minimum and maximum eigenvalues of $M$, $|M|$ be its cardinality, and let $\text{sign}(M)$ be matrix of s of elements of $M$. For any real $x$, let $\text{sign}(x) $ denotes of $x$, and let $|x|$ denotes its absolute value.

Background and Problem Set-up
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Let $X=(X_1, X_2, \cdots, X_p) $ be a zero-mean p-dimensional random vector. The focus of this paper is the estimation of the covariance matrix $\Sigma:=\mathbb{E}(XX^T)$ and its inverse $\Sigma^{-1}$ from a sample of independently and identically distributed data $\{ X^{(k)} \}^{n}_{k=1}$. In this section we provide some background and problem setup more precisely.

The choice of loss function is very crucial in any optimization problem. An optimal estimator for a particular loss function may not be optimal for another choice of loss function. Recent literature in covariance matrix and inverse covariance matrix estimation mostly focuses on estimation based on likelihood function or quadratic loss function \[@Freidman:11, @Banerjee:38, @Bickel:3, @Ravi1:24, @Roth:25, @Maurya:19\]. The maximum likelihood estimation requires a tractable probability distribution of observations whereas quadratic loss function does not have any such requirement and therefore fully non-parametric. The quadratic loss function is convex and due to this analytical tractability, it is a widely applicable choice for many data 