  The goal of ordinal embedding is to represent items as points in a
low-dimensional Euclidean space given a set of constraints in the form of
distance comparisons like "item $i$ is closer to item $j$ than item $k$".
Ordinal constraints like this often come from human judgments. To account for
errors and variation in judgments, we consider the noisy situation in which the
given constraints are independently corrupted by reversing the correct
constraint with some probability. This paper makes several new contributions to
this problem. First, we derive prediction error bounds for ordinal embedding
with noise by exploiting the fact that the rank of a distance matrix of points
in $\mathbb{R}^d$ is at most $d+2$. These bounds characterize how well a
learned embedding predicts new comparative judgments. Second, we investigate
the special case of a known noise model and study the Maximum Likelihood
estimator. Third, knowledge of the noise model enables us to relate prediction
errors to embedding accuracy. This relationship is highly non-trivial since we
show that the linear map corresponding to distance comparisons is
non-invertible, but there exists a nonlinear map that is invertible. Fourth,
two new algorithms for ordinal embedding are proposed and evaluated in
experiments.
