A Riemannian Optimization Approach to Clustering Problems

This paper considers the optimization problem in the form of $\min_{X \in \mathcal{F}_v} f(x) + \lambda \|X\|_1,$ where $f$ is smooth, $\mathcal{F}_v = \{X \in \mathbb{R}^{n \times q} : X^T X = I_q, v \in \mathrm{span}(X)\}$, and $v$ is a given positive vector. The clustering models including but not limited to the models used by $k$-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain $\mathcal{F}_v$ forms a compact embedded submanifold of $\mathbb{R}^{n \times q}$ and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of $\mathcal{F}_v$ are derived. An inexact accelerated Riemannian proximal gradient method that allows adaptive step size is proposed and its global convergence is established. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method

Fix Your Eyes in the Space You Could Reach: Neurons in the Macaque Medial Parietal Cortex Prefer Gaze Positions in Peripersonal Space

Interacting in the peripersonal space requires coordinated arm and eye movements to visual targets in depth. In primates, the medial posterior parietal cortex (PPC) represents a crucial node in the process of visual-to-motor signal transformations. The medial PPC area V6A is a key region engaged in the control of these processes because it jointly processes visual information, eye position and arm movement related signals. However, to date, there is no evidence in the medial PPC of spatial encoding in three dimensions. Here, using single neuron recordings in behaving macaques, we studied the neural signals related to binocular eye position in a task that required the monkeys to perform saccades and fixate targets at different locations in peripersonal and extrapersonal space. A significant proportion of neurons were modulated by both gaze direction and depth, i.e., by the location of the foveated target in 3D space. The population activity of these neurons displayed a strong preference for peripersonal space in a time interval around the saccade that preceded fixation and during fixation as well. This preference for targets within reaching distance during both target capturing and fixation suggests that binocular eye position signals are implemented functionally in V6A to support its role in reaching and grasping

Model reduction via truncation: an interpolation point of view

In this paper, we focus our attention on linear time invariant continuous time linear systems with one input and one output (SISO LTI systems). We consider the problem of constructing a reduced order system via truncation of the original system. Given a SISO strictly proper transfer function T(s) of McMillan degree N and a strictly proper SISO transfer function (T) over cap (s) of McMillan degree n < N, we prove that (T) over cap (s) can always be constructed via truncation of the system T(s). The proof is mainly based on interpolation theory, and more precisely on multipoint Pade interpolation. Moreover, new results about Krylov subspaces are developed. (C) 2003 Elsevier Inc. All rights reserved

CALCULATING THE H∞-NORM OF LARGE SPARSE SYSTEMS VIA CHANDRASEKHAR ITERATIONS AND EXTRAPOLATION

We describe an algorithm for estimating the H∞-norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the H∞-norm and then uses extrapolation to improve these estimates