40,979 research outputs found
Linear-quadratic stochastic differential games for distributed parameter systems
A linear-quadratic differential game with infinite dimensional state space is considered. The system state is affected by disturbance and both players have access to different measurements. Optimal linear strategies for the pursuer and the evader, when they exist, are explicitly determined
Distributed Basis Pursuit
We propose a distributed algorithm for solving the optimization problem Basis
Pursuit (BP). BP finds the least L1-norm solution of the underdetermined linear
system Ax = b and is used, for example, in compressed sensing for
reconstruction. Our algorithm solves BP on a distributed platform such as a
sensor network, and is designed to minimize the communication between nodes.
The algorithm only requires the network to be connected, has no notion of a
central processing node, and no node has access to the entire matrix A at any
time. We consider two scenarios in which either the columns or the rows of A
are distributed among the compute nodes. Our algorithm, named D-ADMM, is a
decentralized implementation of the alternating direction method of
multipliers. We show through numerical simulation that our algorithm requires
considerably less communications between the nodes than the state-of-the-art
algorithms.Comment: Preprint of the journal version of the paper; IEEE Transactions on
Signal Processing, Vol. 60, Issue 4, April, 201
Innovation Pursuit: A New Approach to Subspace Clustering
In subspace clustering, a group of data points belonging to a union of
subspaces are assigned membership to their respective subspaces. This paper
presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of
subspace clustering using a new geometrical idea whereby subspaces are
identified based on their relative novelties. We present two frameworks in
which the idea of innovation pursuit is used to distinguish the subspaces.
Underlying the first framework is an iterative method that finds the subspaces
consecutively by solving a series of simple linear optimization problems, each
searching for a direction of innovation in the span of the data potentially
orthogonal to all subspaces except for the one to be identified in one step of
the algorithm. A detailed mathematical analysis is provided establishing
sufficient conditions for iPursuit to correctly cluster the data. The proposed
approach can provably yield exact clustering even when the subspaces have
significant intersections. It is shown that the complexity of the iterative
approach scales only linearly in the number of data points and subspaces, and
quadratically in the dimension of the subspaces. The second framework
integrates iPursuit with spectral clustering to yield a new variant of
spectral-clustering-based algorithms. The numerical simulations with both real
and synthetic data demonstrate that iPursuit can often outperform the
state-of-the-art subspace clustering algorithms, more so for subspaces with
significant intersections, and that it significantly improves the
state-of-the-art result for subspace-segmentation-based face clustering
Greed is good: algorithmic results for sparse approximation
This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms
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