  Sparse principal component analysis (sPCA) has become one of the most widely
used techniques for dimensionality reduction in high-dimensional datasets. The
main challenge underlying sPCA is to estimate the first vector of loadings of
the population covariance matrix, provided that only a certain number of
loadings are non-zero. In this paper, we propose confidence intervals for
individual loadings and for the largest eigenvalue of the population covariance
matrix. Given an independent sample $X^i \in\mathbb R^p, i = 1,...,n,$
generated from an unknown distribution with an unknown covariance matrix
$\Sigma_0$, our aim is to estimate the first vector of loadings and the largest
eigenvalue of $\Sigma_0$ in a setting where $p\gg n$. Next to the
high-dimensionality, another challenge lies in the inherent non-convexity of
the problem. We base our methodology on a Lasso-penalized M-estimator which,
despite non-convexity, may be solved by a polynomial-time algorithm such as
coordinate or gradient descent. We show that our estimator achieves the minimax
optimal rates in $\ell_1$ and $\ell_2$-norm. We identify the bias in the
Lasso-based estimator and propose a de-biased sparse PCA estimator for the
vector of loadings and for the largest eigenvalue of the covariance matrix
$\Sigma_0$. Our main results provide theoretical guarantees for asymptotic
normality of the de-biased estimator. The major conditions we impose are
sparsity in the first eigenvector of small order $\sqrt{n}/\log p$ and sparsity
of the same order in the columns of the inverse Hessian matrix of the
population risk.
