Blurring the Line Between Structure and Learning to Optimize and Adapt Receptive Fields

The visual world is vast and varied, but its variations divide into structured and unstructured factors. We compose free-form filters and structured Gaussian filters, optimized end-to-end, to factorize deep representations and learn both local features and their degree of locality. Our semi-structured composition is strictly more expressive than free-form filtering, and changes in its structured parameters would require changes in free-form architecture. In effect this optimizes over receptive field size and shape, tuning locality to the data and task. Dynamic inference, in which the Gaussian structure varies with the input, adapts receptive field size to compensate for local scale variation. Optimizing receptive field size improves semantic segmentation accuracy on Cityscapes by 1-2 points for strong dilated and skip architectures and by up to 10 points for suboptimal designs. Adapting receptive fields by dynamic Gaussian structure further improves results, equaling the accuracy of free-form deformation while improving efficiency

Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity

We develop a general duality between neural networks and compositional kernels, striving towards a better understanding of deep learning. We show that initial representations generated by common random initializations are sufficiently rich to express all functions in the dual kernel space. Hence, though the training objective is hard to optimize in the worst case, the initial weights form a good starting point for optimization. Our dual view also reveals a pragmatic and aesthetic perspective of neural networks and underscores their expressive power

Improving Efficiency in Convolutional Neural Network with Multilinear Filters

The excellent performance of deep neural networks has enabled us to solve several automatization problems, opening an era of autonomous devices. However, current deep net architectures are heavy with millions of parameters and require billions of floating point operations. Several works have been developed to compress a pre-trained deep network to reduce memory footprint and, possibly, computation. Instead of compressing a pre-trained network, in this work, we propose a generic neural network layer structure employing multilinear projection as the primary feature extractor. The proposed architecture requires several times less memory as compared to the traditional Convolutional Neural Networks (CNN), while inherits the similar design principles of a CNN. In addition, the proposed architecture is equipped with two computation schemes that enable computation reduction or scalability. Experimental results show the effectiveness of our compact projection that outperforms traditional CNN, while requiring far fewer parameters.Comment: 10 pages, 3 figure

Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes

There is a previously identified equivalence between wide fully connected neural networks (FCNs) and Gaussian processes (GPs). This equivalence enables, for instance, test set predictions that would have resulted from a fully Bayesian, infinitely wide trained FCN to be computed without ever instantiating the FCN, but by instead evaluating the corresponding GP. In this work, we derive an analogous equivalence for multi-layer convolutional neural networks (CNNs) both with and without pooling layers, and achieve state of the art results on CIFAR10 for GPs without trainable kernels. We also introduce a Monte Carlo method to estimate the GP corresponding to a given neural network architecture, even in cases where the analytic form has too many terms to be computationally feasible. Surprisingly, in the absence of pooling layers, the GPs corresponding to CNNs with and without weight sharing are identical. As a consequence, translation equivariance, beneficial in finite channel CNNs trained with stochastic gradient descent (SGD), is guaranteed to play no role in the Bayesian treatment of the infinite channel limit - a qualitative difference between the two regimes that is not present in the FCN case. We confirm experimentally, that while in some scenarios the performance of SGD-trained finite CNNs approaches that of the corresponding GPs as the channel count increases, with careful tuning SGD-trained CNNs can significantly outperform their corresponding GPs, suggesting advantages from SGD training compared to fully Bayesian parameter estimation.Comment: Published as a conference paper at ICLR 201

A Gaussian Process perspective on Convolutional Neural Networks

In this paper we cast the well-known convolutional neural network in a Gaussian process perspective. In this way we hope to gain additional insights into the performance of convolutional networks, in particular understand under what circumstances they tend to perform well and what assumptions are implicitly made in the network. While for fully-connected networks the properties of convergence to Gaussian processes have been studied extensively, little is known about situations in which the output from a convolutional network approaches a multivariate normal distribution

Mellin-Meijer-kernel density estimation on R+\mathbb{R}^+

Nonparametric kernel density estimation is a very natural procedure which simply makes use of the smoothing power of the convolution operation. Yet, it performs poorly when the density of a positive variable is to be estimated (boundary issues, spurious bumps in the tail). So various extensions of the basic kernel estimator allegedly suitable for $\mathbb{R}^+$-supported densities, such as those using Gamma or other asymmetric kernels, abound in the literature. Those, however, are not based on any valid smoothing operation analogous to the convolution, which typically leads to inconsistencies. By contrast, in this paper a kernel estimator for $\mathbb{R}^+$-supported densities is defined by making use of the Mellin convolution, the natural analogue of the usual convolution on $\mathbb{R}^+$. From there, a very transparent theory flows and leads to new type of asymmetric kernels strongly related to Meijer's $G$-functions. The numerous pleasant properties of this `Mellin-Meijer-kernel density estimator' are demonstrated in the paper. Its pointwise and $L_2$-consistency (with optimal rate of convergence) is established for a large class of densities, including densities unbounded at 0 and showing power-law decay in their right tail. Its practical behaviour is investigated further through simulations and some real data analyses

Feature Weight Tuning for Recursive Neural Networks

This paper addresses how a recursive neural network model can automatically leave out useless information and emphasize important evidence, in other words, to perform "weight tuning" for higher-level representation acquisition. We propose two models, Weighted Neural Network (WNN) and Binary-Expectation Neural Network (BENN), which automatically control how much one specific unit contributes to the higher-level representation. The proposed model can be viewed as incorporating a more powerful compositional function for embedding acquisition in recursive neural networks. Experimental results demonstrate the significant improvement over standard neural models

Differentiable Compositional Kernel Learning for Gaussian Processes

The generalization properties of Gaussian processes depend heavily on the choice of kernel, and this choice remains a dark art. We present the Neural Kernel Network (NKN), a flexible family of kernels represented by a neural network. The NKN architecture is based on the composition rules for kernels, so that each unit of the network corresponds to a valid kernel. It can compactly approximate compositional kernel structures such as those used by the Automatic Statistician (Lloyd et al., 2014), but because the architecture is differentiable, it is end-to-end trainable with gradient-based optimization. We show that the NKN is universal for the class of stationary kernels. Empirically we demonstrate pattern discovery and extrapolation abilities of NKN on several tasks that depend crucially on identifying the underlying structure, including time series and texture extrapolation, as well as Bayesian optimization.Comment: ICML 2018; update proo

Towards Interpretable R-CNN by Unfolding Latent Structures

This paper first proposes a method of formulating model interpretability in visual understanding tasks based on the idea of unfolding latent structures. It then presents a case study in object detection using popular two-stage region-based convolutional network (i.e., R-CNN) detection systems. We focus on weakly-supervised extractive rationale generation, that is learning to unfold latent discriminative part configurations of object instances automatically and simultaneously in detection without using any supervision for part configurations. We utilize a top-down hierarchical and compositional grammar model embedded in a directed acyclic AND-OR Graph (AOG) to explore and unfold the space of latent part configurations of regions of interest (RoIs). We propose an AOGParsing operator to substitute the RoIPooling operator widely used in R-CNN. In detection, a bounding box is interpreted by the best parse tree derived from the AOG on-the-fly, which is treated as the qualitatively extractive rationale generated for interpreting detection. We propose a folding-unfolding method to train the AOG and convolutional networks end-to-end. In experiments, we build on R-FCN and test our method on the PASCAL VOC 2007 and 2012 datasets. We show that the method can unfold promising latent structures without hurting the performance.Comment: 9 page

Monotone Increment Processes, Classical Markov Processes, and Loewner Chains

We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments