Lai, Ming-JunYin, Wotao2018-06-192018-06-192012-01Lai, Ming-Jun and Yin, Wotao. "Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm." (2012) <a href="https://hdl.handle.net/1911/102192">https://hdl.handle.net/1911/102192</a>.https://hdl.handle.net/1911/102192This paper studies the models of minimizing $||x||_1+1/(2\alpha)||x||_2^2$ where $x$ is a vector, as well as those of minimizing $||X||_*+1/(2\alpha)||X||_F^2$ where $X$ is a matrix and $||X||_*$ and $||X||_F$ are the nuclear and Frobenius norms of $X$, respectively. We show that they can efficiently recover sparse vectors and low-rank matrices. In particular, they enjoy exact and stable recovery guarantees similar to those known for minimizing $||x||_1$ and $||X||_*$ under the conditions on the sensing operator such as its null-space property, restricted isometry property, spherical section property, or RIPless property. To recover a (nearly) sparse vector $x^0$, minimizing $||x||_1+1/(2\alpha)||x||_2^2$ returns (nearly) the same solution as minimizing $||x||_1$ almost whenever $\alpha\ge 10||x^0||_\infty$. The same relation also holds between minimizing $||X||_*+1/(2\alpha)||X||_F^2$ and minimizing $||X||_*$ for recovering a (nearly) low-rank matrix $X^0$, if $\alpha\ge 10||X^0||_2$. Furthermore, we show that the linearized Bregman algorithm for minimizing $||x||_1+1/(2\alpha)||x||_2^2$ subject to $Ax=b$ enjoys global linear convergence as long as a nonzero solution exists, and we give an explicit rate of convergence. The convergence property does not require a solution solution or any properties on $A$. To our knowledge, this is the best known global convergence result for first-order sparse optimization algorithms.33 ppengTechnical reportTR12-02