arXiv subjects

Maryam Yashtini

Publications and source records attributed to Maryam Yashtini.

4 recordsLinked to original sources

arXiv2022

Variable Metric Composite Proximal Alternating Linearized Minimization for Nonconvex Nonsmooth Optimization

In this paper we propose a proximal algorithm for minimizing an objective function of two block variables consisting of three terms: 1) a smooth function, 2) a nonsmooth function which is a composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function, and 3) a smooth function which couples the two block variables. We propose a variable metric composite proximal alternating linearized minimization (CPALM) to solve this class of problems. Building on the powerful Kurdyka-Łojasiewicz property, we derive the convergence analysis and establish that each bounded sequence generated by CPALM globally converges to a critical point. We demonstrate the CPALM method on parallel magnetic resonance image reconstruction problems. The obtained numerical results shows the viability and effectiveness of the proposed method.

math.OC↗

arXiv2021

Counting Objects by Diffused Index: geometry-free and training-free approach

Counting objects is a fundamental but challenging problem. In this paper, we propose diffusion-based, geometry-free, and learning-free methodologies to count the number of objects in images. The main idea is to represent each object by a unique index value regardless of its intensity or size, and to simply count the number of index values. First, we place different vectors, refer to as seed vectors, uniformly throughout the mask image. The mask image has boundary information of the objects to be counted. Secondly, the seeds are diffused using an edge-weighted harmonic variational optimization model within each object. We propose an efficient algorithm based on an operator splitting approach and alternating direction minimization method, and theoretical analysis of this algorithm is given. An optimal solution of the model is obtained when the distributed seeds are completely diffused such that there is a unique intensity within each object, which we refer to as an index. For computational efficiency, we stop the diffusion process before a full convergence, and propose to cluster these diffused index values. We refer to this approach as Counting Objects by Diffused Index (CODI). We explore scalar and multi-dimensional seed vectors. For Scalar seeds, we use Gaussian fitting in histogram to count, while for vector seeds, we exploit a high-dimensional clustering method for the final step of counting via clustering. The proposed method is flexible even if the boundary of the object is not clear nor fully enclosed. We present counting results in various applications such as biological cells, agriculture, concert crowd, and transportation. Some comparisons with existing methods are presented.

cs.CV↗

arXiv2021

Multi-Block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates under Kurdyka-Łojasiewicz Property

In this paper, we consider a multi-block generalized alternating direction method of multiplier (GADMM) algorithm for minimizing a linearly constrained separable nonconvex and possibly nonsmooth optimization problem. The GADMM generalizes the classical ADMM by including proximal terms in each primal updates and an over-relaxation parameter in the dual update. We prove that any limit point of the sequence is a critical point. By introducing a modified augmented Lagrangian we show that the sequence generated by the GADMM is bounded and the norm of the difference of consecutive terms approaches to zero. Under the powerful {KŁ} properties we show that the GADMM sequence has a finite length and converges to a stationary point, and we drive its convergence rate. Given a proper lower-semicontinuous function $f:\mathbb R^n\to\mathbb R$ and a critical point $x^*\in\mathbb R^n$, the {KŁ} property asserts that there exists a continuous concave monotonically increasing function $ψ$ such that around $x^*$ it holds $ψ'(f(x)-f(x^*))\cdot{\rm dist}(0,\partial f(x))\ge 1$ . When $ψ(s)=s^{1-θ}$ with $θ\in[0,1]$ this is equivalent to $|f(x)-f(x^*)|^θ{\rm dist}(0,\partial f(x))^{-1}$ to remain bounded around $x^*$. We show that if $θ=0$, the sequence generated by GADMM converges in a finite numbers of iterations. If $θ\in(0,1/2]$, then the rate of convergence is $cQ^{k}$ where $c>0$, $Q\in(0,1)$, and $k\in\mathbb N$ is the iteration number. If $θ\in(1/2,1]$ then the rate $\mathcal O(1/k^{r})$ where $r=(1-θ)/(2θ-1)$ will be achieved.

math.OC↗

arXiv2021

Convergence Analysis of A Proximal Linearized ADMM Algorithm for Nonconvex Nonsmooth Optimization

In this paper, we consider a proximal linearized alternating direction method of multipliers (PL-ADMM) for solving linearly constrained nonconvex and possibly nonsmooth optimization problems. The algorithm is generalized by using variable metric proximal terms in the primal updates and an over-relaxation stepsize in the multiplier update. We prove that the sequence generated by this method is bounded and its limit points are critical points. Under the powerful Kurdyka-{Łojasiewicz} properties we prove that the sequence has a finite length thus converges, and we drive its convergence rates.

math.OC↗