text: Augmenting transformation knowledge onto a convolutional neural network's weights has often yielded significant improvements in performance. For rotational transformation augmentation, an important element to recent approaches has been the use of a steerable basis i.e. the circular harmonics. Here, we propose a scale-steerable filter basis for the locally scale-invariant CNN, denoted as log-radial harmonics. By replacing the kernels in the locally scale-invariant CNN lsi_cnn with scale-steered kernels, significant improvements in performance can be observed on the MNIST-Scale and FMNIST-Scale datasets. Training with a scale-steerable basis results in filters which show meaningful structure, and feature maps demonstrate which demonstrate visibly higher spatial-structure preservation of input. Furthermore, the proposed scale-steerable CNN shows on-par generalization to global affine transformation estimation methods such as Spatial Transformers, in response to test-time data distortions.
  @cite_9: Convolutional Neural Networks (ConvNets) have shown excellent results on many visual classification tasks. With the exception of ImageNet, these datasets are carefully crafted such that objects are well-aligned at similar scales. Naturally, the feature learning problem gets more challenging as the amount of variation in the data increases, as the models have to learn to be invariant to certain changes in appearance. Recent results on the ImageNet dataset show that given enough data, ConvNets can learn such invariances producing very discriminative features [1]. But could we do more: use less parameters, less data, learn more discriminative features, if certain invariances were built into the learning process? In this paper we present a simple model that allows ConvNets to learn features in a locally scale-invariant manner without increasing the number of model parameters. We show on a modified MNIST dataset that when faced with scale variation, building in scale-invariance allows ConvNets to learn more discriminative features with reduced chances of over-fitting.
  @cite_3: We study the effect of injecting local scale equivariance into Convolutional Neural Networks. This is done by applying each convolutional filter at multiple scales. The output is a vector field encoding for the maximally activating scale and the scale itself, which is further processed by the following convolutional layers. This allows all the intermediate representations to be locally scale equivariant. We show that this improves the performance of the model by over 20 in the scale equivariant task of regressing the scaling factor applied to randomly scaled MNIST digits. Furthermore, we find it also useful for scale invariant tasks, such as the actual classification of randomly scaled digits. This highlights the usefulness of allowing for a compact representation that can also learn relationships between different local scales by keeping internal scale equivariance.
summary: Scale-transformed weights were proposed in @cite_9 , where it was observed to improve performance over the normal baseline CNN, on MNIST-Scale. On the same dataset (with a 10k, 2k and 50k split), better performance was observed in @cite_3 , where in addition to forwarding the maximum filter response to a range of scales, the actual scale at which the response was obtained was also forwarded. In both works, weight scaling was only indirectly emulated, by rather scaling the input and the resizing back the convolution response to a fix size for max-pooling across scales.