This paper describes how rank pooling, a very recent approach for pooling representations organized in a sequence $\\{{\bf v}_t\\}_{t=1}^T$, can be used in an end-to-end trained neural network architecture.

Rank pooling is an alternative to average and max pooling for sequences, but with the distinctive advantage of maintaining some order information from the sequence. Rank pooling first solves a regularized (linear) support vector regression (SVR) problem where the inputs are the vector representations ${\bf v}_t$ in the sequence and the target is the corresponding index $t$ of that representation in the sequence (see Equation 5). The output of rank pooling is then simply the linear regression parameters $\bf{u}$ learned for that sequence. Because of the way ${\bf u}$ is trained, we can see that ${\bf u}$ will capture order information, as successful training would imply that ${\bf u}^\top {\bf v}_t < {\bf u}^\top {\bf v}_{t'} $ if $t < t'$. See [this paper](https://www.robots.ox.ac.uk/~vgg/rg/papers/videoDarwin.pdf) for more on rank pooling.

While previous work has focused on using rank pooling on hand-designed and fixed representations, this paper proposes to use ConvNet features (pre-trained on ImageNet) for the representation and backpropagate through rank pooling to fine-tune the ConvNet features. Since the output of rank pooling corresponds to an argmin operation, passing gradients through this operation is not as straightforward as for average or max pooling. However, it turns out that if the objective being minimized (in our case regularized SVR) is twice differentiable, gradients with respect to its argmin can be computed (see Lemmas 1 and 2). The authors derive the gradient for rank pooling (Equation 21). Finally, since its gradient requires inverting a matrix (corresponding to a hessian), the authors propose to either use an efficient procedure for computing it by exploiting properties of sums of rank-one matrices (see Lemma 3) or to simply use an approximation based on using a diagonal hessian.

In experiments on two small scale video activity recognition datasets (UCF-Sports and Hollywood2), the authors show that fine-tuning the ConvNet features significantly improves the performance of rank pooling and makes it superior to max and average pooling.

**My two cents**

This paper was eye opening for me, first because I did not realize that one could backpropagate through an operation corresponding to an argmin that doesn't have a closed form solution (though apparently this paper isn't the first to make that observation). Moreover, I did not know about rank pooling, which itself is a really thought provoking approach to pooling representations in a way that preserves some organizational information about the original representations.

I wonder how sensitive the results are to the value of the regularization constant of the SVR problem. The authors mention some theoretical guaranties on the stability of the solution found by SVR in general, but intuitively I would expect that the regularization constant would play a large role in the stability.

I'll be looking forward to any future attempts to increase the speed of rank pooling (or any similar method). Indeed, as the authors mention, it is currently too slow to be used on the larger video datasets that are currently available. 

Code for computing rank pooling (though not for computing its gradients) seems to be available [here](https://bitbucket.org/bfernando/videodarwin).