Graph support measures are functions measuring how frequently a given subgraph pattern occurs in a given database graph. An important class of support measures relies on overlap graphs. A major advantage of the overla...
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While linearprogramming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original fo...
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While linearprogramming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. By modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, we first prove that LP decoding can be performed by solving a number of LP problems that each contains at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate solving such systems iteratively. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.
In this paper, we propose a second order interiorpoint algorithm for symmetric cone programming using a wide neighborhood of the central path. The convergence is shown for commutative class of search directions. The ...
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In this paper, we propose a second order interiorpoint algorithm for symmetric cone programming using a wide neighborhood of the central path. The convergence is shown for commutative class of search directions. The complexity bound is O(r(3/2)log epsilon(-1)) for the NT methods, and O(r(2)log epsilon(-1)) for the XS and SX methods, where r is the rank of the associated Euclidean Jordan algebra and epsilon > 0 is a given tolerance. If the staring point is strictly feasible, then the corresponding bounds can be reduced by a factor of r(3/4). The theory of Euclidean Jordan algebras is a basic tool in our analysis.
The simplex method has proven its efficiency in practice for linearprogramming (LP) problems of various types and sizes. However, its theoretical worst-case complexity in addition to its poor performance for very lar...
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The simplex method has proven its efficiency in practice for linearprogramming (LP) problems of various types and sizes. However, its theoretical worst-case complexity in addition to its poor performance for very large-scale LP problems has driven researchers to develop alternative methods for LP problems. In this paper, we develop the hybrid-LP: a two-phase approach for solving LP problems. Rather than following a path of extreme points on the boundary of the feasible region as in the simplex method, the first phase of the hybrid-LP moves through the interior of the feasible region to obtain an improved and advanced initial basic feasible solution (BFS). Then, in the second phase simplex or other LP methods can be used to find the optimal solution. Since the introduction of polynomial-time methods for LP, a considerable amount of research has focused on interior-pointmethods for solving large-scale LP problems. Although fewer iterations are needed for interior-pointmethods to converge to a solution, the iterations are computationally intensive. Our approach is a hybrid method that uses a computationally efficient pivot to move in the interior of the feasible region in its first phase. This single iteration is able to bypassing several extreme points to an improved BFS, which can then be used as a starting point in any LP method in the second phase of the method. Our approach can also be modified to perform a number of interior pivots in the first phase based on the trade-off between the number of iterations and the running time. The hybrid-LP uses an efficient pivoting iteration which is computationally comparable to the standard simplex iteration. Another feature is adaptability in finding the advanced starting point by avoiding the boundaries of the feasible region. In addition, the hybrid-LP has the ability to start from a feasible point which may not be a BFS. Our computational experiments demonstrate that the hybrid-LP reduces both the number of iterations
Conic optimization is an extension of linear optimization in which the linear inequality constraints are replaced with vector inequalities defined by convex cones. The two most important examples, second-order cone pr...
Conic optimization is an extension of linear optimization in which the linear inequality constraints are replaced with vector inequalities defined by convex cones. The two most important examples, second-order cone programming and semidefinite programming, are used in a wide range of applications and form the basis of modeling tools for convex optimization. In fact, most convex constraints encountered in practice can be formulated as conic inequalities with respect to the second-order or semidefinite cone. Second-order cone programs and semidefinite programs can be solved in polynomial time using interior-pointmethods. However, methods for exploiting structure in semidefinite programming problems are not as well-developed as those for linear optimization, and despite its powerful modeling capabilities, the semidefinite cone does not always provide an efficient representation of convex constraints. We therefore consider methods for linear cone programs with two types of sparse matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. These cones include not only the nonnegative orthant, the second-order cone, and the semidefinite cone as special cases, but also a number of useful lower-dimensional, intermediate cones which are generally not self-dual. Our results show that the sparse matrix cone approach is advantageous for many problems with sparse, chordal sparsity patterns, and for problems that have an efficient chordal embedding. For problems with band or block-arrow structure, for example, the cost of a single interior-point iteration grows only linearly in the order of the matrix variable instead of quadratically or worse for general-purpose semidefinite programming solvers. We also explore other applications in nonlinear optimization. By combining interior-pointmethods and results from matrix completion theory, we formula
We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The m...
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We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation Sigma nor does it need to pre-estimate Sigma. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate Sigma{(s/n) log p}(1/2) in the prediction norm, and thus matching the performance of the lasso with known Sigma. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.
linearprogramming (LP) decoders can outperform currently used message-passing decoders in channel coding applications, but require prohibitively large complexity on even moderately sized codes. Previous works have pr...
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linearprogramming (LP) decoders can outperform currently used message-passing decoders in channel coding applications, but require prohibitively large complexity on even moderately sized codes. Previous works have proposed complexity-reducing algorithms that either relax the problem or modify the number of constraints; however, little work is done in optimizing solver implementation. We show that popular LP solvers like LIPSOL may not be efficient for LP decoding (LPD), and that an equivalent dual LP problem can be solved with equal accuracy but much more quickly. We propose an improved primal-dual method (iPD-MPC) whose overall runtime for both problem formulations outpreform LIPSOL. Additionally, as an alternative for memory-limited systems, we propose an improved hybrid gradient descent and Newton's method (iGD-NM) that further decreases overall runtime. In this way, we make LPD more feasible for channel codes of practical lengths.
Computers with CPUs of multiple computing cores are now widely available. In the paper, we investigate how these modern architectures perform in the practice of interiorpointmethods. In our study, we shall focus on ...
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Computers with CPUs of multiple computing cores are now widely available. In the paper, we investigate how these modern architectures perform in the practice of interiorpointmethods. In our study, we shall focus on the implementation of the Cholesky factorization of large-scale and sparse symmetric positive semidefinite matrices. Our numerical experiments will demonstrate that capabilities of modern processors can be efficiently utilized by special implementation techniques.
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