  Rate-distortion (RD) theory is at the heart of lossy data compression. Here
we aim to model the generalized RD (GRD) trade-off between the visual quality
of a compressed video and its encoding profiles (e.g., bitrate and spatial
resolution). We first define the theoretical functional space $\mathcal{W}$ of
the GRD function by analyzing its mathematical properties.We show that
$\mathcal{W}$ is a convex set in a Hilbert space, inspiring a computational
model of the GRD function, and a method of estimating model parameters from
sparse measurements. To demonstrate the feasibility of our idea, we collect a
large-scale database of real-world GRD functions, which turn out to live in a
low-dimensional subspace of $\mathcal{W}$. Combining the GRD reconstruction
framework and the learned low-dimensional space, we create a low-parameter
eigen GRD method to accurately estimate the GRD function of a source video
content from only a few queries. Experimental results on the database show that
the learned GRD method significantly outperforms state-of-the-art empirical RD
estimation methods both in accuracy and efficiency. Last, we demonstrate the
promise of the proposed model in video codec comparison.
