DeepSphere: Efficient spherical convolutional neural network with HEALPix sampling for cosmological applications

Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. So far, these neural networks (NNs) have mostly been developed for regular Euclidean domains such as those supporting images, audio, or video. Because of their success, CNN-based methods are becoming increasingly popular in Cosmology. Cosmological data often comes as spherical maps, which make the use of the traditional CNNs more complicated. The commonly used pixelization scheme for spherical maps is the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix). We present a spherical CNN for analysis of full and partial HEALPix maps, which we call DeepSphere. The spherical CNN is constructed by representing the sphere as a graph. Graphs are versatile data structures that can represent pairwise relationships between objects or act as a discrete representation of a continuous manifold. Using the graph-based representation, we define many of the standard CNN operations, such as convolution and pooling. With filters restricted to being radial, our convolutions are equivariant to rotation on the sphere, and DeepSphere can be made invariant or equivariant to rotation. This way, DeepSphere is a special case of a graph CNN, tailored to the HEALPix sampling of the sphere. This approach is computationally more efficient than using spherical harmonics to perform convolutions. We demonstrate the method on a classification problem of weak lensing mass maps from two cosmological models and compare its performance with that of three baseline classifiers, two based on the power spectrum and pixel density histogram, and a classical 2D CNN. Our experimental results show that the performance of DeepSphere is always superior or equal to the baselines. For high noise levels and for data covering only a smaller fraction of the sphere, DeepSphere achieves typically 10% better classification accuracy than the baselines. Finally, we show how learned filters can be visualized to introspect the NN. Code and examples are available at https://github.com/SwissDataScienceCenter/DeepSphere. (C) 2019 Elsevier B.V. All rights reserved.

A power-efficient, current-mode, binary-tree min / max circuit for Kohonen self-organizing feature maps and nonlinear filters

Rafal Tomasz Dlugosz

A novel current-mode, binary-tree Min / Max circuit for application in analog neural networks and filters has been presented. In the proposed circuit input currents are first converted to step signals with equal amplitudes and different delays that are proportional to the values of these currents. In the second step these delays are compared using a set of time domain comparators in the binary tree structure that determine Min or Max signal. The circuit realized in the CMOS 0.18 μm process offers a precision of 99.5% at data rate of 2.5 MS/s and energy of 0.5 pJ per input.

2010

Filter Learning for Linear Structure Segmentation

Pascal Fua, Germán González Serrano, Vincent Lepetit, Roberto Rigamonti, Engin Türetken

We introduce an approach to learning convolution filters whose joint output can be fed to a classifier that labels them as belonging to linear structures or not. The filters are learned using sparse synthesis techniques but we show that enforcing constraints is not required at run-time to achieve good classification performance. In practice, this is important as it drastically reduces the computational cost. We show that our approach outperforms the state-of-the-art on difficult, and very different, images of roads, retinal scans, and dendritic networks.

2011

Kernel methods and Model predictive approaches for Learning and Control

Sanket Sanjay Diwale

Data-driven modeling and feedback control play a vital role in several application areas ranging from robotics, control theory, manufacturing to management of assets, financial portfolios and supply chains. Many such problems in one way or another are related to variational problems in optimal control and machine learning. The following work first presents, a generalized representer theorem approach to solving such variational problems when closed, densely defined operators, like the differential operators, are involved. Furthermore, loss functionals on infinite dimensional Hilbert spaces are considered to allow for greater freedom in problem formulations. The statement of the theorem presents a necessary and sufficient condition for the existence of linear representer for optimal solutions of such problems. Finally, examples, applying the theorem to neural networks, stochastic regression, and sparsity-inducing regularization problems are presented. The second part of the thesis deals with applications of variational optimization in control problems. Examples from optimal control and model predictive control are presented for applications in the domain of autonomous vehicles and airborne wind energy systems. First, a combination of manifold learning and model predictive control is presented for obstacle avoidance in autonomous driving. Manifold learning is presented as a means to describe boundaries of star-shaped sets for which a single inequality constraint is sufficient to check containment of a point in the set's interior. The approach presented, learns the largest star-shaped set within a circular range such that all obstacle points remain outside the set. The inequality condition for checking containment in such sets is incorporated into a multi-phase, free-end-time optimal control problem to plan trajectories and control inputs moving the vehicle from one point to another while remaining within a given collection of star-shaped sets. The multi-phase, free-end-time problem is adapted to a moving horizon form to give a model predictive path following controller that avoids obstacles by virtue of the manifold learning scheme. A real-time, dynamically updated manifold is learned using point cloud data from a lidar-like sensor on the vehicle to avoid any apriori unknown or moving obstacles. Convergence and recursive feasibility guarantees for the MPC scheme are provided under mild assumptions on the behavior of the obstacles and dynamics of the vehicle. An automated parking scenario in the presence of static and dynamic obstacles is demonstrated in simulation for the complete process of optimal trajectory planning and path following. Next, a continuous time, path following model predictive control scheme is shown for an Airborne Wind Energy (AWE) system. Here stability and convergence guarantees are provided by combining the model predictive controller with terminal constraints inspired from a convergent vector field design problem. A formal stability proof relying on Lyapunov stability arguments is presented to show that for such a design of vector field terminal constraints the path following controller converges to a zero tracking error on the desired path. The last part of the thesis deals with uncertainty in AWE systems due to wind conditions and unknown aerodynamic characteristics. A Gaussian process data-driven optimisation technique and a direct adaptive nonlinear controller design are presented for the same.

EPFL2019

Kernel methods and Model predictive approaches for Learning and Control

Sanket Sanjay Diwale

EPFL2019