Differentially Private Convex Optimization with Piecewise Affine Objectives

Differential privacy is a recently proposed notion of privacy that provides strong privacy guarantees without any assumptions on the adversary. The paper studies the problem of computing a differentially private solution to convex optimization problems whose objective function is piecewise affine. Such problem is motivated by applications in which the affine functions that define the objective function contain sensitive user information. We propose several privacy preserving mechanisms and provide analysis on the trade-offs between optimality and the level of privacy for these mechanisms. Numerical experiments are also presented to evaluate their performance in practice

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto \lambda_k(M,g)$. We propose a computational method for finding the conformal spectrum $\Lambda^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $\lambda_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $\gamma$. This is known as the topological spectrum for genus $\gamma$ and denoted by $\Lambda^t_k(\gamma)$. Our computations support a conjecture of N. Nadirashvili (2002) that $\Lambda^t_k(0) = 8 \pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that $\Lambda^t_k(1) = \frac{8\pi^2}{\sqrt{3}} + 8\pi (k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$-th Laplace-Beltrami eigenvalue has a local maximum with value $\lambda_k = 4\pi^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure