Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTopos and Stacks of Deep Neural Networks
119
0
0.0
(
0
)
Added by
Jean-Claude Belfiore
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendiecks topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Girauds stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannons entropy (P.Baudot and D.B. 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendiecks derivators.
rate research
Read More
Deep learning is a group of exciting new technologies for neural networks. Through a combination of advanced training techniques and neural network architectural components, it is now possible to create neural networks that can handle tabular data, images, text, and audio as both input and output. Deep learning allows a neural network to learn hierarchies of information in a way that is like the function of the human brain. This course will introduce the student to classic neural network structures, Convolution Neural Networks (CNN), Long Short-Term Memory (LSTM), Gated Recurrent Neural Networks (GRU), General Adversarial Networks (GAN), and reinforcement learning. Application of these architectures to computer vision, time series, security, natural language processing (NLP), and data generation will be covered. High-Performance Computing (HPC) aspects will demonstrate how deep learning can be leveraged both on graphical processing units (GPUs), as well as grids. Focus is primarily upon the application of deep learning to problems, with some introduction to mathematical foundations. Readers will use the Python programming language to implement deep learning using Google TensorFlow and Keras. It is not necessary to know Python prior to this book; however, familiarity with at least one programming language is assumed.
This paper is a continuations of the project initiated in the book string topology for stacks. We construct string operations on the SO(2)-equivariant homology of the (free) loop space $L(X)$ of an oriented differentiable stack $X$ and show that $H^{SO(2)}_{*+dim(X) -2}(L(X))$ is a graded Lie algebra. In the particular case where $X$ is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on.
The goal of this paper is to analyze the geometric properties of deep neural network classifiers in the input space. We specifically study the topology of classification regions created by deep networks, as well as their associated decision boundary. Through a systematic empirical investigation, we show that state-of-the-art deep nets learn connected classification regions, and that the decision boundary in the vicinity of datapoints is flat along most directions. We further draw an essential connection between two seemingly unrelated properties of deep networks: their sensitivity to additive perturbations in the inputs, and the curvature of their decision boundary. The directions where the decision boundary is curved in fact remarkably characterize the directions to which the classifier is the most vulnerable. We finally leverage a fundamental asymmetry in the curvature of the decision boundary of deep nets, and propose a method to discriminate between original images, and images perturbed with small adversarial examples. We show the effectiveness of this purely geometric approach for detecting small adversarial perturbations in images, and for recovering the labels of perturbed images.
As neural network model sizes have dramatically increased, so has the interest in various techniques to reduce their parameter counts and accelerate their execution. An active area of research in this field is sparsity - encouraging zero values in parameters that can then be discarded from storage or computations. While most research focuses on high levels of sparsity, there are challenges in universally maintaining model accuracy as well as achieving significant speedups over modern matrix-math hardware. To make sparsity adoption practical, the NVIDIA Ampere GPU architecture introduces sparsity support in its matrix-math units, Tensor Cores. We present the design and behavior of Sparse Tensor Cores, which exploit a 2:4 (50%) sparsity pattern that leads to twice the math throughput of dense matrix units. We also describe a simple workflow for training networks that both satisfy 2:4 sparsity pattern requirements and maintain accuracy, verifying it on a wide range of common tasks and model architectures. This workflow makes it easy to prepare accurate models for efficient deployment on Sparse Tensor Cores.
Deep neural networks can suffer from the exploding and vanishing activation problem, in which the networks fail to train properly because the neural signals either amplify or attenuate across the layers and become saturated. While other normalization methods aim to fix the stated problem, most of them have inference speed penalties in those applications that require running averages of the neural activations. Here we extend the unitary framework based on Lie algebra to neural networks of any dimensionalities, overcoming the major constraints of the prior arts that limit synaptic weights to be square matrices. Our proposed unitary convolutional neural networks deliver up to 32% faster inference speeds and up to 50% reduction in permanent hard disk space while maintaining competitive prediction accuracy.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
