In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU ( x ) = x + = max ( 0 , x ) = x + | x | 2 = { x if x > 0 , 0 x ≤ 0 {\displaystyle \operatorname {ReLU} (x)=x^{+}=\max(0,x)={\frac {x+|x|}{2}}={\begin{cases}x&{\text{if }}x>0,\\0&x\leq 0\end{cases}}} where x {\displaystyle x} is the input to a neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognition using deep neural nets and computational neuroscience. == History == The ReLU was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used ReLU in the context of visual feature extraction in hierarchical neural networks. In 1998, Gregory Woodbury demonstrated that the rectified linear function could account for a broad range of emergent properties in the visual cortex. His work showed that a single unified model could drive the joint development of refined retinotopic maps, ocular dominance columns, and orientation selectivity. By utilizing the rectifier's "cutoff" property, Woodbury achieved a close quantitative fit to biological data, matching the spatial periodicities and topographic refinement patterns observed in macaque and cat cortical maps. Furthermore, he extended this framework to adult plasticity, accurately replicating the spatial and temporal dynamics of lesion-induced cortical reorganization. This research established that the rectified linear response was a necessary mechanism for the stable self-organisation and maintenance of complex, multi-feature neural maps. In 2000, Hahnloser et al. argued that ReLU approximates the biological relationship between neural firing rates and input current, in addition to enabling recurrent neural network dynamics to stabilise under weaker criteria. Prior to 2010, most activation functions used were the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more numerically efficient counterpart, the hyperbolic tangent. Around 2010, the use of ReLU became common again. Jarrett et al. (2009) noted that rectification by either absolute or ReLU (which they called "positive part") was critical for object recognition in convolutional neural networks (CNNs), specifically because it allows average pooling without neighboring filter outputs cancelling each other out. They hypothesized that the use of sigmoid or tanh was responsible for poor performance in previous CNNs. Nair and Hinton (2010) made a theoretical argument that the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by considering a single binary neuron in a Boltzmann machine that takes x {\displaystyle x} as input, and produces 1 as output with probability σ ( x ) = 1 1 + e − x {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}} . They then considered extending its range of output by making infinitely many copies of it X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } , that all take the same input, offset by an amount 0.5 , 1.5 , 2.5 , … {\displaystyle 0.5,1.5,2.5,\dots } , then their outputs are added together as ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} . They then demonstrated that ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} is approximately equal to N ( log ( 1 + e x ) , σ ( x ) ) {\displaystyle {\mathcal {N}}(\log(1+e^{x}),\sigma (x))} , which is also approximately equal to ReLU ( N ( x , σ ( x ) ) ) {\displaystyle \operatorname {ReLU} ({\mathcal {N}}(x,\sigma (x)))} , where N {\displaystyle {\mathcal {N}}} stands for the gaussian distribution. They also argued for another reason for using ReLU: that it allows "intensity equivariance" in image recognition. That is, multiplying input image by a constant k {\displaystyle k} multiplies the output also. In contrast, this is false for other activation functions like sigmoid or tanh. They found that ReLU activation allowed good empirical performance in restricted Boltzmann machines. Glorot et al (2011) argued that ReLU has the following advantages over sigmoid or tanh: ReLU is more similar to biological neurons' responses in their main operating regime. ReLU avoids vanishing gradients. ReLU is cheaper to compute. ReLU creates sparse representation naturally, because many hidden units output exactly zero for a given input. They also found empirically that deep networks trained with ReLU can achieve strong performance without unsupervised pre-training, especially on large, purely supervised tasks. In 2017, the rectified linear function became a central component of the transformer architecture introduced in the Vaswani et al paper "Attention Is All You Need". Within every transformer layer, ReLU is utilized in the position-wise feed-forward networks (FFN), defined by Equation 2 of their paper: FFN ( x ) = max ( 0 , x W 1 + b 1 ) W 2 + b 2 {\displaystyle \operatorname {FFN} (x)=\max(0,xW_{1}+b_{1})W_{2}+b_{2}} This equation is foundational to the model's capacity; while the attention mechanism determines the relationships between tokens, the ReLU-based FFN performs the majority of the numerical computation and houses the bulk of the model's parameters. The efficiency and scalability of this rectified framework triggered a global technological revolution, enabling the development of Large Language Models that have had a profound economic impact. The industrial response to this architecture—including the massive expansion of AI-specific hardware and the birth of the generative AI sector—has positioned the Transformer as a cornerstone of 21st-century infrastructure. During the post 2017 period of rapid AI advancement, the rectified linear unit function has been key to achieving increased model performance and scaling due to the fact that it zeros out responses that are immaterial for a given stimuli, preventing them from accumulating in massive scale models. It is the complete silencing of the parts of the model found to be stimuli-irrelevant during learning that allows for scaling. As the stimuli-irrelevant proportion of the model becomes more massive, these highly numerous connections within the model would inevitably accumulate during scaling no matter how small each individual response is. Therefore, the rectified linear unit function, with its absolute zeroing property, enabled the scaling to hundred billion parameter models and beyond. Early Transformer scaling giants like GPT-3 (2020) and Falcon-180B (2023) relied on the rectified linear unit function explicitly, while successors such as GPT-4 (2023) and Llama 3 (2024) utilized smoother variants like GELU or SwiGLU. These variants were used to improve training stability while fundamentally preserving the rectified principle of zeroing low responses. At the centre of modern artificial intelligence ReLU and its variants maintain absolute zero response across the bulk of the model at any one time, while maintaining approximately linear reponses for stimuli-relevant connections enabling high performance on each specific cognitive task. This feature of activation sparsity has been critical for massive scaling and performance gains of AI models right up to the present day. == Advantages == Advantages of ReLU include: Sparse activation: for example, in a randomly initialized network, only about 50% of hidden units are activated (i.e. have a non-zero output). Better gradient propagation: fewer vanishing gradient problems compared to sigmoidal activation functions that saturate in both directions. Efficiency: only requires comparison and addition. Scale-invariant (homogeneous, or "intensity equivariance"): max ( 0 , a x ) = a max ( 0 , x ) for a ≥ 0 {\displaystyle \max(0,ax)=a\max(0,x){\text{ for }}a\geq 0} . == Potential problems == Possible downsides can include: Non-differentiability at zero (however, it is differentiable anywhere else, and the value of the derivative at zero can be chosen to be 0 or 1 arbitrarily). Not zero-centered: ReLU outputs are always non-negative. This can make it harder for the network to learn during backpropagation, because gradient updates tend to push weights in one direction (positive or negative). Batch normalization can help address this. ReLU is unbounded. Redundancy of the parametrization: Because ReLU is scale-invariant, the network computes the exact same function by scaling the weights and biases in front of a ReLU activation by k {\displaystyle k} , and the weights after by 1 / k {\displaystyle 1/k} . Dying ReLU: ReLU neurons can sometimes be pushed into states
Computational semantics
Computational semantics is a subfield of computational linguistics. Its goal is to elucidate the cognitive mechanisms supporting the generation and interpretation of meaning in humans. It usually involves the creation of computational models that simulate particular semantic phenomena, and the evaluation of those models against data from human participants. While computational semantics is a scientific field, it has many applications in real-world settings and substantially overlaps with Artificial Intelligence. Broadly speaking, the discipline can be subdivided into areas that mirror the internal organization of linguistics. For example, lexical semantics and frame semantics have active research communities within computational linguistics. Some popular methodologies are also strongly inspired by traditional linguistics. Most prominently, the area of distributional semantics, which underpins investigations into embeddings and the internals of Large Language Models, has roots in the work of Zellig Harris. Some traditional topics of interest in computational semantics are: construction of meaning representations, semantic underspecification, anaphora resolution, presupposition projection, and quantifier scope resolution. Methods employed usually draw from formal semantics or statistical semantics. Computational semantics has points of contact with the areas of lexical semantics (word-sense disambiguation and semantic role labeling), discourse semantics, knowledge representation and automated reasoning (in particular, automated theorem proving). Since 1999 there has been an ACL special interest group on computational semantics, SIGSEM.
Physical neural network
A physical neural network is a type of artificial neural network in which an electrically adjustable material is used to emulate the function of a neural synapse or a higher-order (dendritic) neuron model. "Physical" neural network is used to emphasize the reliance on physical hardware used to emulate neurons as opposed to software-based approaches. More generally the term is applicable to other artificial neural networks in which a memristor or other electrically adjustable resistance material is used to emulate a neural synapse. == Types of physical neural networks == === ADALINE === In the 1960s Bernard Widrow and Ted Hoff developed ADALINE (Adaptive Linear Neuron) which used electrochemical cells called memistors (memory resistors) to emulate synapses of an artificial neuron. The memistors were implemented as 3-terminal devices operating based on the reversible electroplating of copper such that the resistance between two of the terminals is controlled by the integral of the current applied via the third terminal. The ADALINE circuitry was briefly commercialized by the Memistor Corporation in the 1960s enabling some applications in pattern recognition. However, since the memistors were not fabricated using integrated circuit fabrication techniques the technology was not scalable and was eventually abandoned as solid-state electronics became mature. === Analog VLSI === In 1989 Carver Mead published his book Analog VLSI and Neural Systems, which spun off perhaps the most common variant of analog neural networks. The physical realization is implemented in analog VLSI. This is often implemented as field effect transistors in low inversion. Such devices can be modelled as translinear circuits. This is a technique described by Barrie Gilbert in several papers around mid 1970th, and in particular his Translinear Circuits from 1981. With this method circuits can be analyzed as a set of well-defined functions in steady-state, and such circuits assembled into complex networks. === Physical Neural Network === Alex Nugent describes a physical neural network as one or more nonlinear neuron-like nodes used to sum signals and nanoconnections formed from nanoparticles, nanowires, or nanotubes which determine the signal strength input to the nodes. Alignment or self-assembly of the nanoconnections is determined by the history of the applied electric field performing a function analogous to neural synapses. Numerous applications for such physical neural networks are possible. For example, a temporal summation device can be composed of one or more nanoconnections having an input and an output thereof, wherein an input signal provided to the input causes one or more of the nanoconnection to experience an increase in connection strength thereof over time. Another example of a physical neural network is taught by U.S. Patent No. 7,039,619 entitled "Utilized nanotechnology apparatus using a neural network, a solution and a connection gap," which issued to Alex Nugent by the U.S. Patent & Trademark Office on May 2, 2006. A further application of physical neural network is shown in U.S. Patent No. 7,412,428 entitled "Application of hebbian and anti-hebbian learning to nanotechnology-based physical neural networks," which issued on August 12, 2008. Nugent and Molter have shown that universal computing and general-purpose machine learning are possible from operations available through simple memristive circuits operating the AHaH plasticity rule. More recently, it has been argued that also complex networks of purely memristive circuits can serve as neural networks. === Phase change neural network === In 2002, Stanford Ovshinsky described an analog neural computing medium in which phase-change material has the ability to cumulatively respond to multiple input signals. An electrical alteration of the resistance of the phase change material is used to control the weighting of the input signals. === Memristive neural network === Greg Snider of HP Labs describes a system of cortical computing with memristive nanodevices. The memristors (memory resistors) are implemented by thin film materials in which the resistance is electrically tuned via the transport of ions or oxygen vacancies within the film. DARPA's SyNAPSE project has funded IBM Research and HP Labs, in collaboration with the Boston University Department of Cognitive and Neural Systems (CNS), to develop neuromorphic architectures which may be based on memristive systems. === Protonic artificial synapses === In 2022, researchers reported the development of nanoscale brain-inspired artificial synapses, using the ion proton (H+), for 'analog deep learning'.
Multifactor dimensionality reduction
Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression. The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute. This process of constructing a new attribute changes the representation space of the data. The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data. == Illustrative example == Consider the following simple example using the exclusive OR (XOR) function. XOR is a logical operator that is commonly used in data mining and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relationship between the attributes (X1 and X2) and the class variable (Y) is defined by the XOR function such that Y = X1 XOR X2. Table 1 A machine learning algorithm would need to discover or approximate the XOR function in order to accurately predict Y using information about X1 and X2. An alternative strategy would be to first change the representation of the data using constructive induction to facilitate predictive modeling. The MDR algorithm would change the representation of the data (X1 and X2) in the following manner. MDR starts by selecting two attributes. In this simple example, X1 and X2 are selected. Each combination of values for X1 and X2 are examined and the number of times Y=1 and/or Y=0 is counted. In this simple example, Y=1 occurs zero times and Y=0 occurs once for the combination of X1=0 and X2=0. With MDR, the ratio of these counts is computed and compared to a fixed threshold. Here, the ratio of counts is 0/1 which is less than our fixed threshold of 1. Since 0/1 < 1 we encode a new attribute (Z) as a 0. When the ratio is greater than one we encode Z as a 1. This process is repeated for all unique combinations of values for X1 and X2. Table 2 illustrates our new transformation of the data. Table 2 The machine learning algorithm now has much less work to do to find a good predictive function. In fact, in this very simple example, the function Y = Z has a classification accuracy of 1. A nice feature of constructive induction methods such as MDR is the ability to use any data mining or machine learning method to analyze the new representation of the data. Decision trees, neural networks, or a naive Bayes classifier could be used in combination with measures of model quality such as balanced accuracy and mutual information. == Machine learning with MDR == As illustrated above, the basic constructive induction algorithm in MDR is very simple. However, its implementation for mining patterns from real data can be computationally complex. As with any machine learning algorithm there is always concern about overfitting. That is, machine learning algorithms are good at finding patterns in completely random data. It is often difficult to determine whether a reported pattern is an important signal or just chance. One approach is to estimate the generalizability of a model to independent datasets using methods such as cross-validation. Models that describe random data typically don't generalize. Another approach is to generate many random permutations of the data to see what the data mining algorithm finds when given the chance to overfit. Permutation testing makes it possible to generate an empirical p-value for the result. Replication in independent data may also provide evidence for an MDR model but can be sensitive to difference in the data sets. These approaches have all been shown to be useful for choosing and evaluating MDR models. An important step in a machine learning exercise is interpretation. Several approaches have been used with MDR including entropy analysis and pathway analysis. Tips and approaches for using MDR to model gene-gene interactions have been reviewed. == Extensions to MDR == Numerous extensions to MDR have been introduced. These include family-based methods, fuzzy methods, covariate adjustment, odds ratios, risk scores, survival methods, robust methods, methods for quantitative traits, and many others. == Applications of MDR == MDR has mostly been applied to detecting gene-gene interactions or epistasis in genetic studies of common human diseases such as atrial fibrillation, autism, bladder cancer, breast cancer, cardiovascular disease, hypertension, obesity, pancreatic cancer, prostate cancer and tuberculosis. It has also been applied to other biomedical problems such as the genetic analysis of pharmacology outcomes. A central challenge is the scaling of MDR to big data such as that from genome-wide association studies (GWAS). Several approaches have been used. One approach is to filter the features prior to MDR analysis. This can be done using biological knowledge through tools such as BioFilter. It can also be done using computational tools such as ReliefF. Another approach is to use stochastic search algorithms such as genetic programming to explore the search space of feature combinations. Yet another approach is a brute-force search using high-performance computing. == Implementations == www.epistasis.org provides an open-source and freely-available MDR software package. An R package for MDR. An sklearn-compatible Python implementation. An R package for Model-Based MDR. MDR in Weka. Generalized MDR.
GraphLab
Turi is a graph-based, high performance, distributed computation framework written in C++. The GraphLab project was started by Prof. Carlos Guestrin of Carnegie Mellon University in 2009. It is an open source project that uses the Apache License. While GraphLab was originally developed for machine learning tasks, it has also been developed for other data-mining tasks. == Motivation == As the amounts of collected data and computing power grow (multicore, GPUs, clusters, clouds), modern datasets no longer fit into one computing node. Efficient distributed parallel algorithms for handling large-scale data are required. The GraphLab framework is a parallel programming abstraction targeted for sparse iterative graph algorithms. GraphLab provides a programming interface, allowing deployment of distributed machine learning algorithms. The main design considerations behind the design of GraphLab are: Sparse data with local dependencies Iterative algorithms Potentially asynchronous execution == GraphLab toolkits == On top of GraphLab, several implemented libraries of algorithms: Topic modeling - contains applications like LDA, which can be used to cluster documents and extract topical representations. Graph analytics - contains applications like pagerank and triangle counting, which can be applied to general graphs to estimate community structure. Clustering - contains standard data clustering tools such as Kmeans Collaborative filtering - contains a collection of applications used to make predictions about users interests and factorize large matrices. Graphical models - contains tools for making joint predictions about collections of related random variables. Computer vision - contains a collection of tools for reasoning about images. == Turi == Turi (formerly called Dato and before that GraphLab Inc.) is a company that was founded by Prof. Carlos Guestrin from University of Washington in May 2013 to continue development support of the GraphLab open source project. Dato Inc. raised a $6.75M Series A from Madrona Venture Group and New Enterprise Associates (NEA). They raised a $18.5M Series B from Vulcan Capital and Opus Capital, with participation from Madrona and NEA. On August 5, 2016, Turi was acquired by Apple Inc. for $200,000,000.
Geometric hashing
In computer science, geometric hashing is a method for efficiently finding two-dimensional objects represented by discrete points that have undergone an affine transformation, though extensions exist to other object representations and transformations. In an off-line step, the objects are encoded by treating each pair of points as a geometric basis. The remaining points can be represented in an invariant fashion with respect to this basis using two parameters. For each point, its quantized transformed coordinates are stored in the hash table as a key, and indices of the basis points as a value. Then a new pair of basis points is selected, and the process is repeated. In the on-line (recognition) step, randomly selected pairs of data points are considered as candidate bases. For each candidate basis, the remaining data points are encoded according to the basis and possible correspondences from the object are found in the previously constructed table. The candidate basis is accepted if a sufficiently large number of the data points index a consistent object basis. Geometric hashing was originally suggested in computer vision for object recognition in 2D and 3D, but later was applied to different problems such as structural alignment of proteins. == Geometric hashing in computer vision == Geometric hashing is a method used for object recognition. Let’s say that we want to check if a model image can be seen in an input image. This can be accomplished with geometric hashing. The method could be used to recognize one of the multiple objects in a base, in this case the hash table should store not only the pose information but also the index of object model in the base. === Example === For simplicity, this example will not use too many point features and assume that their descriptors are given by their coordinates only (in practice local descriptors such as SIFT could be used for indexing). ==== Training Phase ==== Find the model's feature points. Assume that 5 feature points are found in the model image with the coordinates ( 12 , 17 ) ; {\displaystyle (12,17);} ( 45 , 13 ) ; {\displaystyle (45,13);} ( 40 , 46 ) ; {\displaystyle (40,46);} ( 20 , 35 ) ; {\displaystyle (20,35);} ( 35 , 25 ) {\displaystyle (35,25)} , see the picture. Introduce a basis to describe the locations of the feature points. For 2D space and similarity transformation the basis is defined by a pair of points. The point of origin is placed in the middle of the segment connecting the two points (P2, P4 in our example), the x ′ {\displaystyle x'} axis is directed towards one of them, the y ′ {\displaystyle y'} is orthogonal and goes through the origin. The scale is selected such that absolute value of x ′ {\displaystyle x'} for both basis points is 1. Describe feature locations with respect to that basis, i.e. compute the projections to the new coordinate axes. The coordinates should be discretised to make recognition robust to noise, we take the bin size 0.25. We thus get the coordinates ( − 0.75 , − 1.25 ) ; {\displaystyle (-0.75,-1.25);} ( 1.00 , 0.00 ) ; {\displaystyle (1.00,0.00);} ( − 0.50 , 1.25 ) ; {\displaystyle (-0.50,1.25);} ( − 1.00 , 0.00 ) ; {\displaystyle (-1.00,0.00);} ( 0.00 , 0.25 ) {\displaystyle (0.00,0.25)} Store the basis in a hash table indexed by the features (only transformed coordinates in this case). If there were more objects to match with, we should also store the object number along with the basis pair. Repeat the process for a different basis pair (Step 2). It is needed to handle occlusions. Ideally, all the non-colinear pairs should be enumerated. We provide the hash table after two iterations, the pair (P1, P3) is selected for the second one. Hash Table: Most hash tables cannot have identical keys mapped to different values. So in real life one won’t encode basis keys (1.0, 0.0) and (-1.0, 0.0) in a hash table. ==== Recognition Phase ==== Find interesting feature points in the input image. Choose an arbitrary basis. If there isn't a suitable arbitrary basis, then it is likely that the input image does not contain the target object. Describe coordinates of the feature points in the new basis. Quantize obtained coordinates as it was done before. Compare all the transformed point features in the input image with the hash table. If the point features are identical or similar, then increase the count for the corresponding basis (and the type of object, if any). For each basis such that the count exceeds a certain threshold, verify the hypothesis that it corresponds to an image basis chosen in Step 2. Transfer the image coordinate system to the model one (for the supposed object) and try to match them. If successful, the object is found. Otherwise, go back to Step 2. === Finding mirrored pattern === It seems that this method is only capable of handling scaling, translation, and rotation. However, the input image may contain the object in mirror transform. Therefore, geometric hashing should be able to find the object, too. There are two ways to detect mirrored objects. For the vector graph, make the left side positive, and the right side negative. Multiplying the x position by -1 will give the same result. Use 3 points for the basis. This allows detecting mirror images (or objects). Actually, using 3 points for the basis is another approach for geometric hashing. === Geometric hashing in higher-dimensions === Similar to the example above, hashing applies to higher-dimensional data. For three-dimensional data points, three points are also needed for the basis. The first two points define the x-axis, and the third point defines the y-axis (with the first point). The z-axis is perpendicular to the created axis using the right-hand rule. Notice that the order of the points affects the resulting basis
Ni1000
The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.