In deep learning, weight initialization or parameter initialization describes the initial step in creating a neural network. A neural network contains trainable parameters that are modified during training: weight initialization is the pre-training step of assigning initial values to these parameters. The choice of weight initialization method affects the speed of convergence, the scale of neural activation within the network, the scale of gradient signals during backpropagation, and the quality of the final model. Proper initialization is necessary for avoiding issues such as vanishing and exploding gradients and activation function saturation. Note that even though this article is titled "weight initialization", both weights and biases are used in a neural network as trainable parameters, so this article describes how both of these are initialized. Similarly, trainable parameters in convolutional neural networks (CNNs) are called kernels and biases, and this article also describes these. == Constant initialization == We discuss the main methods of initialization in the context of a multilayer perceptron (MLP). Specific strategies for initializing other network architectures are discussed in later sections. For an MLP, there are only two kinds of trainable parameters, called weights and biases. Each layer l {\displaystyle l} contains a weight matrix W ( l ) ∈ R n l − 1 × n l {\displaystyle W^{(l)}\in \mathbb {R} ^{n_{l-1}\times n_{l}}} and a bias vector b ( l ) ∈ R n l {\displaystyle b^{(l)}\in \mathbb {R} ^{n_{l}}} , where n l {\displaystyle n_{l}} is the number of neurons in that layer. A weight initialization method is an algorithm for setting the initial values for W ( l ) , b ( l ) {\displaystyle W^{(l)},b^{(l)}} for each layer l {\displaystyle l} . The simplest form is zero initialization: W ( l ) = 0 , b ( l ) = 0 {\displaystyle W^{(l)}=0,b^{(l)}=0} Zero initialization is usually used for initializing biases, but it is not used for initializing weights, as it leads to symmetry in the network, causing all neurons to learn the same features. In this page, we assume b = 0 {\displaystyle b=0} unless otherwise stated. Recurrent neural networks typically use activation functions with bounded range, such as sigmoid and tanh, since unbounded activation may cause exploding values. (Le, Jaitly, Hinton, 2015) suggested initializing weights in the recurrent parts of the network to identity and zero bias, similar to the idea of residual connections and LSTM with no forget gate. In most cases, the biases are initialized to zero, though some situations can use a nonzero initialization. For example, in multiplicative units, such as the forget gate of LSTM, the bias can be initialized to 1 to allow good gradient signal through the gate. For neurons with ReLU activation, one can initialize the bias to a small positive value like 0.1, so that the gradient is likely nonzero at initialization, avoiding the dying ReLU problem. == Random initialization == Random initialization means sampling the weights from a normal distribution or a uniform distribution, usually independently. === LeCun initialization === LeCun initialization, popularized in (LeCun et al., 1998), is designed to preserve the variance of neural activations during the forward pass. It samples each entry in W ( l ) {\displaystyle W^{(l)}} independently from a distribution with mean 0 and variance 1 / n l − 1 {\displaystyle 1/n_{l-1}} . For example, if the distribution is a continuous uniform distribution, then the distribution is U ( ± 3 / n l − 1 ) {\displaystyle {\mathcal {U}}(\pm {\sqrt {3/n_{l-1}}})} . === Glorot initialization === Glorot initialization (or Xavier initialization) was proposed by Xavier Glorot and Yoshua Bengio. It was designed as a compromise between two goals: to preserve activation variance during the forward pass and to preserve gradient variance during the backward pass. For uniform initialization, it samples each entry in W ( l ) {\displaystyle W^{(l)}} independently and identically from U ( ± 6 / ( n l + 1 + n l − 1 ) ) {\displaystyle {\mathcal {U}}(\pm {\sqrt {6/(n_{l+1}+n_{l-1})}})} . In the context, n l − 1 {\displaystyle n_{l-1}} is also called the "fan-in", and n l + 1 {\displaystyle n_{l+1}} the "fan-out". When the fan-in and fan-out are equal, then Glorot initialization is the same as LeCun initialization. === He initialization === As Glorot initialization performs poorly for ReLU activation, He initialization (or Kaiming initialization) was proposed by Kaiming He et al. for networks with ReLU activation. It samples each entry in W ( l ) {\displaystyle W^{(l)}} from N ( 0 , 2 / n l − 1 ) {\displaystyle {\mathcal {N}}(0,2/n_{l-1})} . === Orthogonal initialization === (Saxe et al. 2013) proposed orthogonal initialization: initializing weight matrices as uniformly random (according to the Haar measure) semi-orthogonal matrices, multiplied by a factor that depends on the activation function of the layer. It was designed so that if one initializes a deep linear network this way, then its training time until convergence is independent of depth. Sampling a uniformly random semi-orthogonal matrix can be done by initializing X {\displaystyle X} by IID sampling its entries from a standard normal distribution, then calculate ( X X ⊤ ) − 1 / 2 X {\displaystyle \left(XX^{\top }\right)^{-1/2}X} or its transpose, depending on whether X {\displaystyle X} is tall or wide. For CNN kernels with odd widths and heights, orthogonal initialization is done this way: initialize the central point by a semi-orthogonal matrix, and fill the other entries with zero. As an illustration, a kernel K {\displaystyle K} of shape 3 × 3 × c × c ′ {\displaystyle 3\times 3\times c\times c'} is initialized by filling K [ 2 , 2 , : , : ] {\displaystyle K[2,2,:,:]} with the entries of a random semi-orthogonal matrix of shape c × c ′ {\displaystyle c\times c'} , and the other entries with zero. (Balduzzi et al., 2017) used it with stride 1 and zero-padding. This is sometimes called the Orthogonal Delta initialization. Related to this approach, unitary initialization proposes to parameterize the weight matrices to be unitary matrices, with the result that at initialization they are random unitary matrices (and throughout training, they remain unitary). This is found to improve long-sequence modelling in LSTM. Orthogonal initialization has been generalized to layer-sequential unit-variance (LSUV) initialization. It is a data-dependent initialization method, and can be used in convolutional neural networks. It first initializes weights of each convolution or fully connected layer with orthonormal matrices. Then, proceeding from the first to the last layer, it runs a forward pass on a random minibatch, and divides the layer's weights by the standard deviation of its output, so that its output has variance approximately 1. === Fixup initialization === In 2015, the introduction of residual connections allowed very deep neural networks to be trained, much deeper than the ~20 layers of the previous state of the art (such as the VGG-19). Residual connections gave rise to their own weight initialization problems and strategies. These are sometimes called "normalization-free" methods, since using residual connection could stabilize the training of a deep neural network so much that normalizations become unnecessary. Fixup initialization is designed specifically for networks with residual connections and without batch normalization, as follows: Initialize the classification layer and the last layer of each residual branch to 0. Initialize every other layer using a standard method (such as He initialization), and scale only the weight layers inside residual branches by L − 1 2 m − 2 {\displaystyle L^{-{\frac {1}{2m-2}}}} . Add a scalar multiplier (initialized at 1) in every branch and a scalar bias (initialized at 0) before each convolution, linear, and element-wise activation layer. Similarly, T-Fixup initialization is designed for Transformers without layer normalization. === Others === Instead of initializing all weights with random values on the order of O ( 1 / n ) {\displaystyle O(1/{\sqrt {n}})} , sparse initialization initialized only a small subset of the weights with larger random values, and the other weights zero, so that the total variance is still on the order of O ( 1 ) {\displaystyle O(1)} . Random walk initialization was designed for MLP so that during backpropagation, the L2 norm of gradient at each layer performs an unbiased random walk as one moves from the last layer to the first. Looks linear initialization was designed to allow the neural network to behave like a deep linear network at initialization, since W R e L U ( x ) − W R e L U ( − x ) = W x {\displaystyle W\;\mathrm {ReLU} (x)-W\;\mathrm {ReLU} (-x)=Wx} . It initializes a matrix W {\displaystyle W} of shape R n 2 × m {\displaystyle \mathbb {R} ^{{\frac {n}{2}}\times m}} by any method, such as orthogonal initialization, t
Language Server Protocol
The Language Server Protocol (LSP) is an open, JSON-RPC-based protocol for use between source-code editors or integrated development environments (IDEs) and servers that provide "language intelligence tools": programming language-specific features like code completion, syntax highlighting and marking of warnings and errors, as well as refactoring routines. The goal of the protocol is to allow programming language support to be implemented and distributed independently of any given editor or IDE. In the early 2020s, LSP quickly became a "norm" for language intelligence tools providers. == History == LSP was originally developed for Microsoft Visual Studio Code and is now an open standard. On June 27, 2016, Microsoft announced a collaboration with Red Hat and Codenvy to standardize the protocol's specification. Its specification is hosted and developed on GitHub. == Background == Modern IDEs provide programmers with sophisticated features like code completion, refactoring, navigating to a symbol's definition, syntax highlighting, and error and warning markers. For example, in a text-based programming language, a programmer might want to rename a method read. The programmer could either manually edit the respective source code files and change the appropriate occurrences of the old method name into the new name, or instead use an IDE's refactoring capabilities to make all the necessary changes automatically. To be able to support this style of refactoring, an IDE needs a sophisticated understanding of the programming language that the program's source is written in. A programming tool without such an understanding—for example, one that performs a naive search-and-replace instead—could introduce errors. When renaming a read method, for example, the tool should not replace the partial match in a variable that might be called readyState, nor should it replace the portion of a code comment containing the word "already". Neither should renaming a local variable read, for example, end up altering identically-named variables in other scopes. Conventional compilers or interpreters for a specific programming language are typically unable to provide these language services, because they are written with the goal of either transforming the source code into object code or immediately executing the code. Additionally, language services must be able to handle source code that is not well-formed, e.g. because the programmer is in the middle of editing and has not yet finished typing a statement, procedure, or other construct. Additionally, small changes to a source code file which are done during typing usually change the semantics of the program. In order to provide instant feedback to the user, the editing tool must be able to very quickly evaluate the syntactical and semantical consequences of a specific modification. Compilers and interpreters therefore provide a poor candidate for producing the information needed for an editing tool to consume. Prior to the design and implementation of the Language Server Protocol for the development of Visual Studio Code, most language services were generally tied to a given IDE or other editor. In the absence of the Language Server Protocol, language services are typically implemented by using a tool-specific extension API. Providing the same language service to another editing tool requires effort to adapt the existing code so that the service may target the second editor's extension interfaces. The Language Server Protocol allows for decoupling language services from the editor so that the services may be contained within a general-purpose language server. Any editor can inherit sophisticated support for many different languages by making use of existing language servers. Similarly, a programmer involved with the development of a new programming language can make services for that language available to existing editing tools. Making use of language servers via the Language Server Protocol thus also reduces the burden on vendors of editing tools, because vendors do not need to develop language services of their own for the languages the vendor intends to support, as long as the language servers have already been implemented. The Language Server Protocol also enables the distribution and development of servers contributed by an interested third party, such as end users, without additional involvement by either the vendor of the compiler for the programming language in use or the vendor of the editor to which the language support is being added. LSP is not restricted to programming languages. It can be used for any kind of text-based language, like specifications or domain-specific languages (DSL). == Technical overview == When a user edits one or more source code files using a language server protocol-enabled tool, the tool acts as a client that consumes the language services provided by a language server. The tool may be a text editor or IDE and the language services could be refactoring, code completion, etc. The client informs the server about what the user is doing, e.g., opening a file or inserting a character at a specific text position. The client can also request the server to perform a language service, e.g. to format a specified range in the text document. The server answers a client's request with an appropriate response. For example, the formatting request is answered either by a response that transfers the formatted text to the client or by an error response containing details about the error. The Language Server Protocol defines the messages to be exchanged between client and language server. They are JSON-RPC preceded by headers similar to HTTP. Messages may originate from the server or client. The protocol does not make any provisions about how requests, responses and notifications are transferred between client and server. For example, client and server could be components within the same process exchanging JSON strings via method calls. They could also be different processes on the same or on different machines communicating via network sockets. == Registry == There are lists of LSP-compatible implementations, maintained by the community-driven Langserver.org or Microsoft.
Markov model
In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property. == Introduction == Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made: == Markov chain == The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution. == Hidden Markov model == A hidden Markov model is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist. For example, given a sequence of observations, the Viterbi algorithm will compute the most-likely corresponding sequence of states, the forward algorithm will compute the probability of the sequence of observations, and the Baum–Welch algorithm will estimate the starting probabilities, the transition function, and the observation function of a hidden Markov model. One common use is for speech recognition, where the observed data is the speech audio waveform and the hidden state is the spoken text. In this example, the Viterbi algorithm finds the most likely sequence of spoken words given the speech audio. == Markov decision process == A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards. == Partially observable Markov decision process == A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. POMDPs are known to be NP complete, but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots. == Markov random field == A Markov random field, or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensions. In a Markov chain, state depends only on the previous state in time, whereas in a Markov random field, each state depends on its neighbors in any of multiple directions. A Markov random field may be visualized as a field or graph of random variables, where the distribution of each random variable depends on the neighboring variables with which it is connected. More specifically, the joint distribution for any random variable in the graph can be computed as the product of the "clique potentials" of all the cliques in the graph that contain that random variable. Modeling a problem as a Markov random field is useful because it implies that the joint distributions at each vertex in the graph may be computed in this manner. == Hierarchical Markov models == Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model and the Abstract Hidden Markov Model. Both have been used for behavior recognition and certain conditional independence properties between different levels of abstraction in the model allow for faster learning and inference. == Tolerant Markov model == A Tolerant Markov model (TMM) is a probabilistic-algorithmic Markov chain model. It assigns the probabilities according to a conditioning context that considers the last symbol, from the sequence to occur, as the most probable instead of the true occurring symbol. A TMM can model three different natures: substitutions, additions or deletions. Successful applications have been efficiently implemented in DNA sequences compression. == Markov-chain forecasting models == Markov-chains have been used as a forecasting methods for several topics, for example price trends, wind power and solar irradiance. The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series to hidden Markov-models combined with wavelets and the Markov-chain mixture distribution model (MCM).
Multilinear principal component analysis
Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear (tensor) independent component analysis (MICA). In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between multilinear data models that employed 2nd order statistics versus higher order statistics to compute a set of independent components for each mode, such as Multilinear ICA Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized, or whose observations are treated as a collection of column/row observations, an "observation as a matrix", and concatenated into a data tensor. The latter approach is suitable for compression and reducing redundancy in the rows, columns and fibers that are unrelated to the causal factors of data formation. Vasilescu and Terzopoulos in their paper "TensorFaces" introduced the M-mode SVD algorithm which are algorithms misidentified in the literature as the HOSVD or the Tucker which employ the power method or gradient descent, respectively. Vasilescu and Terzopoulos framed the data analysis, recognition and synthesis problems as multilinear tensor problems. Data is viewed as the compositional consequence of several causal factors, that are well suited for multi-modal tensor factor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in the following papers: Human Motion Signatures (CVPR 2001, ICPR 2002), face recognition – TensorFaces, (ECCV 2002, CVPR 2003, etc.) and computer graphics – TensorTextures (Siggraph 2004). == The algorithm == The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent. == Feature selection == MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition while a semi-supervised MPCA feature selection is employed in visualization tasks. == Extensions == Various extension of MPCA: Robust MPCA (RMPCA) Multi-Tensor Factorization, that also finds the number of components automatically (MTF)
Mathematics of neural networks in machine learning
An artificial neural network (ANN) or neural network combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways. == Structure == === Neuron === A neuron with label j {\displaystyle j} receiving an input p j ( t ) {\displaystyle p_{j}(t)} from predecessor neurons consists of the following components: an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter, an optional threshold θ j {\displaystyle \theta _{j}} , which stays fixed unless changed by learning, an activation function f {\displaystyle f} that computes the new activation at a given time t + 1 {\displaystyle t+1} from a j ( t ) {\displaystyle a_{j}(t)} , θ j {\displaystyle \theta _{j}} and the net input p j ( t ) {\displaystyle p_{j}(t)} giving rise to the relation a j ( t + 1 ) = f ( a j ( t ) , p j ( t ) , θ j ) , {\displaystyle a_{j}(t+1)=f(a_{j}(t),p_{j}(t),\theta _{j}),} and an output function f out {\displaystyle f_{\text{out}}} computing the output from the activation o j ( t ) = f out ( a j ( t ) ) . {\displaystyle o_{j}(t)=f_{\text{out}}(a_{j}(t)).} Often the output function is simply the identity function. An input neuron has no predecessor but serves as input interface for the whole network. Similarly an output neuron has no successor and thus serves as output interface of the whole network. === Propagation function === The propagation function computes the input p j ( t ) {\displaystyle p_{j}(t)} to the neuron j {\displaystyle j} from the outputs o i ( t ) {\displaystyle o_{i}(t)} and typically has the form p j ( t ) = ∑ i o i ( t ) w i j . {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}.} === Bias === A bias term can be added, changing the form to the following: p j ( t ) = ∑ i o i ( t ) w i j + w 0 j , {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}+w_{0j},} where w 0 j {\displaystyle w_{0j}} is a bias. == Neural networks as functions == Neural network models can be viewed as defining a function that takes an input (observation) and produces an output (decision) f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} or a distribution over X {\displaystyle \textstyle X} or both X {\displaystyle \textstyle X} and Y {\displaystyle \textstyle Y} . Sometimes models are intimately associated with a particular learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons, number of layers or their connectivity). Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} , that can further be decomposed into other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between functions. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right)} , where K {\displaystyle \textstyle K} (commonly referred to as the activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth transition as input values change, i.e. a small change in input produces a small change in output. The following refers to a collection of functions g i {\displaystyle \textstyle g_{i}} as a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} . This figure depicts such a decomposition of f {\displaystyle \textstyle f} , with dependencies between variables indicated by arrows. These can be interpreted in two ways. The first view is the functional view: the input x {\displaystyle \textstyle x} is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} , which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} , which is finally transformed into f {\displaystyle \textstyle f} . This view is most commonly encountered in the context of optimization. The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} , which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} , which depends upon the random variable X {\displaystyle \textstyle X} . This view is most commonly encountered in the context of graphical models. The two views are largely equivalent. In either case, for this particular architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} are independent of each other given their input h {\displaystyle \textstyle h} ). This naturally enables a degree of parallelism in the implementation. Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} is shown as dependent upon itself. However, an implied temporal dependence is not shown. == Backpropagation == Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant); Levenberg–Marquardt and conjugate gradient (Fletcher–Reeves update, Polak–Ribiére update, Powell–Beale restart, scaled conjugate gradient). === Algorithm === Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displaystyle \mathbb {R} ^{n}} , and w 0 , w 1 , w 2 , … {\displaystyle w_{0},w_{1},w_{2},\ldots } vectors in R e {\displaystyle \mathbb {R} ^{e}} . These are called inputs, outputs and weights, respectively. The network corresponds to a function y = f N ( w , x ) {\displaystyle y=f_{N}(w,x)} which, given a weight w {\displaystyle w} , maps an input x {\displaystyle x} to an output y {\displaystyle y} . In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random. These weights are computed in turn: first compute w i {\displaystyle w_{i}} using only ( x i , y i , w i − 1 ) {\displaystyle (x_{i},y_{i},w_{i-1})} for i = 1 , … , p {\displaystyle i=1,\dots ,p} . The output of the algorithm is then w p {\displaystyle w_{p}} , giving a new function x ↦ f N ( w p , x ) {\displaystyle x\mapsto f_{N}(w_{p},x)} . The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , x 1 ) , y 1 ) {\displaystyle w\mapsto E(f_{N}(w,x_{1}),y_{1})} to find a local minimum, starting at w = w 0 {\displaystyle w=w_{0}} . This makes w 1 {\displaystyle w_{1}} the minimizing weight found by gradient descent. == Learning pseudocode == To implement the algorithm above, explicit formulas are required for the gradient of the function w ↦ E ( f N ( w , x ) , y ) {\displaystyle w\mapsto E(f_{N}(w,x),y)} where the function is E ( y , y ′ ) = | y − y ′ | 2 {\displaystyle E(y,y')=|y-y'|^{2}} . The learning algorithm can be divided into two phases: propagation and weight update. === Propagation === Propagation involves the following steps: Propagation forward through the network to generate the output value(s) Calculation of the cost (error term) Propagation of the output activations back through the network using the training pattern target to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. === Weight update === For each weight: Multiply the weight's output delta and input activation to find the gradient of the weight. Subtract the ratio (percentage) of the weight's gradient from the weight. The learning rate is the ratio (percentage) that influences the speed and quality of learning. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurat
U-Net
U-Net is a convolutional neural network that was developed for image segmentation. The network is based on a fully convolutional neural network whose architecture was modified and extended to work with fewer training images and to yield more precise segmentation. Segmentation of a 512 × 512 image takes less than a second on a modern (2015) GPU using the U-Net architecture. The U-Net architecture has also been employed in diffusion models for iterative image denoising. This technology underlies many modern image generation models, such as DALL-E, Midjourney, and Stable Diffusion. U-Net is also being explored for language models. Tokenization is not a separate step, allowing the model to more easily understand spelling and concurrently vectorizing / tokenizing higher level concepts. == Description == The U-Net architecture stems from the so-called "fully convolutional network". The main idea is to supplement a usual contracting network by successive layers, where pooling operations are replaced by upsampling operators. Hence these layers increase the resolution of the output. A successive convolutional layer can then learn to assemble a precise output based on this information. One important modification in U-Net is that there are a large number of feature channels in the upsampling part, which allow the network to propagate context information to higher resolution layers. As a consequence, the expansive path is more or less symmetric to the contracting part, and yields a u-shaped architecture. The network only uses the valid part of each convolution without any fully connected layers. To predict the pixels in the border region of the image, the missing context is extrapolated by mirroring the input image. This tiling strategy is important to apply the network to large images, since otherwise the resolution would be limited by the GPU memory. Recently, there had also been an interest in receptive field based U-Net models for medical image segmentation. == Network architecture == The network consists of a contracting path and an expansive path, which gives it the u-shaped architecture. The contracting path is a typical convolutional network that consists of repeated application of convolutions, each followed by a rectified linear unit (ReLU) and a max pooling operation. During the contraction, the spatial information is reduced while feature information is increased. The expansive pathway combines the feature and spatial information through a sequence of up-convolutions and concatenations with high-resolution features from the contracting path. == Applications == There are many applications of U-Net in biomedical image segmentation, such as brain image segmentation (''BRATS'') and liver image segmentation ("siliver07") as well as protein binding site prediction. U-Net implementations have also found use in the physical sciences, for example in the analysis of micrographs of materials. Variations of the U-Net have also been applied for medical image reconstruction. Here are some variants and applications of U-Net as follows: Pixel-wise regression using U-Net and its application on pansharpening; 3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation; TernausNet: U-Net with VGG11 Encoder Pre-Trained on ImageNet for Image Segmentation. Image-to-image translation to estimate fluorescent stains In binding site prediction of protein structure. == History == U-Net was created by Olaf Ronneberger, Philipp Fischer, Thomas Brox in 2015 and reported in the paper "U-Net: Convolutional Networks for Biomedical Image Segmentation". It is an improvement and development of FCN: Evan Shelhamer, Jonathan Long, Trevor Darrell (2014). "Fully convolutional networks for semantic segmentation".
Generalized blockmodeling of binary networks
Generalized blockmodeling of binary networks (also relational blockmodeling) is an approach of generalized blockmodeling, analysing the binary network(s). As most network analyses deal with binary networks, this approach is also considered as the fundamental approach of blockmodeling. This is especially noted, as the set of ideal blocks, when used for interpretation of blockmodels, have binary link patterns, which precludes them to be compared with valued empirical blocks. When analysing the binary networks, the criterion function is measuring block inconsistencies, while also reporting the possible errors. The ideal block in binary blockmodeling has only three types of conditions: "a certain cell must be (at least) 1, a certain cell must be 0 and the f {\displaystyle f} over each row (or column) must be at least 1". It is also used as a basis for developing the generalized blockmodeling of valued networks.