arXiv subjects

Ting Tao

Publications and source records attributed to Ting Tao.

8 recordsLinked to original sources

arXiv2025

GLL-type Nonmonotone Descent Methods Revisited under Kurdyka-{\L}ojasiewicz Property

The purpose of this paper is to extend the full convergence results of the classic GLL-type (Grippo-Lampariello-Lucidi) nonmonotone methods to nonconvex and nonsmooth optimization. We propose a novel iterative framework for the minimization of a proper and lower semicontinuous function $\Phi$. The framework consists of the GLL-type nonmonotone decrease condition for a sequence, a relative error condition for its augmented sequence with respect to a Kurdyka-{\L}ojasiewicz (KL) function $\Theta$, and a relative gap condition for the partial maximum objective value sequence. The last condition is shown to be a product of the prox-regularity of $\Phi$ on the set of cluster points, and to hold automatically under a mild condition on the objective value sequence. We prove that for any sequence and its bounded augmented sequence together falling within the framework, the sequence itself is convergent. Furthermore, when $\Theta$ is a KL function of exponent $\theta\in(0, 1)$, the convergence admits a linear rate if $\theta\in(0, 1/2]$ and a sublinear rate if $\theta\in(1/2, 1)$. As applications, we prove, for the first time, that the two existing algorithms, namely the nonmonotone proximal gradient (NPG) method with majorization and NPG with extrapolation both enjoy the full convergence of the iterate sequences for nonconvex and nonsmooth KL composite optimization problems.

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arXiv2024

Convergence of ZH-type nonmonotone descent method for Kurdyka-{\L}ojasiewicz optimization problems

We propose a novel iterative framework for minimizing a proper lower semicontinuous Kurdyka-{\L}ojasiewicz (KL) function $\Phi$. It comprises a Zhang-Hager (ZH-type) nonmonotone decrease condition and a relative error condition. Hence, the sequence generated by the ZH-type nonmonotone descent methods will fall within this framework. Any sequence conforming to this framework is proved to converge to a critical point of $\Phi$. If in addition $\Phi$ has the KL property of exponent $\theta\!\in(0,1)$ at the critical point, the convergence has a linear rate for $\theta\in(0,1/2]$ and a sublinear rate of exponent $\frac{1-\theta}{1-2\theta}$ for $\theta\in(1/2,1)$. To the best of our knowledge, this is the first work to establish the full convergence of the iterate sequence generated by a ZH-type nonmonotone descent method for nonconvex and nonsmooth optimization problems. The obtained results are also applied to achieve the full convergence of the iterate sequences produced by the proximal gradient method and Riemannian gradient method with the ZH-type nonmonotone line-search.

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arXiv2024

Convergence of the majorized PAM method with subspace correction for low-rank composite factorization model

This paper focuses on the convergence certificates of the majorized proximal alternating minimization (PAM) method with subspace correction, proposed in \cite{TaoQianPan22} for the column $\ell_{2,0}$-norm regularized factorization model and now extended to a class of low-rank composite factorization models from matrix completion. The convergence analysis of this PAM method becomes extremely challenging because a subspace correction step is introduced to every proximal subproblem to ensure a closed-form solution. We establish the full convergence of the iterate sequence and column subspace sequences of factor pairs generated by the PAM, under the KL property of the objective function and a condition that holds automatically for the column $\ell_{2,0}$-norm function. Numerical comparison with the popular proximal alternating linearized minimization (PALM) method is conducted on one-bit matrix completion problems, which indicates that the PAM with subspace correction has an advantage in seeking lower relative error within less time.

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arXiv2023

An inexact LPA for DC composite optimization and application to matrix completions with outliers

This paper concerns a class of DC composite optimization problems which, as an extension of convex composite optimization problems and DC programs with nonsmooth components, often arises in robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) by computing at each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions at the current iterate. We establish the full convergence of the generated iterate sequence under the Kurdyka-\L\"ojasiewicz (KL) property of a potential function, and employ the composite structure to provide a verifiable condition for the potential function to satisfy the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate. This condition is weaker than the one provided in \cite[Theorem 3.2]{LiPong18} for identifying the KL property of exponent $p\in[0,1)$ for a general composite function. The proposed iLPA is applied to a robust factorization model for matrix completion with outliers and non-uniform sampling, and numerical comparisons with the Polyak subgradient method and a proximal alternating minimization (PAM) method validate its efficiency.

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arXiv2021

Column $\ell_{2,0}$-norm regularized factorization model of low-rank matrix recovery and its computation

This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimize of a smooth nonconvex loss. We provide the global convergence analysis for the proposed AMM methods and apply them to the matrix completion problem with non-uniform sampling schemes. Numerical experiments are conducted with synthetic and real data examples, and comparison results with the nuclear-norm regularized factorization model and the max-norm regularized convex model show that the column $\ell_{2,0}$-regularized factorization model has an advantage in offering solutions of lower error and rank within less time.

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arXiv2021

Error bound of critical points and KL property of exponent $1/2$ for squared F-norm regularized factorization

This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample setting achieve this property for a class of critical points. These theoretical findings are also confirmed by solving the squared F-norm regularized factorization problem with an accelerated alternating minimization method.

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arXiv2019

KL property of exponent $1/2$ of $\ell_{2,0}$-norm and DC regularized factorizations for low-rank matrix recovery

This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function and its equivalent DC reformulations in the set of their global minimizers. We also confirm the theoretical findings by applying a proximal linearized alternating minimization method to the regularized factorizations.

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arXiv2019

Calibrated zero-norm regularized LS estimator for high-dimensional error-in-variables regression

This paper is concerned with high-dimensional error-in-variables regression that aims at identifying a small number of important interpretable factors for corrupted data from many applications where measurement errors or missing data can not be ignored. Motivated by CoCoLasso due to Datta and Zou \cite{Datta16} and the advantage of the zero-norm regularized LS estimator over Lasso for clean data, we propose a calibrated zero-norm regularized LS (CaZnRLS) estimator by constructing a calibrated least squares loss with a positive definite projection of an unbiased surrogate for the covariance matrix of covariates, and use the multi-stage convex relaxation approach to compute the CaZnRLS estimator. Under a restricted eigenvalue condition on the true matrix of covariates, we derive the $\ell_2$-error bound of every iterate and establish the decreasing of the error bound sequence, and the sign consistency of the iterates after finite steps. The statistical guarantees are also provided for the CaZnRLS estimator under two types of measurement errors. Numerical comparisons with CoCoLasso and NCL (the nonconvex Lasso proposed by Poh and Wainwright \cite{Loh11}) demonstrate that CaZnRLS not only has the comparable or even better relative RSME but also has the least number of incorrect predictors identified.

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