arXiv subjects

Shosuke Kiami

Publications and source records attributed to Shosuke Kiami.

3 recordsLinked to original sources

arXiv2024

Approximately Jumping Towards the Origin

Given an initial point $x_0 \in \mathbb{R}^d$ and a sequence of vectors $v_1, v_2, \dots$ in $\mathbb{R}^d$, we define a greedy sequence by setting $x_{n} = x_{n-1} \pm v_n$ where the sign is chosen so as to minimize $\|x_n\|$. We prove that if the vectors $v_i$ are chosen uniformly at random from $\mathbb{S}^{d-1}$ then elements of the sequence are, on average, approximately at distance $\|x_n\| \sim \sqrt{\pi d/8}$ from the origin. We show that the sequence $(\|x_n\|)_{n=1}^{\infty}$ has an invariant measure $\pi_d$ depending only on $d$ and we determine its mean and study its decay for all $d$. We also investigate a completely deterministic example in $d=2$ where the $v_n$ are derived from the van der Corput sequence. Several additional examples are considered.

math.PR↗

arXiv2024

Lossless Convexification for Linear Systems with Piecewise Linear Controls

Lossless Convexification (LCvx) is a convexification technique that transforms a class of nonconvex optimal control problems$\unicode{x2013}$where the nonconvexity arises from a lower bound on the control norm$\unicode{x2013}$into equivalent convex problems, with the goal being to apply fast polynomial-time solvers. However, to solve these infinite-dimensional problems in practice, they must first be converted into finite-dimensional problems, and it remains an open challenge to ensure the theoretical guarantees of LCvx are maintained across this discretization step. Prior work has proven guarantees for piecewise constant controls, but these methods do not extend to piecewise linear controls, which are more relevant to real world applications. In this work, we present an algorithm that extends LCvx guarantees to piecewise linear controls. Under mild assumptions, our algorithm provably finds a solution violating the nonconvex constraints along at most $2n_x + 2$ trajectory "edges" using $O(\log(\Delta\rho/\varepsilon))$ solver calls (where $n_x$ is the state space dimension and $\Delta\rho = \rho_{\max} - \rho_{\min}$ is the difference in our control norm bounds). A key feature is the perturbation of the control norm lower bound and the addition of rate constraints on the controls, ensuring LCvx holds along the trajectory edges. Finally, we provide numerical results demonstrating the effectiveness of our algorithm.

math.OC↗

arXiv2024

Finding high posterior density phylogenies by systematically extending a directed acyclic graph

Bayesian phylogenetics typically estimates a posterior distribution, or aspects thereof, using Markov chain Monte Carlo methods. These methods integrate over tree space by applying local rearrangements to move a tree through its space as a random walk. Previous work explored the possibility of replacing this random walk with a systematic search, but was quickly overwhelmed by the large number of probable trees in the posterior distribution. In this paper we develop methods to sidestep this problem using a recently introduced structure called the subsplit directed acyclic graph (sDAG). This structure can represent many trees at once, and local rearrangements of trees translate to methods of enlarging the sDAG. Here we propose two methods of introducing, ranking, and selecting local rearrangements on sDAGs to produce a collection of trees with high posterior density. One of these methods successfully recovers the set of high posterior density trees across a range of data sets. However, we find that a simpler strategy of aggregating trees into an sDAG in fact is computationally faster and returns a higher fraction of probable trees.

q-bio.PE↗