Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Deep Neural Networks and Splines

In course

MICRO-512: Image processing II

Study of advanced image processing; mathematical imaging. Development of image-processing software and prototyping in Jupyter Notebooks; application to real-world examples in industrial vision and biomedical imaging. ...

Description

This lecture covers the fundamentals of deep neural networks and splines, starting with feedforward deep neural networks and the ReLU activation function. It then delves into continuous-piecewise linear functions in one and multiple dimensions, discussing their algebraic properties and implications for deep ReLU neural networks. The lecture also explores the universal approximation properties of CPWL functions and their implementation via deep ReLU networks. Additionally, it examines the refinement of activation functions, constraining activation functions, the representer theorem for deep neural networks, and the outcome of this theorem. The lecture concludes with a comparison of linear interpolators, the discussion on deep spline networks, their opportunities, challenges, and their connection with existing schemes.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (4)

Michael Stefan Daniel Liebling

Dimitri Nestor Alice Van De Ville

Michaël Unser

Daniel Sage

Official source

Ontological neighbourhood

Information engineering

Machine learning: Artificial neural networks

Related concepts (32)

Activation function

Activation function of a node in an artificial neural network is a function that calculates the output of the node (based on its inputs and the weights on individual inputs). Nontrivial problems can be solved only using a nonlinear activation function. Modern activation functions include the smooth version of the ReLU, the GELU, which was used in the 2018 BERT model, the logistic (sigmoid) function used in the 2012 speech recognition model developed by Hinton et al, the ReLU used in the 2012 AlexNet computer vision model and in the 2015 ResNet model.

Deep learning

Deep learning is part of a broader family of machine learning methods, which is based on artificial neural networks with representation learning. The adjective "deep" in deep learning refers to the use of multiple layers in the network. Methods used can be either supervised, semi-supervised or unsupervised.

Convolutional neural network

Convolutional neural network (CNN) is a regularized type of feed-forward neural network that learns feature engineering by itself via filters (or kernel) optimization. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over fewer connections. For example, for each neuron in the fully-connected layer 10,000 weights would be required for processing an image sized 100 × 100 pixels.

Feedforward neural network

A feedforward neural network (FNN) is one of the two broad types of artificial neural network, characterized by direction of the flow of information between its layers. Its flow is uni-directional, meaning that the information in the model flows in only one direction—forward—from the input nodes, through the hidden nodes (if any) and to the output nodes, without any cycles or loops, in contrast to recurrent neural networks, which have a bi-directional flow.

Universal approximation theorem

In the mathematical theory of artificial neural networks, universal approximation theorems are results that put limits on what neural networks can theoretically learn, i.e. that establish the density of an algorithmically generated class of functions within a given function space of interest. Typically, these results concern the approximation capabilities of the feedforward architecture on the space of continuous functions between two Euclidean spaces, and the approximation is with respect to the compact convergence topology.

Related lectures (38)

Neural Networks: Two Layers Neural Network

Covers the basics of neural networks, focusing on the development from two layers neural networks to deep neural networks.

Deep Splines: Unifying Framework for Deep Neural Networks

Introduces a functional framework for deep neural networks with adaptive piecewise-linear splines, focusing on biomedical image reconstruction and the challenges of deep splines.

Splines and Machine Learning

Explores supervised learning as an ill-posed problem and the integration of sparse adaptive splines into neural architectures.

Splines and Imaging: From Compressed Sensing to Deep Neural Nets

Explores the optimality of splines for imaging and deep neural networks, demonstrating sparsity and global optimality with spline activations.

Deep Neural Networks: Optimization and Approximation