Artificial neural networkArtificial neural networks (ANNs, also shortened to neural networks (NNs) or neural nets) are a branch of machine learning models that are built using principles of neuronal organization discovered by connectionism in the biological neural networks constituting animal brains. An ANN is based on a collection of connected units or nodes called artificial neurons, which loosely model the neurons in a biological brain. Each connection, like the synapses in a biological brain, can transmit a signal to other neurons.
Types of artificial neural networksThere are many types of artificial neural networks (ANN). Artificial neural networks are computational models inspired by biological neural networks, and are used to approximate functions that are generally unknown. Particularly, they are inspired by the behaviour of neurons and the electrical signals they convey between input (such as from the eyes or nerve endings in the hand), processing, and output from the brain (such as reacting to light, touch, or heat). The way neurons semantically communicate is an area of ongoing research.
Convolutional neural networkConvolutional 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.
Lipschitz continuityIn mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).
Recurrent neural networkA recurrent neural network (RNN) is one of the two broad types of artificial neural network, characterized by direction of the flow of information between its layers. In contrast to uni-directional feedforward neural network, it is a bi-directional artificial neural network, meaning that it allows the output from some nodes to affect subsequent input to the same nodes. Their ability to use internal state (memory) to process arbitrary sequences of inputs makes them applicable to tasks such as unsegmented, connected handwriting recognition or speech recognition.
Activation functionActivation 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.
Neural networkA neural network can refer to a neural circuit of biological neurons (sometimes also called a biological neural network), a network of artificial neurons or nodes in the case of an artificial neural network. Artificial neural networks are used for solving artificial intelligence (AI) problems; they model connections of biological neurons as weights between nodes. A positive weight reflects an excitatory connection, while negative values mean inhibitory connections. All inputs are modified by a weight and summed.
Tomographic reconstructionTomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner.
Residual neural networkA Residual Neural Network (a.k.a. Residual Network, ResNet) is a deep learning model in which the weight layers learn residual functions with reference to the layer inputs. A Residual Network is a network with skip connections that perform identity mappings, merged with the layer outputs by addition. It behaves like a Highway Network whose gates are opened through strongly positive bias weights. This enables deep learning models with tens or hundreds of layers to train easily and approach better accuracy when going deeper.
Modulus of continuityIn mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function f : I → R admits ω as a modulus of continuity if and only if for all x and y in the domain of f. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families.
Medical imagingMedical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to reveal internal structures hidden by the skin and bones, as well as to diagnose and treat disease. Medical imaging also establishes a database of normal anatomy and physiology to make it possible to identify abnormalities.
Image segmentationIn and computer vision, image segmentation is the process of partitioning a into multiple image segments, also known as image regions or image objects (sets of pixels). The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyze. Image segmentation is typically used to locate objects and boundaries (lines, curves, etc.) in images. More precisely, image segmentation is the process of assigning a label to every pixel in an image such that pixels with the same label share certain characteristics.
Digital image processingDigital image processing is the use of a digital computer to process s through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over . It allows a much wider range of algorithms to be applied to the input data and can avoid problems such as the build-up of noise and distortion during processing. Since images are defined over two dimensions (perhaps more) digital image processing may be modeled in the form of multidimensional systems.
Softmax functionThe softmax function, also known as softargmax or normalized exponential function, converts a vector of K real numbers into a probability distribution of K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes, based on Luce's choice axiom.
Image scalingIn computer graphics and digital imaging, image scaling refers to the resizing of a digital image. In video technology, the magnification of digital material is known as upscaling or resolution enhancement. When scaling a vector graphic image, the graphic primitives that make up the image can be scaled using geometric transformations, with no loss of . When scaling a raster graphics image, a new image with a higher or lower number of pixels must be generated.
Magnetic resonance imagingMagnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio waves to generate images of the organs in the body. MRI does not involve X-rays or the use of ionizing radiation, which distinguishes it from computed tomography (CT) and positron emission tomography (PET) scans.
Batch normalizationBatch normalization (also known as batch norm) is a method used to make training of artificial neural networks faster and more stable through normalization of the layers' inputs by re-centering and re-scaling. It was proposed by Sergey Ioffe and Christian Szegedy in 2015. While the effect of batch normalization is evident, the reasons behind its effectiveness remain under discussion. It was believed that it can mitigate the problem of internal covariate shift, where parameter initialization and changes in the distribution of the inputs of each layer affect the learning rate of the network.
Metric mapIn the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the , Met. Such functions are always continuous functions. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that and are metric spaces and is a function from to . Thus we have a metric map when, for any points and in , Here and denote the metrics on and respectively.
Hölder conditionIn mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition.
Banach fixed-point theoremIn mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.
