We consider the problem of rigid registration, where we wish to jointly register multiple point sets via rigid transforms. This arises in applications such as sensor network localization, multiview registration, and protein structure determination. The least-squares estimator for this problem can be reduced to a rank-constrained semidefinite program (REG-SDP). It was recently shown that by formally applying the alternating direction method of multipliers (ADMM), we can derive an iterative solver (REG-ADMM) for REG-SDP, wherein each subproblem admits a simple closed-form solution. The empirical success of REG-ADMM has been demonstrated for multiview registration. However, its convergence does not follow from the existing literature on nonconvex ADMM. In this work, we study the convergence of REG-ADMM and our main findings are as follows. We prove that any fixed point of REG-ADMM is a stationary (KKT) point of REG-SDP. Moreover, for clean measurements, we give an explicit formula for the ADMM parameter @math , for which REG-ADMM is guaranteed to converge to the global optimum (with arbitrary initialization). If the noise is low, we can still show that the iterates converge to the global optimum, provided they are initialized sufficiently close to the optimum. On the other hand, if the noise is high, we explain why REG-ADMM becomes unstable if @math is less than some threshold, irrespective of the initialization. We present simulation results to support our theoretical predictions. The novelty of our analysis lies in the fact that we exploit the notion of tightness of convex relaxation to arrive at our convergence results.
We do not make such smoothness assumptions in our analysis. We can afford to do this since we are analyzing a special class of problems, as opposed to the more general setups in @cite_17 @cite_0 @cite_7 . Instead of showing a monotonic decrease in the augmented Lagrangian, our analysis relies on the phenomenon of tightness of convex relaxation. This provides more insights into the convergence behavior of the algorithm. For instance, our explanation in Section shows that the instability of the algorithm (in the high noise regime) for low values of @math is fundamental, while suggesting why this instability is not observed in the low noise regime.
@cite_0 @cite_7 @cite_17
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis We discuss a new robust convergence analysis of the well-known subspace iteration algorithm for computing the dominant singular vectors of a matrix, also known as simultaneous iteration or power method. The result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. While interesting in its own right, the main motivation comes from the problem of privacy-preserving spectral analysis where noise is added in order to achieve the privacy guarantee known as differential privacy. This result leads to nearly tight worst-case bounds for the problem of computing a differentially private low-rank approximation in the spectral norm. Our results extend to privacy-preserving principal component analysis. We obtain improvements for several variants of differential privacy that have been considered. The running time of our algorithm is nearly linear in the input sparsity leading to strong improvements in running time over previous work. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by a tight dependence on the coherence of the matrix. This parameter is always bounded by the matrix dimension but often much smaller. Indeed, the assumption of low coherence is essential in several machine learning and signal processing applications. We study learning formulations with non-convex regularizaton that are natural for sparse linear models. There are two approaches to this problem: • Heuristic methods such as gradient descent that only find a local minimum. A drawback of this approach is the lack of theoretical guarantee showing that the local minimum gives a good solution. • Convex relaxation such as L1-regularization that solves the problem under some conditions. However it often leads to sub-optimal sparsity in reality. This paper tries to remedy the above gap between theory and practice. In particular, we investigate a multi-stage convex relaxation scheme for solving problems with non-convex regularization. Theoretically, we analyze the behavior of a resulting two-stage relaxation scheme for the capped-L1 regularization. Our performance bound shows that the procedure is superior to the standard L1 convex relaxation for learning sparse targets. Experiments confirm the effectiveness of this method on some simulation and real data. This paper proves best known guarantees for exact reconstruction of a sparse signal f from few non-adaptive universal linear measurements. We consider Fourier measurements (random sample of frequencies of f) and random Gaussian measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly non-convex problem to a convex problem, and then solve it as a linear program. What are best guarantees for the reconstruction problem to be equivalent to its convex relaxation is an open question. Recent work shows that the number of measurements k(r,n) needed to exactly reconstruct any r-sparse signal f of length n from its linear measurements with convex relaxation is usually O(r poly log (n)). However, known guarantees involve huge constants, in spite of very good performance of the algorithms in practice. In attempt to reconcile theory with practice, we prove the first guarantees for universal measurements (i.e. which work for all sparse functions) with reasonable constants. For Gaussian measurements, k(r,n) lsim 11.7 r [1.5 + log(n r)], which is optimal up to constants. For Fourier measurements, we prove the best known bound k(r, n) = O(r log(n) middot log2(r) log(r log n)), which is optimal within the log log n and log3 r factors. Our arguments are based on the technique of geometric functional analysis and probability in Banach spaces.