Compact convolutional neural networks gain efficiency mainly through depthwise convolutions, expanded channels and complex topologies, which contrarily aggravate the training efforts. In this work, we identify the shift problem occurs in even-sized kernel (2x2, 4x4) convolutions, and eliminate it by proposing symmetric padding on each side of the feature maps (C2sp, C4sp). Symmetric padding enlarges the receptive fields of even-sized kernels with little computational cost. In classification tasks, C2sp outperforms the conventional 3x3 convolution and obtains comparable accuracies to existing compact convolution blocks, but consumes less memory and time during training. In generation tasks, C2sp and C4sp both achieve improved image qualities and stabilized training. Symmetric padding coupled with even-sized convolution is easy to be implemented into deep learning frameworks, providing promising building units for architecture designs that emphasize training efforts on online and continual learning occasions.
Since even-sized kernels are integer multiples of 2, they are mostly applied together with stride 2 to rescale images, which can avoid the checkerboard artifact @cite_35 . For example, the shallower models of @cite_34 use 4 @math 4 kernels and stride 2 in the discriminators and generators to down-sample and up-sample images. However, 3 @math 3 kernel is mostly preferred when it comes to deep and large-scale GANs @cite_40 @cite_32 @cite_29 . In segmentation tasks, U-Net @cite_41 and its followers use 2 @math 2 kernels to up-sample images. Except for scaling, very few works have implemented even-sized kernels as basic building blocks for their CNN models. In relational reinforcement learning @cite_7 , two C2 layers are adopted to achieve reasoning and planning of objects represented by 4 pixels. It is discussed that factorizing a C3 into two C2s only provides 11
@cite_35 @cite_7 @cite_41 @cite_29 @cite_32 @cite_40 @cite_34
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We show that the output of a (residual) convolutional neural network (CNN) with an appropriate prior over the weights and biases is a Gaussian process (GP) in the limit of infinitely many convolutional filters, extending similar results for dense networks. For a CNN, the equivalent kernel can be computed exactly and, unlike "deep kernels", has very few parameters: only the hyperparameters of the original CNN. Further, we show that this kernel has two properties that allow it to be computed efficiently; the cost of evaluating the kernel for a pair of images is similar to a single forward pass through the original CNN with only one filter per layer. The kernel equivalent to a 32-layer ResNet obtains 0.84 classification error on MNIST, a new record for GPs with a comparable number of parameters. We introduce a kernel approximation strategy that enables computation of the Gaussian process log marginal likelihood and all hyperparameter derivatives in @math time. Our GRIEF kernel consists of @math eigenfunctions found using a Nystrom approximation from a dense Cartesian product grid of inducing points. By exploiting algebraic properties of Kronecker and Khatri-Rao tensor products, computational complexity of the training procedure can be practically independent of the number of inducing points. This allows us to use arbitrarily many inducing points to achieve a globally accurate kernel approximation, even in high-dimensional problems. The fast likelihood evaluation enables type-I or II Bayesian inference on large-scale datasets. We benchmark our algorithms on real-world problems with up to two-million training points and @math inducing points. Modern learning problems in computer vision, natural language processing, computational biology, and other areas are often based on large data sets of tens of thousands to millions of training instances. However, several standard learning algorithms, such as kernel-based algorithms, e.g., Support Vector Machines, Kernel Ridge Regression, Kernel PCA, do not easily scale to such orders of magnitude. This thesis focuses on sampling-based matrix approximation techniques that help scale kernel-based algorithms to large-scale datasets. We address several fundamental theoretical and empirical questions including: (1) What approximation should be used? We discuss two common sampling-based methods, providing novel theoretical insights regarding their suitability for various applications and experimental results motivated by this theory. Our results show that one of these methods, the Nystrom method, is superior in the context of large-scale learning. (2) Do these approximations work in practice? We show the effectiveness of approximation techniques on a variety of problems. In the largest study to-date for manifold learning, we use the Nystrom method to extract low-dimensional structure from high-dimensional data to effectively cluster face images. We also report good empirical results for Kernel Ridge Regression and Kernel Logistic Regression. (3) How should we sample columns? A key aspect of sampling-based algorithms is the distribution according to which columns are sampled. We study both fixed and adaptive sampling schemes as well as a promising ensemble technique that can be easily parallelized and generates superior approximations, both in theory and in practice. (4) How well do these approximations work in theory? We provide theoretical analyses of the Nystrom method to understand when this technique should be used. We present guarantees on approximation accuracy based on various matrix properties and analyze the effect of matrix approximation on actual kernel-based algorithms. This work has important consequences for the machine learning community since it extends to large-scale applications the benefits of kernel-based algorithms. The crucial aspect of this research, involving low-rank matrix approximation, is of independent interest within the field of linear algebra. One approach to improving the running time of kernel-based methods is to build a small sketch of the kernel matrix and use it in lieu of the full matrix in the machine learning task of interest. Here, we describe a version of this approach that comes with running time guarantees as well as improved guarantees on its statistical performance. By extending the notion of statistical leverage scores to the setting of kernel ridge regression, we are able to identify a sampling distribution that reduces the size of the sketch (i.e., the required number of columns to be sampled) to the effective dimensionality of the problem. This latter quantity is often much smaller than previous bounds that depend on the maximal degrees of freedom. We give an empirical evidence supporting this fact. Our second contribution is to present a fast algorithm to quickly compute coarse approximations to these scores in time linear in the number of samples. More precisely, the running time of the algorithm is O(np2) with p only depending on the trace of the kernel matrix and the regularization parameter. This is obtained via a variant of squared length sampling that we adapt to the kernel setting. Lastly, we discuss how this new notion of the leverage of a data point captures a fine notion of the difficulty of the learning problem.