The strength of collisional effects in a plasma can be summarized by the non-dimensional Knudsen number Kn. For very small values of Kn, plasma behavior is well-described by the 5N moment fluid model, including the Br...
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A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if h(x) = Af(x) + b where A is a linear map and f is the output of the penultimate l...
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In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singulari...
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We develop Banach spaces for ReLU neural networks of finite depth L and infinite width. The spaces contain all finite fully connected L-layer networks and their L2-limiting objects under bounds on the natural path-nor...
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We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solut...
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We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain ...
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The first computer implementation of the Dantzig-Fulkerson- Johnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s ...
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Most existing theoretical analysis of reinforcement learning (RL) is limited to the tabular setting or linear models due to the difficulty in dealing with function approximation in high dimensional space with an uncer...
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In single particle reconstruction (SPR) from cryo-electron microscopy (EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed alrea...
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In single particle reconstruction (SPR) from cryo-electron microscopy (EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix of size (2l+1)×(2l +1) for each l = 0, 1, 2, · · ·. In this paper we show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data.
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