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Obuchi, Tomoyuki, Sakata, Ayaka
We investigate the signal reconstruction performance of sparse linear regression in the presence of noise when piecewise continuous nonconvex penalties are used. Among such penalties, we focus on the smoothly clipped absolute deviation (SCAD) penalty. The contributions of this study are three-fold: We first present a theoretical analysis of a typical reconstruction performance, using the replica method, under the assumption that each component of the design matrix is given as an independent and identically distributed (i.i.d.) Gaussian variable. This clarifies the superiority of the SCAD estimator compared with $\ell_1$ in a wide parameter range, although the nonconvex nature of the penalty tends to lead to solution multiplicity in certain regions. This multiplicity is shown to be connected to replica symmetry breaking in the spin-glass theory, and associated phase diagrams are given. We also show that the global minimum of the mean square error between the estimator and the true signal is located in the replica symmetric phase. Second, we develop an approximate formula efficiently computing the cross-validation error without actually conducting the cross-validation, which is also applicable to the non-i.i.d. design matrices. It is shown that this formula is only applicable to the unique solution region and tends to be unstable in the multiple solution region. We implement instability detection procedures, which allows the approximate formula to stand alone and resultantly enables us to draw phase diagrams for any specific dataset. Third, we propose an annealing procedure, called nonconvexity annealing, to obtain the solution path efficiently. Numerical simulations are conducted on simulated datasets to examine these results to verify the consistency of the theoretical results and the efficiency of the approximate formula and nonconvexity annealing.
Sakata, Ayaka, Obuchi, Tomoyuki
We consider compressed sensing formulated as a minimization problem of nonconvex sparse penalties, Smoothly Clipped Absolute deviation (SCAD) and Minimax Concave Penalty (MCP). The nonconvexity of these penalties is controlled by nonconvexity parameters, and L1 penalty is contained as a limit with respect to these parameters. The analytically derived reconstruction limit overcomes that of L1 and the algorithmic limit in the Bayes-optimal setting, when the nonconvexity parameters have suitable values. For the practical usage, we apply the approximate message passing (AMP) to these nonconvex penalties. We show that the performance of AMP is considerably improved by controlling nonconvexity parameters.
We propose an estimator of prediction error using an approximate message passing (AMP) algorithm that can be applied to a broad range of sparse penalties. Following Stein's lemma, the estimator of the generalized degrees of freedom, which is a key quantity for the construction of the estimator of the prediction error, is calculated at the AMP fixed point. The resulting form of the AMP-based estimator does not depend on the penalty function, and its value can be further improved by considering the correlation between predictors. The proposed estimator is asymptotically unbiased when the components of the predictors and response variables are independently generated according to a Gaussian distribution. We examine the behaviour of the estimator for real data under nonconvex sparse penalties, where Akaike's information criterion does not correspond to an unbiased estimator of the prediction error. The model selected by the proposed estimator is close to that which minimizes the true prediction error.
Sakata, Ayaka, Xu, Yingying
We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida--Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric solution. Numerical experiments confirm that for a sufficiently large system size, SCAD-AMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric (RS) and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of L1-based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated.
We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics - the cavity and replica methods - to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable in principle in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error, and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.