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Understanding and mitigating noise in quantum systems is a fundamental challenge in achieving scalable and fault-tolerant quantum computation. Error modeling for quantum systems can be formulated in many ways, some of which are very…
We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish…
We consider a class of nonsmooth optimization problems over the Stiefel manifold, in which the objective function is weakly convex in the ambient Euclidean space. Such problems are ubiquitous in engineering applications but still largely…
Matrix-valued optimization tasks, including those involving symmetric positive definite (SPD) matrices, arise in a wide range of applications in machine learning, data science and statistics. Classically, such problems are solved via…
A many-body Hamiltonian can be block-diagonalized by expressing it in terms of symmetry-adapted basis states. Finding the group orbit representatives of these basis states and their corresponding symmetries is currently a…
We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn from…
In this paper, we develop a regularized higher-order Taylor based method for solving composite (e.g., nonlinear least-squares) problems. At each iteration, we replace each smooth component of the objective function by a higher-order Taylor…
The effectiveness of dimensionality reduction with quadratic manifolds hinges on the choice of a reduced basis and the associated quadratic correction terms. Existing approaches typically rely on subspaces spanned by the leading principal…
In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., $f + h$, where $f$ is continuously differentiable,…
Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science and becomes especially challenging for high-dimensional complex structure problems. One specific issue is that of quantum process…
This work considers optimization of composition of functions in a nested form over Riemannian manifolds where each function contains an expectation. This type of problems is gaining popularity in applications such as policy evaluation in…
We advocate a new approach of addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the Hidden Symmetry Subgroup Problem (HSSP), which is a generalization of the well-studied Hidden…
The paper addresses the problem of optimizing a class of composite functions on Riemannian manifolds and a new first order optimization algorithm (FOA) with a fast convergence rate is proposed. Through the theoretical analysis for FOA, it…
While classical data analysis has addressed observations that are real numbers or elements of a real vector space, at present many statistical problems of high interest in the sciences address the analysis of data that consist of more…
Rotation averaging is a key subproblem in applications of computer vision and robotics. Many methods for solving this problem exist, and there are also several theoretical results analyzing difficulty and optimality. However, one aspect…
We give robust recovery results for synchronization on the rotation group, $\mathrm{SO}(D)$. In particular, we consider an adversarial corruption setting, where a limited percentage of the observations are arbitrarily corrupted. We give a…
We study inverse boundary problems for semilinear Schr\"odinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities…
The multi-reference alignment (MRA) problem involves reconstructing a signal from multiple noisy observations, each transformed by a random group element. In this paper, we focus on the group \(\mathrm{SO}(2)\) of in-plane rotations and…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
A landmark result of non-smooth convex optimization is that gradient descent is an optimal algorithm whenever the number of computed gradients is smaller than the dimension $d$. In this paper we study the extension of this result to the…