143 research outputs found
Boosting Deep Neural Networks with Geometrical Prior Knowledge: A Survey
While Deep Neural Networks (DNNs) achieve state-of-the-art results in many
different problem settings, they are affected by some crucial weaknesses. On
the one hand, DNNs depend on exploiting a vast amount of training data, whose
labeling process is time-consuming and expensive. On the other hand, DNNs are
often treated as black box systems, which complicates their evaluation and
validation. Both problems can be mitigated by incorporating prior knowledge
into the DNN.
One promising field, inspired by the success of convolutional neural networks
(CNNs) in computer vision tasks, is to incorporate knowledge about symmetric
geometrical transformations of the problem to solve. This promises an increased
data-efficiency and filter responses that are interpretable more easily. In
this survey, we try to give a concise overview about different approaches to
incorporate geometrical prior knowledge into DNNs. Additionally, we try to
connect those methods to the field of 3D object detection for autonomous
driving, where we expect promising results applying those methods.Comment: Survey Pape
Implicit Neural Convolutional Kernels for Steerable CNNs
Steerable convolutional neural networks (CNNs) provide a general framework
for building neural networks equivariant to translations and other
transformations belonging to an origin-preserving group , such as
reflections and rotations. They rely on standard convolutions with
-steerable kernels obtained by analytically solving the group-specific
equivariance constraint imposed onto the kernel space. As the solution is
tailored to a particular group , the implementation of a kernel basis does
not generalize to other symmetry transformations, which complicates the
development of group equivariant models. We propose using implicit neural
representation via multi-layer perceptrons (MLPs) to parameterize -steerable
kernels. The resulting framework offers a simple and flexible way to implement
Steerable CNNs and generalizes to any group for which a -equivariant MLP
can be built. We apply our method to point cloud (ModelNet-40) and molecular
data (QM9) and demonstrate a significant improvement in performance compared to
standard Steerable CNNs
Equivariant Neural Networks for Indirect Measurements
In the recent years, deep learning techniques have shown great success in
various tasks related to inverse problems, where a target quantity of interest
can only be observed through indirect measurements by a forward operator.
Common approaches apply deep neural networks in a post-processing step to the
reconstructions obtained by classical reconstruction methods. However, the
latter methods can be computationally expensive and introduce artifacts that
are not present in the measured data and, in turn, can deteriorate the
performance on the given task. To overcome these limitations, we propose a
class of equivariant neural networks that can be directly applied to the
measurements to solve the desired task. To this end, we build appropriate
network structures by developing layers that are equivariant with respect to
data transformations induced by well-known symmetries in the domain of the
forward operator. We rigorously analyze the relation between the measurement
operator and the resulting group representations and prove a representer
theorem that characterizes the class of linear operators that translate between
a given pair of group actions. Based on this theory, we extend the existing
concepts of Lie group equivariant deep learning to inverse problems and
introduce new representations that result from the involved measurement
operations. This allows us to efficiently solve classification, regression or
even reconstruction tasks based on indirect measurements also for very sparse
data problems, where a classical reconstruction-based approach may be hard or
even impossible. We illustrate the effectiveness of our approach in numerical
experiments and compare with existing methods.Comment: 22 pages, 6 figure
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