1,609 research outputs found
Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering
Semidefinite programs (SDPs) often arise in relaxations of some NP-hard
problems, and if the solution of the SDP obeys certain rank constraints, the
relaxation will be tight. Decomposition methods based on chordal sparsity have
already been applied to speed up the solution of sparse SDPs, but methods for
dealing with rank constraints are underdeveloped. This paper leverages a
minimum rank completion result to decompose the rank constraint on a single
large matrix into multiple rank constraints on a set of smaller matrices. The
re-weighted heuristic is used as a proxy for rank, and the specific form of the
heuristic preserves the sparsity pattern between iterations. Implementations of
rank-minimized SDPs through interior-point and first-order algorithms are
discussed. The problem of subspace clustering is used to demonstrate the
computational improvement of the proposed method.Comment: 6 pages, 6 figure
Joint Centrality Distinguishes Optimal Leaders in Noisy Networks
We study the performance of a network of agents tasked with tracking an
external unknown signal in the presence of stochastic disturbances and under
the condition that only a limited subset of agents, known as leaders, can
measure the signal directly. We investigate the optimal leader selection
problem for a prescribed maximum number of leaders, where the optimal leader
set minimizes total system error defined as steady-state variance about the
external signal. In contrast to previously established greedy algorithms for
optimal leader selection, our results rely on an expression of total system
error in terms of properties of the underlying network graph. We demonstrate
that the performance of any given set of leaders depends on their influence as
determined by a new graph measure of centrality of a set. We define the of a set of nodes in a network graph such that a leader set with
maximal joint centrality is an optimal leader set. In the case of a single
leader, we prove that the optimal leader is the node with maximal information
centrality. In the case of multiple leaders, we show that the nodes in the
optimal leader set balance high information centrality with a coverage of the
graph. For special cases of graphs, we solve explicitly for optimal leader
sets. We illustrate with examples.Comment: Conditionally accepted to IEEE TCN
Approximating Non-Uniform Sparsest Cut via Generalized Spectra
We give an approximation algorithm for non-uniform sparsest cut with the
following guarantee: For any , given cost and demand
graphs with edge weights respectively, we can find a set
with at most
times the optimal non-uniform sparsest cut value,
in time 2^{r/(\delta\epsilon)}\poly(n) provided . Here is the 'th smallest generalized
eigenvalue of the Laplacian matrices of cost and demand graphs; (resp. ) is the weight of edges crossing the
cut in cost (resp. demand) graph and is the
sparsity of the optimal cut. In words, we show that the non-uniform sparsest
cut problem is easy when the generalized spectrum grows moderately fast. To the
best of our knowledge, there were no results based on higher order spectra for
non-uniform sparsest cut prior to this work.
Even for uniform sparsest cut, the quantitative aspects of our result are
somewhat stronger than previous methods. Similar results hold for other
expansion measures like edge expansion, normalized cut, and conductance, with
the 'th smallest eigenvalue of the normalized Laplacian playing the role of
in the latter two cases.
Our proof is based on an l1-embedding of vectors from a semi-definite program
from the Lasserre hierarchy. The embedded vectors are then rounded to a cut
using standard threshold rounding. We hope that the ideas connecting
-embeddings to Lasserre SDPs will find other applications. Another
aspect of the analysis is the adaptation of the column selection paradigm from
our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather
than vertices. This feature is important in order to extend the algorithms to
non-uniform sparsest cut.Comment: 16 page
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