  Recently, methods have been developed to accurately predict the testing
performance of a Deep Neural Network (DNN) on a particular task, given
statistics of its underlying topological structure. However, further leveraging
this newly found insight for practical applications is intractable due to the
high computational cost in terms of time and memory. In this work, we define a
new class of topological features that accurately characterize the progress of
learning while being quick to compute during running time. Additionally, our
proposed topological features are readily equipped for backpropagation, meaning
that they can be incorporated in end-to-end training. Our newly developed
practical topological characterization of DNNs allows for an additional set of
applications. We first show we can predict the performance of a DNN without a
testing set and without the need for high-performance computing. We also
demonstrate our topological characterization of DNNs is effective in estimating
task similarity. Lastly, we show we can induce learning in DNNs by actively
constraining the DNN's topological structure. This opens up new avenues in
constricting the underlying structure of DNNs in a meta-learning framework.
