27,630 research outputs found
Sparse Inverse Covariance Estimation for Chordal Structures
In this paper, we consider the Graphical Lasso (GL), a popular optimization
problem for learning the sparse representations of high-dimensional datasets,
which is well-known to be computationally expensive for large-scale problems.
Recently, we have shown that the sparsity pattern of the optimal solution of GL
is equivalent to the one obtained from simply thresholding the sample
covariance matrix, for sparse graphs under different conditions. We have also
derived a closed-form solution that is optimal when the thresholded sample
covariance matrix has an acyclic structure. As a major generalization of the
previous result, in this paper we derive a closed-form solution for the GL for
graphs with chordal structures. We show that the GL and thresholding
equivalence conditions can significantly be simplified and are expected to hold
for high-dimensional problems if the thresholded sample covariance matrix has a
chordal structure. We then show that the GL and thresholding equivalence is
enough to reduce the GL to a maximum determinant matrix completion problem and
drive a recursive closed-form solution for the GL when the thresholded sample
covariance matrix has a chordal structure. For large-scale problems with up to
450 million variables, the proposed method can solve the GL problem in less
than 2 minutes, while the state-of-the-art methods converge in more than 2
hours
Distributed Robustness Analysis of Interconnected Uncertain Systems Using Chordal Decomposition
Large-scale interconnected uncertain systems commonly have large state and
uncertainty dimensions. Aside from the heavy computational cost of solving
centralized robust stability analysis techniques, privacy requirements in the
network can also introduce further issues. In this paper, we utilize IQC
analysis for analyzing large-scale interconnected uncertain systems and we
evade these issues by describing a decomposition scheme that is based on the
interconnection structure of the system. This scheme is based on the so-called
chordal decomposition and does not add any conservativeness to the analysis
approach. The decomposed problem can be solved using distributed computational
algorithms without the need for a centralized computational unit. We further
discuss the merits of the proposed analysis approach using a numerical
experiment.Comment: 3 figures. Submitted to the 19th IFAC world congres
Optimal Control for LQG Systems on Graphs---Part I: Structural Results
In this two-part paper, we identify a broad class of decentralized
output-feedback LQG systems for which the optimal control strategies have a
simple intuitive estimation structure and can be computed efficiently. Roughly,
we consider the class of systems for which the coupling of dynamics among
subsystems and the inter-controller communication is characterized by the same
directed graph. Furthermore, this graph is assumed to be a multitree, that is,
its transitive reduction can have at most one directed path connecting each
pair of nodes. In this first part, we derive sufficient statistics that may be
used to aggregate each controller's growing available information. Each
controller must estimate the states of the subsystems that it affects (its
descendants) as well as the subsystems that it observes (its ancestors). The
optimal control action for a controller is a linear function of the estimate it
computes as well as the estimates computed by all of its ancestors. Moreover,
these state estimates may be updated recursively, much like a Kalman filter
Robust Stability Analysis of Sparsely Interconnected Uncertain Systems
In this paper, we consider robust stability analysis of large-scale sparsely
interconnected uncertain systems. By modeling the interconnections among the
subsystems with integral quadratic constraints, we show that robust stability
analysis of such systems can be performed by solving a set of sparse linear
matrix inequalities. We also show that a sparse formulation of the analysis
problem is equivalent to the classical formulation of the robustness analysis
problem and hence does not introduce any additional conservativeness. The
sparse formulation of the analysis problem allows us to apply methods that rely
on efficient sparse factorization techniques, and our numerical results
illustrate the effectiveness of this approach compared to methods that are
based on the standard formulation of the analysis problem.Comment: Provisionally accepted to appear in IEEE Transactions on Automatic
Contro
Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers
We design sparse and block sparse feedback gains that minimize the variance
amplification (i.e., the norm) of distributed systems. Our approach
consists of two steps. First, we identify sparsity patterns of feedback gains
by incorporating sparsity-promoting penalty functions into the optimal control
problem, where the added terms penalize the number of communication links in
the distributed controller. Second, we optimize feedback gains subject to
structural constraints determined by the identified sparsity patterns. In the
first step, the sparsity structure of feedback gains is identified using the
alternating direction method of multipliers, which is a powerful algorithm
well-suited to large optimization problems. This method alternates between
promoting the sparsity of the controller and optimizing the closed-loop
performance, which allows us to exploit the structure of the corresponding
objective functions. In particular, we take advantage of the separability of
the sparsity-promoting penalty functions to decompose the minimization problem
into sub-problems that can be solved analytically. Several examples are
provided to illustrate the effectiveness of the developed approach.Comment: To appear in IEEE Trans. Automat. Contro
A Fast Algorithm for Sparse Controller Design
We consider the task of designing sparse control laws for large-scale systems
by directly minimizing an infinite horizon quadratic cost with an
penalty on the feedback controller gains. Our focus is on an improved algorithm
that allows us to scale to large systems (i.e. those where sparsity is most
useful) with convergence times that are several orders of magnitude faster than
existing algorithms. In particular, we develop an efficient proximal Newton
method which minimizes per-iteration cost with a coordinate descent active set
approach and fast numerical solutions to the Lyapunov equations. Experimentally
we demonstrate the appeal of this approach on synthetic examples and real power
networks significantly larger than those previously considered in the
literature
Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations
It is well-known that any sum of squares (SOS) program can be cast as a
semidefinite program (SDP) of a particular structure and that therein lies the
computational bottleneck for SOS programs, as the SDPs generated by this
procedure are large and costly to solve when the polynomials involved in the
SOS programs have a large number of variables and degree. In this paper, we
review SOS optimization techniques and present two new methods for improving
their computational efficiency. The first method leverages the sparsity of the
underlying SDP to obtain computational speed-ups. Further improvements can be
obtained if the coefficients of the polynomials that describe the problem have
a particular sparsity pattern, called chordal sparsity. The second method
bypasses semidefinite programming altogether and relies instead on solving a
sequence of more tractable convex programs, namely linear and second order cone
programs. This opens up the question as to how well one can approximate the
cone of SOS polynomials by second order representable cones. In the last part
of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201
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