In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Input: Consider Input: The paper addresses the problem of learning a regression model parameterized by a fixedrank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks.

Output: Regression on fixed-rank positive semidefinite matrices: a Riemannian approach


Input: Consider Input: Despite being so vital to success of Support Vector Machines, the principle of separating margin maximisation is not used in deep learning. We show that minimisation of margin variance and not maximisation of the margin is more suitable for improving generalisation in deep architectures. We propose the Halfway loss function that minimises the Normalised Margin Variance (NMV) at the output of a deep learning models and evaluate its performance against the Softmax Cross-Entropy loss on the MNIST, smallNORB and CIFAR-10 datasets.

Output: Effects of the optimisation of the margin distribution on generalisation in deep architectures


Input: Consider Input: We present DataGrad, a general back-propagation style training procedure for deep neural architectures that uses regularization of a deep Jacobian-based penalty. It can be viewed as a deep extension of the layerwise contractive auto-encoder penalty. More importantly, it unifies previous proposals for adversarial training of deep neural nets – this list includes directly modifying the gradient, training on a mix of original and adversarial examples, using contractive penalties, and approximately optimizing constrained adversarial objective functions. In an experiment using a Deep Sparse Rectifier Network, we find that the deep Jacobian regularization of DataGrad (which also has L1 and L2 flavors of regularization) outperforms traditional L1 and L2 regularization both on the original dataset as well as on adversarial examples.


Output: Unifying Adversarial Training Algorithms with Flexible Deep Data Gradient Regularization