized NN problems can be solved to globalmore »

We describe the convex semiinfinite dual of the twolayer vectoroutput ReLU neural network training problem. This semiinfinite dual admits a finite dimensional representation, but its support is over a convex set which is difficult to characterize. In particular, we demonstrate that the nonconvex neural network training problem is equivalent to a finitedimensional convex copositive program. Our work is the first to identify this strong connection between the global optima of neural networks and those of copositive programs. We thus demonstrate how neural networks implicitly attempt to solve copositive programs via seminonnegative matrix factorization, and draw key insights from this formulation.more »

We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons, and data dimension. More specifically, we develop a convex analytic framework utilizing semiinfinite duality to obtain equivalent convex optimization problems for several two and threelayer CNN architectures. We first prove that twolayer CNNs can be globally optimized via an `2 norm regularized convex program. We then show that multilayer circular CNN training problems with a single ReLU layer are equivalent to an `1 regularized convex program thatmore »