Related papers: Hessian Operators on Constraint Manifolds

Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By…

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…

Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth…

This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based…

In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the…

We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization algorithm. In order to do this, we…

We develop \emph{geometric optimisation} on the manifold of Hermitian positive definite (HPD) matrices. In particular, we consider optimising two types of cost functions: (i) geodesically convex (g-convex); and (ii) log-nonexpansive (LN).…

We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…

We will propose a new algorithm for finding critical points of cost functions defined on a differential manifold. We will lift the initial cost function to a manifold that can be embedded in a Riemannian manifold (Euclidean space) and will…

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate…

We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group $O(d)$ and naturally reductive homogeneous manifolds obtained from the action of the rotation group $SO(d)$. We theoretically and…

We consider optimization problems with manifold-valued constraints. These generalize classical equality and inequality constraints to a setting in which both the domain and the codomain of the constraint mapping are smooth manifolds. We…

In this work, we introduce an algorithm to compute the derivatives of physical observables along the constrained subspace when flexible constraints are imposed on the system (i.e., constraints in which the hard coordinates are fixed to…

In this paper, an efficient modified Newton type algorithm is proposed for nonlinear unconstrianed optimization problems. The modified Hessian is a convex combination of the identity matrix (for steepest descent algorithm) and the Hessian…

This paper proposes several novel optimization algorithms for minimizing a nonlinear objective function. The algorithms are enlightened by the optimal state trajectory of an optimal control problem closely related to the minimized objective…

We introduce a novel approach to perform first-order optimization with orthogonal and unitary constraints. This approach is based on a parametrization stemming from Lie group theory through the exponential map. The parametrization…

In Riemannian optimization, it is well known that the condition number of the Riemannian Hessian at an optimum strongly influences the asymptotic convergence behavior of optimization algorithms. On the manifold of symmetric positive…

The problem of ensuring constraints satisfaction on the output of machine learning models is critical for many applications, especially in safety-critical domains. Modern approaches rely on penalty-based methods at training time, which do…

Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical…

In the first part, we obtain sharp results for L^2 boundedness of strongly singular operators on the Heisenberg group. We also define the oscillating convolution operators on the Heisenberg group and study their boundedness properties. In…