We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(\epsilon^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $\epsilon$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use.

翻译：我们提出了一种从样本中估计$\mathbb R^d$上对数凹密度的方法，其假设是存在一个正交变换，使得随机向量的各个分量相互独立。尽管对数凹密度估计在计算和统计上都很困难，但独立分量假设缓解了这两个问题，同时仍能保持一个较大的非参数类别。我们证明，在温和条件下，最多仅需$\tilde{\mathcal{O}}(\epsilon^{-4})$个样本（忽略常数和对数因子），我们提出的估计量就能在平方Hellinger距离下与原密度相差不超过$\epsilon$。在计算方面，通常的对数凹最大似然估计可以通过有限维凸规划获得，但计算速度缓慢——尤其是在更高维度上。我们通过数值实验证明，我们的估计量可以被高效计算，使其更具实用价值。