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Data augmentation has been proven to be an effective technique for developing machine learning models that are robust to known classes of distributional shifts (e.g., rotations of images), and alignment regularization is a technique often…
Randomized iterative algorithms for solving a factorized linear system, $\mathbf A\mathbf B\mathbf x=\mathbf b$ with $\mathbf A\in{\mathbb{R}}^{m\times \ell}$, $\mathbf B\in{\mathbb{R}}^{\ell\times n}$, and $\mathbf b\in{\mathbb{R}}^m$,…
Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may…
Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems as smooth unconstrained problems over the set of symmetric matrices which are then solved via the cubic-regularized Newton method. A…
The Hubbard model at finite chemical potential is a cornerstone for understanding doped correlated systems, but simulations are severely limited by the sign problem. In the auxiliary-field formulation, the spin basis mitigates the sign…
In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic…
Low rank approximation is a commonly occurring problem in many computer vision and machine learning applications. There are two common ways of optimizing the resulting models. Either the set of matrices with a given rank can be explicitly…
Nonconvex methods have emerged as a dominant approach for low-rank matrix estimation, a problem that arises widely in machine learning and AI for learning and representing high-dimensional data. Existing analyses for these methods often…
The typical methods for symbolic regression produce rather abrupt changes in solution candidates. In this work, we have tried to transform symbolic regression from an optimization problem, with a landscape that is so rugged that typical…
A general method for solving nonlinear ill-posed problems is developed. The method consists of solving a Cauchy problem with a regularized operator and proving that the solution of this problem tends, as time grows, to a solution of the…
We discuss a method for regularizing chiral gauge theories. The idea is to formulate the gauge fields on the lattice, while the fermion determinant is regularized and computed in the continuum. A simple effective action emerges which lends…
We present a solution to the sign problem in dynamical random matrix simulations of a two-matrix model at nonzero chemical potential. The sign problem, caused by the complex fermion determinants, is solved by gathering the matrices into…
Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their…
In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…
This paper presents a method for alleviating sign problems in lattice path integrals, including those associated with finite fermion density in relativistic systems. The method makes use of information gained from some systematic expansion…
The quantum Monte Carlo method on asymptotic Lefschetz thimbles is a numerical algorithm devised specifically for alleviation of the sign problem appearing in the simulations of quantum many-body systems. In this method, the sign problem is…
This paper presents two novel regularization methods motivated in part by the geometric significance of biorthogonal bases in signal processing applications. These methods, in particular, draw upon the structural relevance of orthogonality…
We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization…
In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares…
We consider first the zero-nonzero determination problem, which consists in determining the list of zero-nonzero conditions realized by a finite list of polynomials on a finite set Z included in C^k with C an algebraic closed field. We…