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，其假设存在一个正交变换使得随机向量的各分量相互独立。尽管对数凹密度估计在计算和统计两个层面均存在困难，但独立成分假设在维持一个大型非参数类的同时缓解了这两个问题。我们证明了在温和条件下，对于平方Hellinger距离内与原始密度相差 $\epsilon$ 的估计，最多需要 $\tilde{\mathcal{O}}(\epsilon^{-4})$ 个样本（抑制常数和对数因子）。在计算方面，尽管通常的对数凹极大似然估计可以通过有限维凸规划获得，但其计算速度缓慢——尤其在较高维度中。我们通过数值实验表明，我们的估计器可以高效计算，从而使其更便于实际应用。