Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks

The symmetry and geometry of input data are considered to be encoded in the internal data representation inside the neural network, but the specific encoding rule has been less investigated. In this study, we present a systematic method to induce a generalized neural network and its right inverse operator, called the ridgelet transform, from a joint group invariant function on the data-parameter domain. Since the ridgelet transform is an inverse, (1) it can describe the arrangement of parameters for the network to represent a target function, which is understood as the encoding rule, and (2) it implies the universality of the network. Based on the group representation theory, we present a new simple proof of the universality by using Schur's lemma in a unified manner covering a wide class of networks, for example, the original ridgelet transform, formal deep networks, and the dual voice transform. Since traditional universality theorems were demonstrated based on functional analysis, this study sheds light on the group theoretic aspect of the approximation theory, connecting geometric deep learning to abstract harmonic analysis.Comment: NeurReps 202

Convolutional Neural Networks as 2-D systems

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible

Machine Learning for Informed Representation Learning

The way we view reality and reason about the processes surrounding us is intimately connected to our perception and the representations we form about our observations and experiences. The popularity of machine learning and deep learning techniques in that regard stems from their ability to form useful representations by learning from large sets of observations. Typical application examples include image recognition or language processing for which artificial neural networks are powerful tools to extract regularity patterns or relevant statistics. In this thesis, we leverage and further develop this representation learning capability to address relevant but challenging real-world problems in geoscience and chemistry, to learn representations in an informed manner relevant to the task at hand, and reason about representation learning in neural networks, in general. Firstly, we develop an approach for efficient and scalable semantic segmentation of degraded soil in alpine grasslands in remotely-sensed images based on convolutional neural networks. To this end, we consider different grassland erosion phenomena in several Swiss valleys. We find that we are able to monitor soil degradation consistent with state-of-the-art methods in geoscience and can improve detection of affected areas. Furthermore, our approach provides a scalable method for large-scale analysis which is infeasible with established methods. Secondly, we address the question of how to identify suitable latent representations to enable generation of novel objects with selected properties. For this, we introduce a new deep generative model in the context of manifold learning and disentanglement. Our model improves targeted generation of novel objects by making use of property cycle consistency in property-relevant and property-invariant latent subspaces. We demonstrate the improvements on the generation of molecules with desired physical or chemical properties. Furthermore, we show that our model facilitates interpretability and exploration of the latent representation. Thirdly, in the context of recent advances in deep learning theory and the neural tangent kernel, we empirically investigate the learning of feature representations in standard convolutional neural networks and corresponding random feature models given by the linearisation of the neural networks. We find that performance differences between standard and linearised networks generally increase with the difficulty of the task but decrease with the considered width or over-parametrisation of these networks. Our results indicate interesting implications for feature learning and random feature models as well as the generalisation performance of highly over-parametrised neural networks. In summary, we employ and study feature learning in neural networks and review how we may use informed representation learning for challenging tasks

Group Invariant Deep Representations for Image Instance Retrieval

Most image instance retrieval pipelines are based on comparison of vectors known as global image descriptors between a query image and the database images. Due to their success in large scale image classification, representations extracted from Convolutional Neural Networks (CNN) are quickly gaining ground on Fisher Vectors (FVs) as state-of-the-art global descriptors for image instance retrieval. While CNN-based descriptors are generally remarked for good retrieval performance at lower bitrates, they nevertheless present a number of drawbacks including the lack of robustness to common object transformations such as rotations compared with their interest point based FV counterparts. In this paper, we propose a method for computing invariant global descriptors from CNNs. Our method implements a recently proposed mathematical theory for invariance in a sensory cortex modeled as a feedforward neural network. The resulting global descriptors can be made invariant to multiple arbitrary transformation groups while retaining good discriminativeness. Based on a thorough empirical evaluation using several publicly available datasets, we show that our method is able to significantly and consistently improve retrieval results every time a new type of invariance is incorporated. We also show that our method which has few parameters is not prone to overfitting: improvements generalize well across datasets with different properties with regard to invariances. Finally, we show that our descriptors are able to compare favourably to other state-of-the-art compact descriptors in similar bitranges, exceeding the highest retrieval results reported in the literature on some datasets. A dedicated dimensionality reduction step --quantization or hashing-- may be able to further improve the competitiveness of the descriptors