Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations

The success of deep convolutional architectures is often attributed in part to their ability to learn multiscale and invariant representations of natural signals. However, a precise study of these properties and how they affect learning guarantees is still missing. In this paper, we consider deep convolutional representations of signals; we study their invariance to translations and to more general groups of transformations, their stability to the action of diffeomorphisms, and their ability to preserve signal information. This analysis is carried by introducing a multilayer kernel based on convolutional kernel networks and by studying the geometry induced by the kernel mapping. We then characterize the corresponding reproducing kernel Hilbert space (RKHS), showing that it contains a large class of convolutional neural networks with homogeneous activation functions. This analysis allows us to separate data representation from learning, and to provide a canonical measure of model complexity, the RKHS norm, which controls both stability and generalization of any learned model. In addition to models in the constructed RKHS, our stability analysis also applies to convolutional networks with generic activations such as rectified linear units, and we discuss its relationship with recent generalization bounds based on spectral norms

A Deep Representation for Invariance And Music Classification

Representations in the auditory cortex might be based on mechanisms similar to the visual ventral stream; modules for building invariance to transformations and multiple layers for compositionality and selectivity. In this paper we propose the use of such computational modules for extracting invariant and discriminative audio representations. Building on a theory of invariance in hierarchical architectures, we propose a novel, mid-level representation for acoustical signals, using the empirical distributions of projections on a set of templates and their transformations. Under the assumption that, by construction, this dictionary of templates is composed from similar classes, and samples the orbit of variance-inducing signal transformations (such as shift and scale), the resulting signature is theoretically guaranteed to be unique, invariant to transformations and stable to deformations. Modules of projection and pooling can then constitute layers of deep networks, for learning composite representations. We present the main theoretical and computational aspects of a framework for unsupervised learning of invariant audio representations, empirically evaluated on music genre classification.Comment: 5 pages, CBMM Memo No. 002, (to appear) IEEE 2014 International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2014

Geometric deep learning: going beyond Euclidean data

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field

A Kernel Perspective for Regularizing Deep Neural Networks

We propose a new point of view for regularizing deep neural networks by using the norm of a reproducing kernel Hilbert space (RKHS). Even though this norm cannot be computed, it admits upper and lower approximations leading to various practical strategies. Specifically, this perspective (i) provides a common umbrella for many existing regularization principles, including spectral norm and gradient penalties, or adversarial training, (ii) leads to new effective regularization penalties, and (iii) suggests hybrid strategies combining lower and upper bounds to get better approximations of the RKHS norm. We experimentally show this approach to be effective when learning on small datasets, or to obtain adversarially robust models.Comment: ICM

Learning An Invariant Speech Representation

Recognition of speech, and in particular the ability to generalize and learn from small sets of labelled examples like humans do, depends on an appropriate representation of the acoustic input. We formulate the problem of finding robust speech features for supervised learning with small sample complexity as a problem of learning representations of the signal that are maximally invariant to intraclass transformations and deformations. We propose an extension of a theory for unsupervised learning of invariant visual representations to the auditory domain and empirically evaluate its validity for voiced speech sound classification. Our version of the theory requires the memory-based, unsupervised storage of acoustic templates -- such as specific phones or words -- together with all the transformations of each that normally occur. A quasi-invariant representation for a speech segment can be obtained by projecting it to each template orbit, i.e., the set of transformed signals, and computing the associated one-dimensional empirical probability distributions. The computations can be performed by modules of filtering and pooling, and extended to hierarchical architectures. In this paper, we apply a single-layer, multicomponent representation for phonemes and demonstrate improved accuracy and decreased sample complexity for vowel classification compared to standard spectral, cepstral and perceptual features.Comment: CBMM Memo No. 022, 5 pages, 2 figure

Visual Representations: Defining Properties and Deep Approximations

Visual representations are defined in terms of minimal sufficient statistics of visual data, for a class of tasks, that are also invariant to nuisance variability. Minimal sufficiency guarantees that we can store a representation in lieu of raw data with smallest complexity and no performance loss on the task at hand. Invariance guarantees that the statistic is constant with respect to uninformative transformations of the data. We derive analytical expressions for such representations and show they are related to feature descriptors commonly used in computer vision, as well as to convolutional neural networks. This link highlights the assumptions and approximations tacitly assumed by these methods and explains empirical practices such as clamping, pooling and joint normalization.Comment: UCLA CSD TR140023, Nov. 12, 2014, revised April 13, 2015, November 13, 2015, February 28, 201

On the Inductive Bias of Neural Tangent Kernels

State-of-the-art neural networks are heavily over-parameterized, making the optimization algorithm a crucial ingredient for learning predictive models with good generalization properties. A recent line of work has shown that in a certain over-parameterized regime, the learning dynamics of gradient descent are governed by a certain kernel obtained at initialization, called the neural tangent kernel. We study the inductive bias of learning in such a regime by analyzing this kernel and the corresponding function space (RKHS). In particular, we study smoothness, approximation, and stability properties of functions with finite norm, including stability to image deformations in the case of convolutional networks, and compare to other known kernels for similar architectures.Comment: NeurIPS 201

Basic Filters for Convolutional Neural Networks Applied to Music: Training or Design?

When convolutional neural networks are used to tackle learning problems based on music or, more generally, time series data, raw one-dimensional data are commonly pre-processed to obtain spectrogram or mel-spectrogram coefficients, which are then used as input to the actual neural network. In this contribution, we investigate, both theoretically and experimentally, the influence of this pre-processing step on the network's performance and pose the question, whether replacing it by applying adaptive or learned filters directly to the raw data, can improve learning success. The theoretical results show that approximately reproducing mel-spectrogram coefficients by applying adaptive filters and subsequent time-averaging is in principle possible. We also conducted extensive experimental work on the task of singing voice detection in music. The results of these experiments show that for classification based on Convolutional Neural Networks the features obtained from adaptive filter banks followed by time-averaging perform better than the canonical Fourier-transform-based mel-spectrogram coefficients. Alternative adaptive approaches with center frequencies or time-averaging lengths learned from training data perform equally well.Comment: Completely revised version; 21 pages, 4 figure