24,780 research outputs found
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
Deep SimNets
We present a deep layered architecture that generalizes convolutional neural
networks (ConvNets). The architecture, called SimNets, is driven by two
operators: (i) a similarity function that generalizes inner-product, and (ii) a
log-mean-exp function called MEX that generalizes maximum and average. The two
operators applied in succession give rise to a standard neuron but in "feature
space". The feature spaces realized by SimNets depend on the choice of the
similarity operator. The simplest setting, which corresponds to a convolution,
realizes the feature space of the Exponential kernel, while other settings
realize feature spaces of more powerful kernels (Generalized Gaussian, which
includes as special cases RBF and Laplacian), or even dynamically learned
feature spaces (Generalized Multiple Kernel Learning). As a result, the SimNet
contains a higher abstraction level compared to a traditional ConvNet. We argue
that enhanced expressiveness is important when the networks are small due to
run-time constraints (such as those imposed by mobile applications). Empirical
evaluation validates the superior expressiveness of SimNets, showing a
significant gain in accuracy over ConvNets when computational resources at
run-time are limited. We also show that in large-scale settings, where
computational complexity is less of a concern, the additional capacity of
SimNets can be controlled with proper regularization, yielding accuracies
comparable to state of the art ConvNets
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