Deep SimNets

We present a deep layered architecture that generalizes convolutional neural networks (ConvNets). The architecture, called SimNets, is driven by two operators: (i) a similarity function that generalizes inner-product, and (ii) a log-mean-exp function called MEX that generalizes maximum and average. The two operators applied in succession give rise to a standard neuron but in "feature space". The feature spaces realized by SimNets depend on the choice of the similarity operator. The simplest setting, which corresponds to a convolution, realizes the feature space of the Exponential kernel, while other settings realize feature spaces of more powerful kernels (Generalized Gaussian, which includes as special cases RBF and Laplacian), or even dynamically learned feature spaces (Generalized Multiple Kernel Learning). As a result, the SimNet contains a higher abstraction level compared to a traditional ConvNet. We argue that enhanced expressiveness is important when the networks are small due to run-time constraints (such as those imposed by mobile applications). Empirical evaluation validates the superior expressiveness of SimNets, showing a significant gain in accuracy over ConvNets when computational resources at run-time are limited. We also show that in large-scale settings, where computational complexity is less of a concern, the additional capacity of SimNets can be controlled with proper regularization, yielding accuracies comparable to state of the art ConvNets

Adaptive Normalized Risk-Averting Training For Deep Neural Networks

This paper proposes a set of new error criteria and learning approaches, Adaptive Normalized Risk-Averting Training (ANRAT), to attack the non-convex optimization problem in training deep neural networks (DNNs). Theoretically, we demonstrate its effectiveness on global and local convexity lower-bounded by the standard $L_p$-norm error. By analyzing the gradient on the convexity index $\lambda$, we explain the reason why to learn $\lambda$ adaptively using gradient descent works. In practice, we show how this method improves training of deep neural networks to solve visual recognition tasks on the MNIST and CIFAR-10 datasets. Without using pretraining or other tricks, we obtain results comparable or superior to those reported in recent literature on the same tasks using standard ConvNets + MSE/cross entropy. Performance on deep/shallow multilayer perceptrons and Denoised Auto-encoders is also explored. ANRAT can be combined with other quasi-Newton training methods, innovative network variants, regularization techniques and other specific tricks in DNNs. Other than unsupervised pretraining, it provides a new perspective to address the non-convex optimization problem in DNNs.Comment: AAAI 2016, 0.39%~0.4% ER on MNIST with single 32-32-256-10 ConvNets, code available at https://github.com/cauchyturing/ANRA

Notes on Hierarchical Splines, DCLNs and i-theory

We define an extension of classical additive splines for multivariate function approximation that we call hierarchical splines. We show that the case of hierarchical, additive, piece-wise linear splines includes present-day Deep Convolutional Learning Networks (DCLNs) with linear rectifiers and pooling (sum or max). We discuss how these observations together with i-theory may provide a framework for a general theory of deep networks.This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF - 1231216

Entangled q-convolutional neural nets

We introduce a machine learning model, the q-CNN model, sharing key features with convolutional neural networks and admitting a tensor network description. As examples, we apply q-CNN to the MNIST and Fashion MNIST classification tasks. We explain how the network associates a quantum state to each classification label, and study the entanglement structure of these network states. In both our experiments on the MNIST and Fashion-MNIST datasets, we observe a distinct increase in both the left/right as well as the up/down bipartition entanglement entropy (EE) during training as the network learns the fine features of the data. More generally, we observe a universal negative correlation between the value of the EE and the value of the cost function, suggesting that the network needs to learn the entanglement structure in order the perform the task accurately. This supports the possibility of exploiting the entanglement structure as a guide to design the machine learning algorithm suitable for given tasks