Realizing Data Features by Deep Nets

Construction of Deep ReLU Nets for Spatially Sparse Learning

Training an interpretable deep net to embody its theoretical advantages is difficult but extremely important in the community of machine learning. In this article, noticing the importance of spatial sparseness in signal and image processing, we develop a constructive approach to generate a deep net to capture the spatial sparseness feature. We conduct both theoretical analysis and numerical verifications to show the power of the constructive approach. Theoretically, we prove that the constructive approach can yield a deep net estimate that achieves the optimal generalization error bounds in the framework of learning theory. Numerically, we show that the constructive approach is essentially better than shallow learning in the sense that it provides better prediction accuracy with less training time.

Depth Selection for Deep ReLU Nets in Feature Extraction and Generalization

Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantages of human ingenuity and prior knowledge. Thus it has triggered enormous research activities in machine learning and pattern recognition. One of the most important challenges of deep learning is to figure out relations between a feature and the depth of deep neural networks (deep nets for short) to reflect the necessity of depth. Our purpose is to quantify this feature-depth correspondence in feature extraction and generalization. We present the adaptivity of features to depths and vice-verse via showing a depth-parameter trade-off in extracting both single feature and composite features. Based on these results, we prove that implementing the classical empirical risk minimization on deep nets can achieve the optimal generalization performance for numerous learning tasks. Our theoretical results are verified by a series of numerical experiments including toy simulations and a real application of earthquake seismic intensity prediction.

Random Sketching for Neural Networks With ReLU

Training neural networks is recently a hot topic in machine learning due to its great success in many applications. Since the neural networks' training usually involves a highly nonconvex optimization problem, it is difficult to design optimization algorithms with perfect convergence guarantees to derive a neural network estimator of high quality. In this article, we borrow the well-known random sketching strategy from kernel methods to transform the training of shallow rectified linear unit (ReLU) nets into a linear least-squares problem. Using the localized approximation property of shallow ReLU nets and a recently developed dimensionality-leveraging scheme, we succeed in equipping shallow ReLU nets with a specific random sketching scheme. The efficiency of the suggested random sketching strategy is guaranteed by theoretical analysis and also verified via a series of numerical experiments. Theoretically, we show that the proposed random sketching is almost optimal in terms of both approximation capability and learning performance. This implies that random sketching does not degenerate the performance of shallow ReLU nets. Numerically, we show that random sketching can significantly reduce the computational burden of numerous backpropagation (BP) algorithms while maintaining their learning performance.