In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Ex Input:
Empirical risk minimization (ERM), with proper loss function and regularization, is the common practice of supervised classification. In this paper, we study training arbitrary (from linear to deep) binary classifier from only unlabeled (U) data by ERM. We prove that it is impossible to estimate the risk of an arbitrary binary classifier in an unbiased manner given a single set of U data, but it becomes possible given two sets of U data with different class priors. These two facts answer a fundamental question---what the minimal supervision is for training any binary classifier from only U data. Following these findings, we propose an ERM-based learning method from two sets of U data, and then prove it is consistent. Experiments demonstrate the proposed method could train deep models and outperform state-of-the-art methods for learning from two sets of U data.

Ex Output:
Three class priors are all you need to train deep models from only U data, while any two should not be enough.


Ex Input:
The vast majority of natural sensory data is temporally redundant. For instance, video frames or audio samples which are sampled at nearby points in time tend to have similar values.   Typically, deep learning algorithms take no advantage of this redundancy to reduce computations.   This can be an obscene waste of energy.   We present a variant on backpropagation for neural networks in which computation scales with the rate of change of the data - not the rate at which we process the data.   We do this by implementing a form of Predictive Coding wherein neurons communicate a combination of their state, and their temporal change in state, and quantize this signal using Sigma-Delta modulation.   Intriguingly, this simple communication rule give rise to units that resemble biologically-inspired leaky integrate-and-fire neurons, and to a spike-timing-dependent weight-update similar to Spike-Timing Dependent Plasticity (STDP), a synaptic learning rule observed in the brain.   We demonstrate that on MNIST, on a temporal variant of MNIST, and on Youtube-BB, a dataset with videos in the wild, our algorithm performs about as well as a standard deep network trained with backpropagation, despite only communicating discrete values between layers.  

Ex Output:
An algorithm for training neural networks efficiently on temporally redundant data.


Ex Input:
Recently various neural networks have been proposed for irregularly structured data such as graphs and manifolds. To our knowledge, all existing graph networks have discrete depth. Inspired by neural ordinary differential equation (NODE) for data in the Euclidean domain, we extend the idea of continuous-depth models to graph data, and propose graph ordinary differential equation (GODE). The derivative of hidden node states are parameterized with a graph neural network, and the output states are the solution to this ordinary differential equation. We demonstrate two end-to-end methods for efficient training of GODE: (1) indirect back-propagation with the adjoint method; (2) direct back-propagation through the ODE solver, which accurately computes the gradient. We demonstrate that direct backprop outperforms the adjoint method in experiments. We then introduce a family of bijective blocks, which enables $\mathcal{O}(1)$ memory consumption. We demonstrate that GODE can be easily adapted to different existing graph neural networks and improve accuracy. We validate the performance of GODE in both semi-supervised node classification tasks and graph classification tasks. Our GODE model achieves a continuous model in time, memory efficiency, accurate gradient estimation, and generalizability with different graph networks.

Ex Output:

Apply ordinary differential equation model on graph structured data