  We propose a method that substantially improves the efficiency of deep
distance metric learning based on the optimization of the triplet loss
function. One epoch of such training process based on a naive optimization of
the triplet loss function has a run-time complexity O(N^3), where N is the
number of training samples. Such optimization scales poorly, and the most
common approach proposed to address this high complexity issue is based on
sub-sampling the set of triplets needed for the training process. Another
approach explored in the field relies on an ad-hoc linearization (in terms of
N) of the triplet loss that introduces class centroids, which must be optimized
using the whole training set for each mini-batch - this means that a naive
implementation of this approach has run-time complexity O(N^2). This complexity
issue is usually mitigated with poor, but computationally cheap, approximate
centroid optimization methods. In this paper, we first propose a solid theory
on the linearization of the triplet loss with the use of class centroids, where
the main conclusion is that our new linear loss represents a tight upper-bound
to the triplet loss. Furthermore, based on the theory above, we propose a
training algorithm that no longer requires the centroid optimization step,
which means that our approach is the first in the field with a guaranteed
linear run-time complexity. We show that the training of deep distance metric
learning methods using the proposed upper-bound is substantially faster than
triplet-based methods, while producing competitive retrieval accuracy results
on benchmark datasets (CUB-200-2011 and CAR196).
