There exist many well-established techniques to recover sparse signals from compressed measurements with known performance guarantees in the static case. However, only a few methods have been proposed to tackle the recovery of time-varying signals, and even fewer benefit from a theoretical analysis. In this paper, we study the capacity of the Iterative Soft-Thresholding Algorithm (ISTA) and its continuous-time analogue the Locally Competitive Algorithm (LCA) to perform this tracking in real time. ISTA is a well-known digital solver for static sparse recovery, whose iteration is a first-order discretization of the LCA differential equation. Our analysis shows that the outputs of both algorithms can track a time-varying signal while compressed measurements are streaming, even when no convergence criterion is imposed at each time step. The L2-distance between the target signal and the outputs of both discrete- and continuous-time solvers is shown to decay to a bound that is essentially optimal. Our analyses is supported by simulations on both synthetic and real data.
Finally, the last class of methods is based on optimization. For instance, in @cite_25 @cite_11 , an optimization program is set up to account for the temporal correlation in the target, and the recovery is performed in batches. @cite_21 , the best dynamical model is chosen among a family of possible dynamics or parameters. In @cite_34 , a continuation approach is used to update the estimate of the target using the solution at the previous time-step. @cite_22 @cite_20 @cite_24 @cite_19 , the optimization is solved using low-complexity iterative schemes. Unfortunately, all of the above methods lack strong theoretical convergence and accuracy guarantees in the dynamic case. Finally, in @cite_8 , a very general projection-based approach is studied. A convergence result is given but it is not clear how the necessary assumptions apply in the time-varying setting and it does not come with an accuracy result.
@cite_22 @cite_8 @cite_21 @cite_24 @cite_19 @cite_34 @cite_25 @cite_20 @cite_11
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In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, ( (x_0, ,x_p,y) ), subject to coupled linear equality constraints. Our ADMM updates each of the primal variables (x_0, ,x_p,y ), followed by updating the dual variable. We separate the variable y from (x_i )’s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, ( _q ) quasi-norm, Schatten-q quasi-norm ( (0<q<1 )), minimax concave penalty (MCP), and smoothly clipped absolute deviation penalty. It also allows nonconvex constraints such as compact manifolds (e.g., spherical, Stiefel, and Grassman manifolds) and linear complementarity constraints. Also, the (x_0 )-block can be almost any lower semi-continuous function. By applying our analysis, we show, for the first time, that several ADMM algorithms applied to solve nonconvex models in statistical learning, optimization on manifold, and matrix decomposition are guaranteed to converge. Our results provide sufficient conditions for ADMM to converge on (convex or nonconvex) monotropic programs with three or more blocks, as they are special cases of our model. ADMM has been regarded as a variant to the augmented Lagrangian method (ALM). We present a simple example to illustrate how ADMM converges but ALM diverges with bounded penalty parameter ( ). Indicated by this example and other analysis in this paper, ADMM might be a better choice than ALM for some nonconvex nonsmooth problems, because ADMM is not only easier to implement, it is also more likely to converge for the concerned scenarios. Many applications in science and engineering lead to models which require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allows to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden’s family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola-Nevanlinna-type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. Recent compilers offer a vast number of multilayered optimizations targeting different code segments of an application. Choosing among these optimizations can significantly impact the performance of the code being optimized. The selection of the right set of compiler optimizations for a particular code segment is a very hard problem, but finding the best ordering of these optimizations adds further complexity. Finding the best ordering represents a long standing problem in compilation research, named the phase-ordering problem. The traditional approach of constructing compiler heuristics to solve this problem simply cannot cope with the enormous complexity of choosing the right ordering of optimizations for every code segment in an application. This article proposes an automatic optimization framework we call MiCOMP, which Mi tigates the Com piler P hase-ordering problem. We perform phase ordering of the optimizations in LLVM’s highest optimization level using optimization sub-sequences and machine learning. The idea is to cluster the optimization passes of LLVM’s O3 setting into different clusters to predict the speedup of a complete sequence of all the optimization clusters instead of having to deal with the ordering of more than 60 different individual optimizations. The predictive model uses (1) dynamic features, (2) an encoded version of the compiler sequence, and (3) an exploration heuristic to tackle the problem. Experimental results using the LLVM compiler framework and the Cbench suite show the effectiveness of the proposed clustering and encoding techniques to application-based reordering of passes, while using a number of predictive models. We perform statistical analysis on the results and compare against (1) random iterative compilation, (2) standard optimization levels, and (3) two recent prediction approaches. We show that MiCOMP’s iterative compilation using its sub-sequences can reach an average performance speedup of 1.31 (up to 1.51). Additionally, we demonstrate that MiCOMP’s prediction model outperforms the -O1, -O2, and -O3 optimization levels within using just a few predictions and reduces the prediction error rate down to only 5 . Overall, it achieves 90 of the available speedup by exploring less than 0.001 of the optimization space.