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  Tesla Dojo

  Tesla Dojo is a series of supercomputers designed and built by Tesla for computer vision video processing and recognition. It was used for training Tesla's machine learning models to improve its Full Self-Driving (FSD) advanced driver-assistance system. It went into production in July 2023. Dojo's goal was to efficiently process millions of terabytes of video data captured from real-life driving situations from Tesla's 4+ million cars. This goal led to a considerably different architecture than conventional supercomputer designs. In August 2025, Bloomberg News reported that the Dojo project had been disbanded, though it was restarted in January 2026. == History == Tesla operates several massively parallel computing clusters for developing its Autopilot advanced driver assistance system. Its primary unnamed cluster using 5,760 Nvidia A100 graphics processing units (GPUs) was touted by Andrej Karpathy in 2021 at the fourth International Joint Conference on Computer Vision and Pattern Recognition (CCVPR 2021) to be "roughly the number five supercomputer in the world" at approximately 81.6 petaflops, based on scaling the performance of the Nvidia Selene supercomputer, which uses similar components. However, the performance of the primary Tesla GPU cluster has been disputed, as it was not clear if this was measured using single-precision or double-precision floating point numbers (FP32 or FP64). Tesla also operates a second 4,032 GPU cluster for training and a third 1,752 GPU cluster for automatic labeling of objects. The primary unnamed Tesla GPU cluster has been used for processing one million video clips, each ten seconds long, taken from Tesla Autopilot cameras operating in Tesla cars in the real world, running at 36 frames per second. Collectively, these video clips contained six billion object labels, with depth and velocity data; the total size of the data set was 1.5 petabytes. This data set was used for training a neural network intended to help Autopilot computers in Tesla cars understand roads. By August 2022, Tesla had upgraded the primary GPU cluster to 7,360 GPUs. Dojo was first mentioned by Elon Musk in April 2019 during Tesla's "Autonomy Investor Day". In August 2020, Musk stated it was "about a year away" due to power and thermal issues. Dojo was officially announced at Tesla's Artificial Intelligence (AI) Day on August 19, 2021. Tesla revealed details of the D1 chip and its plans for "Project Dojo", a datacenter that would house 3,000 D1 chips; the first "Training Tile" had been completed and delivered the week before. In October 2021, Tesla released a "Dojo Technology" whitepaper describing the Configurable Float8 (CFloat8) and Configurable Float16 (CFloat16) floating point formats and arithmetic operations as an extension of Institute of Electrical and Electronics Engineers (IEEE) standard 754. At the follow-up AI Day in September 2022, Tesla announced it had built several System Trays and one Cabinet. During a test, the company stated that Project Dojo drew 2.3 megawatts (MW) of power before tripping a local San Jose, California power substation. At the time, Tesla was assembling one Training Tile per day. In August 2023, Tesla powered on Dojo for production use as well as a new training cluster configured with 10,000 Nvidia H100 GPUs. In January 2024, Musk described Dojo as "a long shot worth taking because the payoff is potentially very high. But it's not something that is a high probability." In June 2024, Musk explained that ongoing construction work at Gigafactory Texas is for a computing cluster claiming that it is planned to comprise an even mix of "Tesla AI" and Nvidia/other hardware with a total thermal design power of at first 130 MW and eventually exceeding 500 MW. In August 2025, Bloomberg News reported that the Dojo project was disbanded, though Musk announced it would be restarted in January 2026 with a new chip iteration. == Technical architecture == The fundamental unit of the Dojo supercomputer is the D1 chip, designed by a team at Tesla led by ex-AMD CPU designer Ganesh Venkataramanan, including Emil Talpes, Debjit Das Sarma, Douglas Williams, Bill Chang, and Rajiv Kurian. The D1 chip is manufactured by the Taiwan Semiconductor Manufacturing Company (TSMC) using 7 nanometer (nm) semiconductor nodes, has 50 billion transistors and a large die size of 645 mm2 (1.0 square inch). Updating at Artificial Intelligence (AI) Day in 2022, Tesla announced that Dojo would scale by deploying multiple ExaPODs, in which there would be: 10 Cabinets per ExaPOD (1,062,000 cores, 3,000 D1 chips) 2 System Trays per Cabinet (106,200 cores, 300 D1 chips) 6 Training Tiles per System Tray (53,100 cores, along with host interface hardware) 25 D1 chips per Training Tile (8,850 cores) 354 computing cores per D1 chip According to Venkataramanan, Tesla's senior director of Autopilot hardware, Dojo will have more than an exaflop (a million teraflops) of computing power. For comparison, according to Nvidia, in August 2021, the (pre-Dojo) Tesla AI-training center used 720 nodes, each with eight Nvidia A100 Tensor Core GPUs for 5,760 GPUs in total, providing up to 1.8 exaflops of performance. === D1 chip === Each node (computing core) of the D1 processing chip is a general purpose 64-bit CPU with a superscalar core. It supports internal instruction-level parallelism, and includes simultaneous multithreading (SMT). It doesn't support virtual memory and uses limited memory protection mechanisms. Dojo software/applications manage chip resources. The D1 instruction set supports both 64-bit scalar and 64-byte single instruction, multiple data (SIMD) vector instructions. The integer unit mixes reduced instruction set computer (RISC-V) and custom instructions, supporting 8, 16, 32, or 64 bit integers. The custom vector math unit is optimized for machine learning kernels and supports multiple data formats, with a mix of precisions and numerical ranges, many of which are compiler composable. Up to 16 vector formats can be used simultaneously. ==== Node ==== Each D1 node uses a 32-byte fetch window holding up to eight instructions. These instructions are fed to an eight-wide decoder which supports two threads per cycle, followed by a four-wide, four-way SMT scalar scheduler that has two integer units, two address units, and one register file per thread. Vector instructions are passed further down the pipeline to a dedicated vector scheduler with two-way SMT, which feeds either a 64-byte SIMD unit or four 8×8×4 matrix multiplication units. The network on-chip (NOC) router links cores into a two-dimensional mesh network. It can send one packet in and one packet out in all four directions to/from each neighbor node, along with one 64-byte read and one 64-byte write to local SRAM per clock cycle. Hardware native operations transfer data, semaphores and barrier constraints across memories and CPUs. System-wide double data rate 4 (DDR4) synchronous dynamic random-access memory (SDRAM) memory works like bulk storage. ==== Memory ==== Each core has a 1.25 megabytes (MB) of SRAM main memory. Load and store speeds reach 400 gigabytes (GB) per second and 270 GB/sec, respectively. The chip has explicit core-to-core data transfer instructions. Each SRAM has a unique list parser that feeds a pair of decoders and a gather engine that feeds the vector register file, which together can directly transfer information across nodes. ==== Die ==== Twelve nodes (cores) are grouped into a local block. Nodes are arranged in an 18×20 array on a single die, of which 354 cores are available for applications. The die runs at 2 gigahertz (GHz) and totals 440 MB of SRAM (360 cores × 1.25 MB/core). It reaches 376 teraflops using 16-bit brain floating point (BF16) numbers or using configurable 8-bit floating point (CFloat8) numbers, which is a Tesla proposal, and 22 teraflops at FP32. Each die comprises 576 bi-directional serializer/deserializer (SerDes) channels along the perimeter to link to other dies, and moves 8 TB/sec across all four die edges. Each D1 chip has a thermal design power of approximately 400 watts. === Training Tile === The water-cooled Training Tile packages 25 D1 chips into a 5×5 array. Each tile supports 36 TB/sec of aggregate bandwidth via 40 input/output (I/O) chips - half the bandwidth of the chip mesh network. Each tile supports 10 TB/sec of on-tile bandwidth. Each tile has 11 GB of SRAM memory (25 D1 chips × 360 cores/D1 × 1.25 MB/core). Each tile achieves 9 petaflops at BF16/CFloat8 precision (25 D1 chips × 376 TFLOP/D1). Each tile consumes 15 kilowatts; 288 amperes at 52 volts. === System Tray === Six tiles are aggregated into a System Tray, which is integrated with a host interface. Each host interface includes 512 x86 cores, providing a Linux-based user environment. Previously, the Dojo System Tray was known as the Training Matrix, which includes six Training Tiles, 20 Dojo Interface Processor cards across four host servers, and Ethernet-l

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  Online machine learning

  In computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., prediction of prices in the financial international markets. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches. Online machine learning algorithms find applications in a wide variety of fields such as sponsored search to maximize ad revenue, portfolio optimization, shortest path prediction (with stochastic weights, e.g. traffic on roads for a maps application), spam filtering, real-time fraud detection, dynamic pricing for e-commerce, etc. There is also growing interest in usage of online learning paradigms for LLMs to enable continuous, real-time adaptation after the initial training. == Introduction == In the setting of supervised learning, a function of f : X → Y {\displaystyle f:X\to Y} is to be learned, where X {\displaystyle X} is thought of as a space of inputs and Y {\displaystyle Y} as a space of outputs, that predicts well on instances that are drawn from a joint probability distribution p ( x , y ) {\displaystyle p(x,y)} on X × Y {\displaystyle X\times Y} . In reality, the learner never knows the true distribution p ( x , y ) {\displaystyle p(x,y)} over instances. Instead, the learner usually has access to a training set of examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} . In this setting, the loss function is given as V : Y × Y → R {\displaystyle V:Y\times Y\to \mathbb {R} } , such that V ( f ( x ) , y ) {\displaystyle V(f(x),y)} measures the difference between the predicted value f ( x ) {\displaystyle f(x)} and the true value y {\displaystyle y} . The ideal goal is to select a function f ∈ H {\displaystyle f\in {\mathcal {H}}} , where H {\displaystyle {\mathcal {H}}} is a space of functions called a hypothesis space, so that some notion of total loss is minimized. Depending on the type of model (statistical or adversarial), one can devise different notions of loss, which lead to different learning algorithms. == Statistical view of online learning == In statistical learning models, the training sample ( x i , y i ) {\displaystyle (x_{i},y_{i})} are assumed to have been drawn from the true distribution p ( x , y ) {\displaystyle p(x,y)} and the objective is to minimize the expected "risk" I [ f ] = E [ V ( f ( x ) , y ) ] = ∫ V ( f ( x ) , y ) d p ( x , y ) . {\displaystyle I[f]=\mathbb {E} [V(f(x),y)]=\int V(f(x),y)\,dp(x,y)\ .} A common paradigm in this situation is to estimate a function f ^ {\displaystyle {\hat {f}}} through empirical risk minimization or regularized empirical risk minimization (usually Tikhonov regularization). The choice of loss function here gives rise to several well-known learning algorithms such as regularized least squares and support vector machines. A purely online model in this category would learn based on just the new input ( x t + 1 , y t + 1 ) {\displaystyle (x_{t+1},y_{t+1})} , the current best predictor f t {\displaystyle f_{t}} and some extra stored information (which is usually expected to have storage requirements independent of training data size). For many formulations, for example nonlinear kernel methods, true online learning is not possible, though a form of hybrid online learning with recursive algorithms can be used where f t + 1 {\displaystyle f_{t+1}} is permitted to depend on f t {\displaystyle f_{t}} and all previous data points ( x 1 , y 1 ) , … , ( x t , y t ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{t},y_{t})} . In this case, the space requirements are no longer guaranteed to be constant since it requires storing all previous data points, but the solution may take less time to compute with the addition of a new data point, as compared to batch learning techniques. A common strategy to overcome the above issues is to learn using mini-batches, which process a small batch of b ≥ 1 {\displaystyle b\geq 1} data points at a time, this can be considered as pseudo-online learning for b {\displaystyle b} much smaller than the total number of training points. Mini-batch techniques are used with repeated passing over the training data to obtain optimized out-of-core versions of machine learning algorithms, for example, stochastic gradient descent. When combined with backpropagation, this is currently the de facto training method for training artificial neural networks. === Example: linear least squares === The simple example of linear least squares is used to explain a variety of ideas in online learning. The ideas are general enough to be applied to other settings, for example, with other convex loss functions. === Batch learning === Consider the setting of supervised learning with f {\displaystyle f} being a linear function to be learned: f ( x j ) = ⟨ w , x j ⟩ = w ⋅ x j {\displaystyle f(x_{j})=\langle w,x_{j}\rangle =w\cdot x_{j}} where x j ∈ R d {\displaystyle x_{j}\in \mathbb {R} ^{d}} is a vector of inputs (data points) and w ∈ R d {\displaystyle w\in \mathbb {R} ^{d}} is a linear filter vector. The goal is to compute the filter vector w {\displaystyle w} . To this end, a square loss function V ( f ( x j ) , y j ) = ( f ( x j ) − y j ) 2 = ( ⟨ w , x j ⟩ − y j ) 2 {\displaystyle V(f(x_{j}),y_{j})=(f(x_{j})-y_{j})^{2}=(\langle w,x_{j}\rangle -y_{j})^{2}} is used to compute the vector w {\displaystyle w} that minimizes the empirical loss I n [ w ] = ∑ j = 1 n V ( ⟨ w , x j ⟩ , y j ) = ∑ j = 1 n ( x j T w − y j ) 2 {\displaystyle I_{n}[w]=\sum _{j=1}^{n}V(\langle w,x_{j}\rangle ,y_{j})=\sum _{j=1}^{n}(x_{j}^{\mathsf {T}}w-y_{j})^{2}} where y j ∈ R . {\displaystyle y_{j}\in \mathbb {R} .} Let X {\displaystyle X} be the i × d {\displaystyle i\times d} data matrix and y ∈ R i {\displaystyle y\in \mathbb {R} ^{i}} is the column vector of target values after the arrival of the first i {\displaystyle i} data points. Assuming that the covariance matrix Σ i = X T X {\displaystyle \Sigma _{i}=X^{\mathsf {T}}X} is invertible (otherwise it is preferential to proceed in a similar fashion with Tikhonov regularization), the best solution f ∗ ( x ) = ⟨ w ∗ , x ⟩ {\displaystyle f^{}(x)=\langle w^{},x\rangle } to the linear least squares problem is given by w ∗ = ( X T X ) − 1 X T y = Σ i − 1 ∑ j = 1 i x j y j . {\displaystyle w^{}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y=\Sigma _{i}^{-1}\sum _{j=1}^{i}x_{j}y_{j}.} Now, calculating the covariance matrix Σ i = ∑ j = 1 i x j x j T {\displaystyle \Sigma _{i}=\sum _{j=1}^{i}x_{j}x_{j}^{\mathsf {T}}} takes time O ( i d 2 ) {\displaystyle O(id^{2})} , inverting the d × d {\displaystyle d\times d} matrix takes time O ( d 3 ) {\displaystyle O(d^{3})} , while the rest of the multiplication takes time O ( d 2 ) {\displaystyle O(d^{2})} , giving a total time of O ( i d 2 + d 3 ) {\displaystyle O(id^{2}+d^{3})} . When there are n {\displaystyle n} total points in the dataset, to recompute the solution after the arrival of every datapoint i = 1 , … , n {\displaystyle i=1,\ldots ,n} , the naive approach will have a total complexity O ( n 2 d 2 + n d 3 ) {\displaystyle O(n^{2}d^{2}+nd^{3})} . Note that when storing the matrix Σ i {\displaystyle \Sigma _{i}} , then updating it at each step needs only adding x i + 1 x i + 1 T {\displaystyle x_{i+1}x_{i+1}^{\mathsf {T}}} , which takes O ( d 2 ) {\displaystyle O(d^{2})} time, reducing the total time to O ( n d 2 + n d 3 ) = O ( n d 3 ) {\displaystyle O(nd^{2}+nd^{3})=O(nd^{3})} , but with an additional storage space of O ( d 2 ) {\displaystyle O(d^{2})} to store Σ i {\displaystyle \Sigma _{i}} . === Online learning: recursive least squares === The recursive least squares (RLS) algorithm considers an online approach to the least squares problem. It can be shown that by initialising w 0 = 0 ∈ R d {\displaystyle \textstyle w_{0}=0\in \mathbb {R} ^{d}} and Γ 0 = I ∈ R d × d {\displaystyle \textstyle \Gamma _{0}=I\in \mathbb {R} ^{d\times d}} , the solution of the linear least squares problem given in the previous section can be computed by the following iteration: Γ i = Γ i − 1 − Γ i − 1 x i x i T Γ i − 1 1 + x i T Γ i − 1 x i {\displaystyle \Gamma _{i}=\Gamma _{i-1}-{\frac {\Gamma _{i-1}x_{i}x_{i}^{\mathsf {T}}\Gamma _{i-1}}{1+x_{i}^{\mathsf {T}}\Gamma _{i-1}x_{i}}}} w i = w i − 1 − Γ i x i ( x i T w i − 1 − y i ) {\displaystyle w_{i}=w_{i-1}-\Gamma _{i}x_{i}\left(x_{i}^{\mathsf {T}}w_{

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  Causal Markov condition

  The Causal Markov (CM) condition states that, conditional on the set of all its direct causes, a node is independent of all variables which are not effects or direct causes of that node. In the event that the structure of a Bayesian network accurately depicts causality, the two conditions are equivalent. This is related to the Markov condition, an assumption made in Bayesian probability theory, that every node in a Bayesian network is conditionally independent of its nondescendants, given its parents. Stated loosely, it is assumed that a node has no bearing on nodes which do not descend from it. In a DAG, this local Markov condition is equivalent to the global Markov condition, which states that d-separations in the graph also correspond to conditional independence relations. This also means that a node is conditionally independent of the entire network, given its Markov blanket. A network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition. == Motivation == Statisticians are enormously interested in the ways in which certain events and variables are connected. The precise notion of what constitutes a cause and effect is necessary to understand the connections between them. The central idea behind the philosophical study of probabilistic causation is that causes raise the probabilities of their effects, all else being equal. A deterministic interpretation of causation means that if A causes B, then A must always be followed by B. In this sense, smoking does not cause cancer because some smokers never develop cancer. On the other hand, a probabilistic interpretation simply means that causes raise the probability of their effects. In this sense, changes in meteorological readings associated with a storm do cause that storm, since they raise its probability. (However, simply looking at a barometer does not change the probability of the storm, for a more detailed analysis, see:). == Examples == In a simple view, releasing one's hand from a hammer causes the hammer to fall. However, doing so in outer space does not produce the same outcome, calling into question if releasing one's fingers from a hammer always causes it to fall. A causal graph could be created to acknowledge that both the presence of gravity and the release of the hammer contribute to its falling. However, it would be very surprising if the surface underneath the hammer affected its falling. This essentially states the Causal Markov Condition, that given the existence of gravity the release of the hammer, it will fall regardless of what is beneath it. == Implications == === Dependence and Causation === It follows from the definition that if X and Y are in V and are probabilistically dependent, then either X causes Y, Y causes X, or X and Y are both effects of some common cause Z in V. This definition was seminally introduced by Hans Reichenbach as the Common Cause Principle (CCP). === Screening === It once again follows from the definition that the parents of X screen X from other "indirect causes" of X (parents of Parents(X)) and other effects of Parents(X) which are not also effects of X.

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  Ho–Kashyap algorithm

  The Ho–Kashyap algorithm is an iterative method in machine learning for finding a linear decision boundary that separates two linearly separable classes. It was developed by Yu-Chi Ho and Rangasami L. Kashyap in 1965, and usually presented as a problem in linear programming. == Setup == Given a training set consisting of samples from two classes, the Ho–Kashyap algorithm seeks to find a weight vector w {\displaystyle \mathbf {w} } and a margin vector b {\displaystyle \mathbf {b} } such that: Y w = b {\displaystyle \mathbf {Yw} =\mathbf {b} } where Y {\displaystyle \mathbf {Y} } is the augmented data matrix with samples from both classes (with appropriate sign conventions, e.g., samples from class 2 are negated), w {\displaystyle \mathbf {w} } is the weight vector to be determined, and b {\displaystyle \mathbf {b} } is a positive margin vector. The algorithm minimizes the criterion function: J ( w , b ) = | | Y w − b | | 2 {\displaystyle J(\mathbf {w} ,\mathbf {b} )=||\mathbf {Yw} -\mathbf {b} ||^{2}} subject to the constraint that b > 0 {\displaystyle \mathbf {b} >\mathbf {0} } (element-wise). Given a problem of linearly separating two classes, we consider a dataset of elements { ( x i , y i ) } i ∈ 1 : N {\displaystyle \{(\mathbf {x_{i}} ,y_{i})\}_{i\in 1:N}} where y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} . Linearly separating them by a perceptron is equivalent to finding weight and bias w , b {\displaystyle \mathbf {w} ,b} for a perceptron, such that: [ y 1 x 1 1 ⋮ ⋮ y N x N 1 ] [ w b ] > 0 {\displaystyle {\begin{bmatrix}y_{1}\mathbf {x} _{1}&1\\\vdots &\vdots \\y_{N}\mathbf {x} _{N}&1\\\end{bmatrix}}{\begin{bmatrix}\mathbf {w} \\b\end{bmatrix}}>0} == Algorithm == The idea of the Ho–Kashyap algorithm is as follows: Given any b {\displaystyle \mathbf {b} } , the corresponding w {\displaystyle \mathbf {w} } is known: It is simply w = Y + b {\displaystyle \mathbf {w} =\mathbf {Y} ^{+}\mathbf {b} } , where Y + {\displaystyle \mathbf {Y} ^{+}} denotes the Moore–Penrose pseudoinverse of Y {\displaystyle \mathbf {Y} } . Therefore, it only remains to find b {\displaystyle \mathbf {b} } by gradient descent. However, the gradient descent may sometimes decrease some of the coordinates of b {\displaystyle \mathbf {b} } , which may cause some coordinates of b {\displaystyle \mathbf {b} } to become negative, which is undesirable. Therefore, whenever some coordinates of b {\displaystyle \mathbf {b} } would have decreased, those coordinates are unchanged instead. As for the coordinates of b {\displaystyle \mathbf {b} } that would increase, those would increase without issue. Formally, the algorithm is as follows: Initialization: Set b ( 0 ) {\displaystyle \mathbf {b} (0)} to an arbitrary positive vector, typically b ( 0 ) = 1 {\displaystyle \mathbf {b} (0)=\mathbf {1} } (a vector of ones). Set the iteration counter k = 0 {\displaystyle k=0} . Set w ( 0 ) = Y + b ( 0 ) {\displaystyle \mathbf {w} (0)=\mathbf {Y} ^{+}\mathbf {b} (0)} Loop until convergence, or until iteration counter exceeds some k m a x {\displaystyle k_{max}} . Error calculation: Compute the error vector: e ( k ) = Y w ( k ) − b ( k ) {\displaystyle \mathbf {e} (k)=\mathbf {Yw} (k)-\mathbf {b} (k)} . Margin update: Update the margin vector: b ( k + 1 ) = b ( k ) + 2 η k ( e ( k ) + | e ( k ) | ) {\displaystyle \mathbf {b} (k+1)=\mathbf {b} (k)+2\eta _{k}(\mathbf {e} (k)+|\mathbf {e} (k)|)} where η k {\displaystyle \eta _{k}} is a positive learning rate parameter, and | e ( k ) | {\displaystyle |\mathbf {e} (k)|} denotes the element-wise absolute value. Weight calculation: Compute the weight vector using the pseudoinverse: w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} . Convergence check: If | | e ( k ) | | ≤ θ {\displaystyle ||\mathbf {e} (k)||\leq \theta } for some predetermined threshold θ {\displaystyle \theta } (close to zero), then return b ( k + 1 ) , w ( k + 1 ) {\displaystyle \mathbf {b} (k+1),\mathbf {w} (k+1)} . if e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } (all components non-positive), return "Samples not separable.". Return "Algorithm failed to converge in time.". == Properties == If the training data is linearly separable, the algorithm converges to a solution (where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } ) in a finite number of iterations. If the data is not linearly separable, the algorithm may or may not ever reach the point where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } . However, if it does happen that e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } at some iteration, this proves non-separability. The convergence rate depends on the choice of the learning rate parameter ρ {\displaystyle \rho } and the degree of linear separability of the data. == Relationship to other algorithms == Perceptron algorithm: Both seek linear separators. The perceptron updates weights incrementally based on individual misclassified samples, while Ho–Kashyap is a batch method that processes all samples to compute the pseudoinverse and updates based on an overall error vector. Linear discriminant analysis (LDA): LDA assumes underlying Gaussian distributions with equal covariances for the classes and derives the decision boundary from these statistical assumptions. Ho–Kashyap makes no explicit distributional assumptions and instead tries to solve a system of linear inequalities directly. Support vector machines (SVM): For linearly separable data, SVMs aim to find the maximum-margin hyperplane. The Ho–Kashyap algorithm finds a separating hyperplane but not necessarily the one with the maximum margin. If the data is not separable, soft-margin SVMs allow for some misclassifications by optimizing a trade-off between margin size and misclassification penalty, while Ho–Kashyap provides a least-squares solution. == Variants == Modified Ho–Kashyap algorithm changes weight calculation step w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} to w ( k + 1 ) = w ( k ) + η k Y + | e ( k ) | {\displaystyle \mathbf {w} (k+1)=\mathbf {w} (k)+\eta _{k}\mathbf {Y} ^{+}|\mathbf {e} (k)|} . Kernel Ho–Kashyap algorithm: Applies kernel methods (the "kernel trick") to the Ho–Kashyap framework to enable non-linear classification by implicitly mapping data to a higher-dimensional feature space.

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  Google Mobile Services

  Google Mobile Services (GMS) is a collection of proprietary applications and application programming interfaces (APIs) services from Google that are typically pre-installed on the majority of Android devices, such as smartphones, tablets, and smart TVs. GMS is not a part of the Android Open Source Project (AOSP), which means an Android manufacturer needs to obtain a license from Google in order to legally pre-install GMS on an Android device. This license is provided by Google without any licensing fees except in the EU. == Core applications == The following are core applications that are part of Google Mobile Services: Google Search Google Chrome YouTube Google Play Google Drive Gmail Google Meet Google Maps Google Photos Google TV YouTube Music === Historically === Google+ Google Hangouts Google Wallet Google Play Magazines Google Play Music Google Play Movies & TV Google Duo == Reception, competitors, and regulators == === FairSearch === Numerous European firms filed a complaint to the European Commission stating that Google had manipulated their power and dominance within the market to push their Services to be used by phone manufacturers. The firms were joined under the name FairSearch, and the main firms included were Microsoft, Expedia, TripAdvisor, Nokia and Oracle. FairSearch's major problem with Google's practices was that they believed Google were forcing phone manufacturers to use their Mobile Services. They claimed Google managed this by asking these manufacturers to sign a contract stating that they must preinstall specific Google Mobile Services, such as Maps, Search and YouTube, in order to get the latest version of Android. Google swiftly responded stating that they "continue to work co-operatively with the European Commission". === Aptoide === The third-party Android app store Aptoide also filed an EU competition complaint against Google once again stating that they are misusing their power within the market. Aptoide alleged that Google was blocking third-party app stores from being on Google Play, as well as blocking Google Chrome from downloading any third-party apps and app stores. As of June 2014, Google had not responded to these allegations. === Abuse of Android dominance === In May 2019, Umar Javeed, Sukarma Thapar, Aaqib Javeed vs. Google LLC & Ors. the Competition Commission of India ordered an antitrust probe against Google for abusing its dominant position with Android to block market rivals. In Prima Facie opinion the commission held that mandatory pre-installation of the entire Google Mobile Services (GMS) suite, under Mobile Application Distribution Agreements (MADA), amounts to the imposition of unfair conditions on the device manufacturers. === EU antitrust ruling === On July 18, 2018, the European Commission fined Google €4.34 billion for breaching EU antitrust rules which resulted in a change of licensing policy for the GMS in the EU. A new paid licensing agreement for smartphones and tablets shipped into the EEA was created. The change is that the GMS is now decoupled from the base Android and will be offered under a separate paid licensing agreement. === Privacy policy === At the same time, Google faced problems with various European data protection agencies, most notably In the United Kingdom and France. The problem they faced was that they had a set of 60 rules merged into one, which allowed Google to "track users more closely". Google once again came out and stated that their new policies still abide by European Union laws. === Android distributions without Google Mobile Services === After surveillance and privacy concerns, several custom android distributions have been implemented, such as GrapheneOS, LineageOS, CalyxOS, iodéOS or /e/OS, and they come either without any GMS installed by default or with microG, that adds a compatibility layer.

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  Time-aware long short-term memory

  Time-aware LSTM (T-LSTM) is a long short-term memory (LSTM) unit capable of handling irregular time intervals in longitudinal patient records. T-LSTM was developed by researchers from Michigan State University, IBM Research, and Cornell University and was first presented in the Knowledge Discovery and Data Mining (KDD) conference. Experiments using real and synthetic data proved that T-LSTM auto-encoder outperformed widely used frameworks including LSTM and MF1-LSTM auto-encoders.

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  Population model (evolutionary algorithm)

  The population model of an evolutionary algorithm (EA) describes the structural properties of its population to which its members are subject. A population is the set of all proposed solutions of an EA considered in one iteration, which are also called individuals according to the biological role model. The individuals of a population can generate further individuals as offspring with the help of the genetic operators of the procedure. The simplest and widely used population model in EAs is the global or panmictic model, which corresponds to an unstructured population. It allows each individual to choose any other individual of the population as a partner for the production of offspring by crossover, whereby the details of the selection are irrelevant as long as the fitness of the individuals plays a significant role. Due to global mate selection, the genetic information of even slightly better individuals can prevail in a population after a few generations (iteration of an EA), provided that no better other offspring have emerged in this phase. If the solution found in this way is not the optimum sought, that is called premature convergence. This effect can be observed more often in panmictic populations. In nature global mating pools are rarely found. What prevails is a certain and limited isolation due to spatial distance. The resulting local neighbourhoods initially evolve independently and mutants have a higher chance of persisting over several generations. As a result, genotypic diversity in the gene pool is preserved longer than in a panmictic population. It is therefore obvious to divide the previously global population by substructures. Two basic models were introduced for this purpose, the island models, which are based on a division of the population into fixed subpopulations that exchange individuals from time to time, and the neighbourhood models, which assign individuals to overlapping neighbourhoods, also known as cellular genetic or evolutionary algorithms (cGA or cEA). The associated division of the population also suggests a corresponding parallelization of the procedure. For this reason, the topic of population models is also frequently discussed in the literature in connection with the parallelization of EAs. == Island models == In the island model, also called the migration model or coarse grained model, evolution takes place in strictly divided subpopulations. These can be organised panmictically, but do not have to be. From time to time an exchange of individuals takes place, which is called migration. The time between an exchange is called an epoch and its end can be triggered by various criteria: E.g. after a given time or given number of completed generations, or after the occurrence of stagnation. Stagnation can be detected, for example, by the fact that no fitness improvement has occurred in the island for a given number of generations. Island models introduce a variety of new strategy parameters: Number of subpopulations Size of the subpopulations Neighbourhood relations between islands: they determine which islands are considered neighbouring and can thus exchange individuals, see picture of a simple unidirectional ring (black arrows) and its extension by additional bidirectional neighbourhood relations (additional green arrows) Criteria for the termination of an epoch, synchronous or asynchronous migration Migration rate: number or proportion of individuals involved in migration. Migrant selection: There are many alternatives for this. E.g. the best individuals can replace the worst or randomly selected ones. Depending on the migration rate, this can affect one or more individuals at a time. With these parameters, the selection pressure can be influenced to a considerable extent. For example, it increases with the interconnectedness of the islands and decreases with the number of subpopulations or the epoch length. == Neighbourhood models or cellular evolutionary algorithms == The neighbourhood model, also called diffusion model or fine grained model, defines a topological neighbouhood relation between the individuals of a population that is independent of their phenotypic properties. The fundamental idea of this model is to provide the EA population with a special structure defined as a connected graph, in which each vertex is an individual that communicates with its nearest neighbours. Particularly, individuals are conceptually set in a toroidal mesh, and are only allowed to recombine with close individuals. This leads to a kind of locality known as isolation by distance. The set of potential mates of an individual is called its neighbourhood or deme. The adjacent figure illustrates that by showing two slightly overlapping neighbourhoods of two individuals marked yellow, through which genetic information can spread between the two demes. It is known that in this kind of algorithm, similar individuals tend to cluster and create niches that are independent of the deme boundaries and, in particular, can be larger than a deme. There is no clear borderline between adjacent groups, and close niches could be easily colonized by competitive ones and maybe merge solution contents during this process. Simultaneously, farther niches can be affected more slowly. EAs with this type of population are also well known as cellular EAs (cEA) or cellular genetic algorithms (cGA). A commonly used structure for arranging the individuals of a population is a 2D toroidal grid, although the number of dimensions can be easily extended (to 3D) or reduced (to 1D, e.g. a ring, see the figure on the right). The neighbourhood of a particular individual in the grid is defined in terms of the Manhattan distance from it to others in the population. In the basic algorithm, all the neighbourhoods have the same size and identical shapes. The two most commonly used neighbourhoods for two-dimensional cEAs are L5 and C9, see the figure on the left. Here, L stands for Linear while C stands for Compact. Each deme represents a panmictic subpopulation within which mate selection and the acceptance of offspring takes place by replacing the parent. The rules for the acceptance of offspring are local in nature and based on the neighbourhood: for example, it can be specified that the best offspring must be better than the parent being replaced or, less strictly, only better than the worst individual in the deme. The first rule is elitist and creates a higher selective pressure than the second non-elitist rule. In elitist EAs, the best individual of a population always survives. In this respect, they deviate from the biological model. The overlap of the neighbourhoods causes a mostly slow spread of genetic information across the neighbourhood boundaries, hence the name diffusion model. A better offspring now needs more generations than in panmixy to spread in the population. This promotes the emergence of local niches and their local evolution, thus preserving genotypic diversity over a longer period of time. The result is a better and dynamic balance between breadth and depth search adapted to the search space during a run. Depth search takes place in the niches and breadth search in the niche boundaries and through the evolution of the different niches of the whole population. For the same neighbourhood size, the spread of genetic information is larger for elongated figures like L9 than for a block like C9, and again significantly larger than for a ring. This means that ring neighbourhoods are well suited for achieving high quality results, even if this requires comparatively long run times. On the other hand, if one is primarily interested in fast and good, but possibly suboptimal results, 2D topologies are more suitable. == Comparison == When applying both population models to genetic algorithms, evolutionary strategy and other EAs, the splitting of a total population into subpopulations usually reduces the risk of premature convergence and leads to better results overall more reliably and faster than would be expected with panmictic EAs. Island models have the disadvantage compared to neighbourhood models that they introduce a large number of new strategy parameters. Despite the existing studies on this topic in the literature, a certain risk of unfavourable settings remains for the user. With neighbourhood models, on the other hand, only the size of the neighbourhood has to be specified and, in the case of the two-dimensional model, the choice of the neighbourhood figure is added. == Parallelism == Since both population models imply population partitioning, they are well suited as a basis for parallelizing an EA. This applies even more to cellular EAs, since they rely only on locally available information about the members of their respective demes. Thus, in the extreme case, an independent execution thread can be assigned to each individual, so that the entire cEA can run on a parallel hardware platform. The island model also supports p

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  Self-organizing map

  A self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher-dimensional data set while preserving the topological structure of the data. For example, a data set with p {\displaystyle p} variables measured in n {\displaystyle n} observations could be represented as clusters of observations with similar values for the variables. These clusters then could be visualized as a two-dimensional "map" such that observations in proximal clusters have more similar values than observations in distal clusters. This can make high-dimensional data easier to visualize and analyze. A SOM is a type of artificial neural network but is trained using competitive learning rather than the error-correction learning (e.g., backpropagation with gradient descent) used by other artificial neural networks. The SOM was introduced by the Finnish professor Teuvo Kohonen in the 1980s and therefore is sometimes called a Kohonen map or Kohonen network. The Kohonen map or network is a computationally convenient abstraction building on biological models of neural systems from the 1970s and morphogenesis models dating back to Alan Turing in the 1950s. SOMs create internal representations reminiscent of the cortical homunculus, a distorted representation of the human body, based on a neurological "map" of the areas and proportions of the human brain dedicated to processing sensory functions, for different parts of the body. == Overview == Self-organizing maps, like most artificial neural networks, operate in two modes: training and mapping. First, training uses an input data set (the "input space") to generate a lower-dimensional representation of the input data (the "map space"). Second, mapping classifies additional input data using the generated map. The goal of training is to represent an input space with p dimensions as a map space with n dimensions, where p > n. Specifically, an input space with p variables is said to have p dimensions. A map space consists of components called "nodes" or "neurons", which are arranged as a hexagonal or rectangular grid with two dimensions. The number of nodes and their arrangement are specified beforehand based on the larger goals of the analysis and exploration of the data. Each node in the map space is associated with a "weight" vector, which is the position of the node in the input space. While nodes in the map space stay fixed, training consists in moving weight vectors toward the input data (reducing a distance metric such as Euclidean distance) without spoiling the topology induced from the map space. After training, the map can be used to classify additional observations for the input space by finding the node with the closest weight vector (smallest distance metric) to the input space vector. == Learning algorithm == The goal of learning in the self-organizing map is to cause different parts of the network to respond similarly to certain input patterns. This is partly motivated by how visual, auditory or other sensory information is handled in separate parts of the cerebral cortex in the human brain. The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the initial weights already give a good approximation of SOM weights. The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times as iterations. The training utilizes competitive learning. When a training example is fed to the network, its Euclidean distance to all weight vectors is computed. The neuron whose weight vector is most similar to the input is called the best matching unit (BMU). The weights of the BMU and neurons close to it in the SOM grid are adjusted towards the input vector. The magnitude of the change decreases with time and with the grid-distance from the BMU. The update formula for a neuron v with weight vector Wv(s) is W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} , where s is the step index, t is an index into the training sample, u is the index of the BMU for the input vector D(t), α(s) is a monotonically decreasing learning coefficient; θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s. Depending on the implementations, t can scan the training data set systematically (t is 0, 1, 2...T-1, then repeat, T being the training sample's size), be randomly drawn from the data set (bootstrap sampling), or implement some other sampling method (such as jackknifing). The neighborhood function θ(u, v, s) (also called function of lateral interaction) depends on the grid-distance between the BMU (neuron u) and neuron v. In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but the Gaussian and Mexican-hat functions are common choices, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons, the weights are converging to local estimates. In some implementations, the learning coefficient α and the neighborhood function θ decrease steadily with increasing s, in others (in particular those where t scans the training data set) they decrease in step-wise fashion, once every T steps. This process is repeated for each input vector for a (usually large) number of cycles λ. The network winds up associating output nodes with groups or patterns in the input data set. If these patterns can be named, the names can be attached to the associated nodes in the trained net. During mapping, there will be one single winning neuron: the neuron whose weight vector lies closest to the input vector. This can be simply determined by calculating the Euclidean distance between input vector and weight vector. While representing input data as vectors has been emphasized in this article, any kind of object which can be represented digitally, which has an appropriate distance measure associated with it, and in which the necessary operations for training are possible can be used to construct a self-organizing map. This includes matrices, continuous functions or even other self-organizing maps. === Algorithm === Randomize the node weight vectors in a map For s = 0 , 1 , 2 , . . . , λ {\displaystyle s=0,1,2,...,\lambda } Randomly pick an input vector D ( t ) {\displaystyle {D}(t)} Find the node in the map closest to the input vector. This node is the best matching unit (BMU). Denote it by u {\displaystyle u} For each node v {\displaystyle v} , update its vector by pulling it closer to the input vector: W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} The variable names mean the following, with vectors in bold, s {\displaystyle s} is the current iteration λ {\displaystyle \lambda } is the iteration limit t {\displaystyle t} is the index of the target input data vector in the input data set D {\displaystyle \mathbf {D} } D ( t ) {\displaystyle {D}(t)} is a target input data vector v {\displaystyle v} is the index of the node in the map W v {\displaystyle \mathbf {W} _{v}} is the current weight vector of node v {\displaystyle v} u {\displaystyle u} is the index of the best matching unit (BMU) in the map θ ( u , v , s ) {\displaystyle \theta (u,v,s)} is the neighbourhood function, α ( s ) {\displaystyle \alpha (s)} is the learning rate schedule. The key design choices are the shape of the SOM, the neighbourhood function, and the learning rate schedule. The idea of the neighborhood function is to make it such that the BMU is updated the most, its immediate neighbors are updated a little less, and so on. The idea of the learning rate schedule is to make it so that the map updates are large at the start, and gradually stop updating. For example, if we want to learn a SOM using a square grid, we can index it using ( i , j ) {\displaystyle (i,j)} where both i , j ∈ 1 : N {\displaystyle i,j\in 1:N} . The neighborhood function can make it so that the BMU updates in full, the nearest neighbors update in half, and their neighbors update in half again, etc. θ ( ( i , j ) , ( i ′ , j ′ ) , s ) = 1 2 | i − i ′ | + | j − j ′ | = { 1 if i = i ′ , j = j ′ 1 / 2 if | i − i ′ | + | j − j ′ | = 1 1 / 4 if | i − i ′ | + | j − j ′ | = 2 ⋯ ⋯ {\displaystyle \theta ((i,j),(i',j'),s)={\frac {1}{2^{|i-i'|+|j-j'|}}}={\begin{cases}1&{\text{if }}i=i',j=j'\\1/2&{\text{if

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  Automation integrator

  An automation integrator is a systems integrator company or individual who makes different versions of automation hardware and software work together, generally combining several subsystems to work together as one large system. The title may refer to those who only integrate hardware, although these will often work with software integrators. Software created by automation integrators allows devices to communicate with each other, as well as collecting and reporting data. The magazine Control Engineering publishes an annual “Automation Integrator Guide” which lists over 2,000 automation integrators. They also give an annual system integrator of the year award to three automation integration firms. The Control System Integrators Association (CSIA) maintains a buyers' guide of over 1200 member and nonmember systems integrators known as the Industrial Automation Exchange, or CSIA Exchange for short. == Certification == The Control System Integrators Association (CSIA) certifies automation integrators, through an audit based on 79 critical criteria from the best practices manual. Companies must be associate members of the CSIA to be eligible for certification. Integrators can also receive certification through a program launched in 2012 by the Robotics Industries Association. == Industries == Automation Integrators work in a wide variety of industries which use robotics and automation. Some of the most common include:

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  Universal approximation theorem

  In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data. The best-known version of the theorem applies to feedforward networks with a single hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU), then the network can act as a "universal approximator." Universality is achieved by increasing the number of neurons in the hidden layer, making the network "wider." Other versions of the theorem show that universality can also be achieved by keeping the network's width fixed but increasing its number of layers, making it "deeper." These are existence theorems. They guarantee that a network with the right structure exists, but they do not provide a method for finding the network's parameters (training it), nor do they specify exactly how large the network must be for a given function. Finding a suitable network remains a practical challenge that is typically addressed with optimization algorithms like backpropagation. == Setup == Artificial neural networks are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued vectors to real-valued vectors. The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces. Most universal approximation theorems are in one of two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). == History == === Arbitrary width === The first results concerned the arbitrary width case. Ken-ichi Funahashi (May 1989) showed that Rumelhart–Hinton–Williams type backpropagation networks possess universal approximation capability with a class of sigmoidal activation functions, extending the result to multi-output mappings as well. Kurt Hornik, Maxwell Stinchcombe, and Halbert White (July 1989) showed that multilayer feed-forward networks with as few as one hidden layer are universal approximators, provided that the activation function satisfies certain conditions. George Cybenko (December 1989) independently established a related result for sigmoid activation functions using functional-analytic methods. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function. === Arbitrary depth === The arbitrary depth case was also studied by a number of authors such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu et al in 2017, Boris Hanin and Mark Sellke in 2018 who focused on neural networks with ReLU activation function. In 2020, Patrick Kidger and Terry Lyons extended those results to neural networks with general activation functions such, e.g. tanh or GeLU. One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings. === Bounded depth and bounded width === The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators. In 2018, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with fewer units in hidden layers. In 2018, they also constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions. In 2022, Shen et al. obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks. === Quantitative bounds === The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of Lp functions using feed-forward neural networks with ReLU as activation functions. Similar results that can be directly applied to residual neural networks were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using control-theoretic arguments. In 2023, Cai obtained the optimal minimum width bound for the universal approximation. For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function. === Kolmogorov network === The Kolmogorov–Arnold representation theorem is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. Robert Hecht-Nielsen showed that a three-layer neural network can approximate any continuous multivariate function. This was extended to the discontinuous case by Vugar Ismailov. In 2024, Ziming Liu and co-authors showed a practical application. === Reservoir computing and quantum reservoir computing === In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons and spiking neurons, respectively. In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. === Variants === Variants include discontinuous activation functions, noncompact domains, certifiable networks, random neural networks, and alternative network architectures and topologies. The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. For input dimension d x {\displaystyle d_{x}} and output dimension d y {\displaystyle d_{y}} the minimum width required for the universal approximation of the Lp functions is exactly m a x { d x + 1 , d y } {\displaystyle max\{d_{x}+1,d_{y}\}} (for a ReLU network). More generally this also holds if both ReLU and a threshold activation function are used. Universal function approximation on graphs (or rather on graph isomorphism classes) by popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅ | E | ) {\displaystyle {\mathcal {O}}(\left|V\right|\cdot \left|E\right|)} -runtime method that performed at state of the art on a collection of benchmarks (where V {\displaystyle V} and E {\displaystyle E} are the sets of nodes and edges of the graph respectively). There are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. == Arbitrary-width case == A universal approximation theorem formally states that a family of neural network funct

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  Rule-based machine learning

  Rule-based machine learning (RBML) is a term in computer science intended to encompass any machine learning method that identifies, learns, or evolves 'rules' to store, manipulate or apply. The defining characteristic of a rule-based machine learner is the identification and utilization of a set of relational rules that collectively represent the knowledge captured by the system. Rule-based machine learning approaches include learning classifier systems, association rule learning, artificial immune systems, and any other method that relies on a set of rules, each covering contextual knowledge. While rule-based machine learning is conceptually a type of rule-based system, it is distinct from traditional rule-based systems, which are often hand-crafted, and other rule-based decision makers. This is because rule-based machine learning applies some form of learning algorithm such as Rough sets theory to identify and minimise the set of features and to automatically identify useful rules, rather than a human needing to apply prior domain knowledge to manually construct rules and curate a rule set. == Rules == Rules typically take the form of an '{IF:THEN} expression', (e.g. {IF 'condition' THEN 'result'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign}). An individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Therefore rule-based machine learning methods typically comprise a set of rules, or knowledge base, that collectively make up the prediction model usually known as decision algorithm. Rules can also be interpreted in various ways depending on the domain knowledge, data types(discrete or continuous) and in combinations. == RIPPER == Repeated incremental pruning to produce error reduction (RIPPER) is a propositional rule learner proposed by William W. Cohen as an optimized version of IREP.

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  Multiclass classification

  In machine learning and statistical classification, multiclass classification or multinomial classification is the problem of classifying instances into one of three or more classes (classifying instances into one of two classes is called binary classification). For example, deciding on whether an image is showing a banana, peach, orange, or an apple is a multiclass classification problem, with four possible classes (banana, peach, orange, apple), while deciding on whether an image contains an apple or not is a binary classification problem (with the two possible classes being: apple, no apple). While many classification algorithms (e.g., decision trees, k-NN, neural networks and multinomial logistic regression) naturally permit the use of more than two classes, some are by nature binary algorithms (e.g., classical binary support vector machine) and require decomposition strategies such as one-vs-all, one-vs-one, or ECOC to solve multiclass problems. Multiclass classification should not be confused with multi-label classification, where multiple labels are to be predicted for each instance (e.g., predicting that an image contains both an apple and an orange, in the previous example). == Better-than-random multiclass models == From the confusion matrix of a multiclass model, we can determine whether a model does better than chance. Let K ≥ 3 {\displaystyle K\geq 3} be the number of classes, O {\displaystyle {\mathcal {O}}} a set of observations, y ^ : O → { 1 , . . . , K } {\displaystyle {\hat {y}}:{\mathcal {O}}\to \{1,...,K\}} a model of the target variable y : O → { 1 , . . . , K } {\displaystyle y:{\mathcal {O}}\to \{1,...,K\}} and n i , j {\displaystyle n_{i,j}} be the number of observations in the set { y = i } ∩ { y ^ = j } {\displaystyle \{y=i\}\cap \{{\hat {y}}=j\}} . We note n i . = ∑ j n i , j {\displaystyle n_{i.}=\sum _{j}n_{i,j}} , n . j = ∑ i n i , j {\displaystyle n_{.j}=\sum _{i}n_{i,j}} , n = ∑ j n . j = ∑ i n i . {\displaystyle n=\sum _{j}n_{.j}=\sum _{i}n_{i.}} , λ i = n i . n {\displaystyle \lambda _{i}={\frac {n_{i.}}{n}}} and μ j = n . j n {\displaystyle \mu _{j}={\frac {n_{.j}}{n}}} . It is assumed that the confusion matrix ( n i , j ) i , j {\displaystyle (n_{i,j})_{i,j}} contains at least one non-zero entry in each row, that is λ i > 0 {\displaystyle \lambda _{i}>0} for any i {\displaystyle i} . Finally we call "normalized confusion matrix" the matrix of conditional probabilities ( P ( y ^ = j ∣ y = i ) ) i , j = ( n i , j n i . ) i , j {\displaystyle (\mathbb {P} ({\hat {y}}=j\mid y=i))_{i,j}=\left({\frac {n_{i,j}}{n_{i.}}}\right)_{i,j}} . === Intuitive explanation === The lift is a way of measuring the deviation from independence of two events A {\displaystyle A} and B {\displaystyle B} : L i f t ( A , B ) = P ( A ∩ B ) P ( A ) P ( B ) = P ( A ∣ B ) P ( A ) = P ( B ∣ A ) P ( B ) {\displaystyle \mathrm {Lift} (A,B)={\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (A)\mathbb {P} (B)}}={\frac {\mathbb {P} (A\mid B)}{\mathbb {P} (A)}}={\frac {\mathbb {P} (B\mid A)}{\mathbb {P} (B)}}} We have L i f t ( A , B ) > 1 {\displaystyle \mathrm {Lift} (A,B)>1} if and only if events A {\displaystyle A} and B {\displaystyle B} occur simultaneously with a greater probability than if they were independent. In other words, if one of the two events occurs, the probability of observing the other event increases. A first condition to satisfy is to have L i f t ( y = i , y ^ = i ) ≥ 1 {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)\geq 1} for any i {\displaystyle i} . And the quality of a model (better or worse than chance) does not change if we over- or undersample the dataset, that is if we multiply each row R i {\displaystyle R_{i}} of the confusion matrix by a constant c i {\displaystyle c_{i}} . Thus the second condition is that the necessary and sufficient conditions for doing better than chance need only depend on the normalized confusion matrix. The condition on lifts can be reformulated with One versus Rest binary models : for any i {\displaystyle i} , we define the binary target variable y i {\displaystyle y_{i}} which is the indicator of event { y = i } {\displaystyle \{y=i\}} , and the binary model y ^ i {\displaystyle {\hat {y}}_{i}} of y i {\displaystyle y_{i}} which is the indicator of event { y ^ = i } {\displaystyle \{{\hat {y}}=i\}} . Each of the y ^ i {\displaystyle {\hat {y}}_{i}} models is a "One versus Rest" model. L i f t ( y = i , y ^ = i ) {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)} only depends on the events { y = i } {\displaystyle \{y=i\}} and { y ^ = i } {\displaystyle \{{\hat {y}}=i\}} , so merging or not merging the other classes doesn't change its value. We therefore have L i f t ( y = i , y ^ = i ) = L i f t ( y i = 1 , y ^ i = 1 ) {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)=\mathrm {Lift} (y_{i}=1,{\hat {y}}_{i}=1)} and the first condition is that all binary One versus Rest models are better than chance. ==== Example ==== If K = 2 {\displaystyle K=2} and 2 is the class of interest , the normalized confusion matrix is ( s p e c i f i c i t y 1 − s p e c i f i c i t y 1 − s e n s i t i v i t y s e n s i t i v i t y ) {\displaystyle {\begin{pmatrix}\mathrm {specificity} &1-\mathrm {specificity} \\1-\mathrm {sensitivity} &\mathrm {sensitivity} \end{pmatrix}}} and we have L i f t ( y = 1 , y ^ = 1 ) − 1 = P ( y = y ^ = 1 ) λ 1 μ 1 − 1 = n 1 , 1 n n 1. n .1 − 1 {\displaystyle \mathrm {Lift} (y=1,{\hat {y}}=1)-1={\frac {\mathbb {P} (y={\hat {y}}=1)}{\lambda _{1}\mu _{1}}}-1={\frac {n_{1,1}n}{n_{1.}n_{.1}}}-1} = n 1 , 1 ( n 1 , 1 + n 1 , 2 + n 2 , 1 + n 2 , 2 ) − ( n 1 , 1 + n 1 , 2 ) ( n 1 , 1 + n 2 , 1 ) n 1. n .1 = n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 n 1. n .1 {\displaystyle ={\frac {n_{1,1}(n_{1,1}+n_{1,2}+n_{2,1}+n_{2,2})-(n_{1,1}+n_{1,2})(n_{1,1}+n_{2,1})}{n_{1.}n_{.1}}}={\frac {n_{1,1}n_{2,2}-n_{1,2}n_{2,1}}{n_{1.}n_{.1}}}} . Thus L i f t ( y = 1 , y ^ = 1 ) ≥ 1 ⟺ n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 ≥ 0 {\displaystyle \mathrm {Lift} (y=1,{\hat {y}}=1)\geq 1\iff n_{1,1}n_{2,2}-n_{1,2}n_{2,1}\geq 0} . Similarly, by swapping the roles of 1 and 2, we find that L i f t ( y = 2 , y ^ = 2 ) ≥ 1 ⟺ n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 ≥ 0 {\displaystyle \mathrm {Lift} (y=2,{\hat {y}}=2)\geq 1\iff n_{1,1}n_{2,2}-n_{1,2}n_{2,1}\geq 0} . Dividing by n 1. n 2. {\displaystyle n_{1.}n_{2.}} we find that the necessary and sufficient condition on the normalized confusion matrix is s e n s i t i v i t y s p e c i f i c i t y − ( 1 − s e n s i t i v i t y ) ( 1 − s p e c i f i c i t y ) ≥ 0 ⟺ s e n s i t i v i t y + s p e c i f i c i t y − 1 ≥ 0 ⟺ J ≥ 0 {\displaystyle \mathrm {sensitivity} \ \mathrm {specificity} -(1-\mathrm {sensitivity} )(1-\mathrm {specificity} )\geq 0\iff \mathrm {sensitivity} +\mathrm {specificity} -1\geq 0\iff J\geq 0} . This brings us back to the classical binary condition: Youden's J must be positive (or zero for random models). === Random models === A random model is a model that is independent of the target variable. This property is easily reformulated with the confusion matrix. This proposition shows that the model y ^ {\displaystyle {\hat {y}}} of y {\displaystyle y} is uninformative if and only if there are two families of numbers ( α i ) i {\displaystyle (\alpha _{i})_{i}} and ( β j ) j {\displaystyle (\beta _{j})_{j}} such that P ( { y = i } ∩ { y ^ = j } ) = α i β j {\displaystyle \mathbb {P} (\{y=i\}\cap \{{\hat {y}}=j\})=\alpha _{i}\beta _{j}} for any i {\displaystyle i} and j {\displaystyle j} . === Multiclass likelihood ratios and diagnostic odds ratios === We define generalized likelihood ratios calculated from the normalized confusion matrix: for any i {\displaystyle i} and j ≠ i {\displaystyle j\not =i} , let L R i , j = P ( y ^ = j ∣ y = j ) P ( y ^ = j ∣ y = i ) {\displaystyle \mathrm {LR} _{i,j}={\frac {\mathbb {P} ({\hat {y}}=j\mid y=j)}{\mathbb {P} ({\hat {y}}=j\mid y=i)}}} . When K = 2 {\displaystyle K=2} , if 2 is the class of interest,, we find the classical likelihood ratios L R 1 , 2 = L R + {\displaystyle \mathrm {LR} _{1,2}=\mathrm {LR} _{+}} and L R 2 , 1 = 1 L R − {\displaystyle \mathrm {LR} _{2,1}={\frac {1}{\mathrm {LR} _{-}}}} . Multiclass diagnostic odds ratios can also be defined using the formula D O R i , j = D O R j , i = L R i , j L R j , i = n i , i n j , j n i , j n j , i = P ( y ^ = j ∣ y = j ) / P ( y ^ = i ∣ y = j ) P ( y ^ = j ∣ y = i ) / P ( y ^ = i ∣ y = i ) {\displaystyle \mathrm {DOR} _{i,j}=\mathrm {DOR} _{j,i}=\mathrm {LR} _{i,j}\mathrm {LR} _{j,i}={\frac {n_{i,i}n_{j,j}}{n_{i,j}n_{j,i}}}={\frac {\mathbb {P} ({\hat {y}}=j\mid y=j)/\mathbb {P} ({\hat {y}}=i\mid y=j)}{\mathbb {P} ({\hat {y}}=j\mid y=i)/\mathbb {P} ({\hat {y}}=i\mid y=i)}}} We saw above that a better-than-chance model (or a random model) must verify L i f t ( y = i , y ^ = i ) ≥ 1 {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)\geq 1} for any i {\displaystyle i} and λ i {\displaystyle \lambda _{i}} . According to the previous corollary, likelihood ratios are thus greater

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  Region Based Convolutional Neural Networks

  Region-based Convolutional Neural Networks (R-CNN) are a family of machine learning models for computer vision, and specifically object detection and localization. The original goal of R-CNN was to take an input image and produce a set of bounding boxes as output, where each bounding box contains an object and also the category (e.g. car or pedestrian) of the object. In general, R-CNN architectures perform selective search over feature maps outputted by a CNN. R-CNN has been extended to perform other computer vision tasks, such as: tracking objects from a drone-mounted camera, locating text in an image, and enabling object detection in Google Lens. Mask R-CNN is also one of seven tasks in the MLPerf Training Benchmark, which is a competition to speed up the training of neural networks. == History == The following covers some of the versions of R-CNN that have been developed. November 2013: R-CNN. April 2015: Fast R-CNN. June 2015: Faster R-CNN. March 2017: Mask R-CNN. December 2017: Cascade R-CNN is trained with increasing Intersection over Union (IoU, also known as the Jaccard index) thresholds, making each stage more selective against nearby false positives. June 2019: Mesh R-CNN adds the ability to generate a 3D mesh from a 2D image. == Architecture == For review articles see. === Selective search === Given an image (or an image-like feature map), selective search (also called Hierarchical Grouping) first segments the image by the algorithm in (Felzenszwalb and Huttenlocher, 2004), then performs the following: Input: (colour) image Output: Set of object location hypotheses L Segment image into initial regions R = {r1, ..., rn} using Felzenszwalb and Huttenlocher (2004) Initialise similarity set S = ∅ foreach Neighbouring region pair (ri, rj) do Calculate similarity s(ri, rj) S = S ∪ s(ri, rj) while S ≠ ∅ do Get highest similarity s(ri, rj) = max(S) Merge corresponding regions rt = ri ∪ rj Remove similarities regarding ri: S = S \ s(ri, r∗) Remove similarities regarding rj: S = S \ s(r∗, rj) Calculate similarity set St between rt and its neighbours S = S ∪ St R = R ∪ rt Extract object location boxes L from all regions in R === R-CNN === With R-CNN, prediction follows a two-step process. A preprocessing selective search step generates a large set of candidate objects (typically as many as 2000), known as regions of interest (ROI). These are forwarded to a CNN, which predicts an object class score and bounding box estimate, independently for each ROI. Importantly, the ROIs are heavily filtered to remove excess candidates. This is achieved using two mechanism. Filtering begins by removing ROIs assigned to the background category. This is a specialized category, which is scored by the CNN alongside other categories. An unfortunate reality is that remaining ROIs typically suffer from heavy duplication. Namely, multiple ROIs that cover same objects in the image are all assigned non-background categories. This is resolved by a heuristic non-maximum suppression (NMS) step. === Fast R-CNN === While the original R-CNN independently computed the neural network features on each of as many as two thousand regions of interest, Fast R-CNN runs the neural network once on the whole image. At the end of the network is a ROIPooling module, which slices out each ROI from the network's output tensor, reshapes it, and classifies it. As in the original R-CNN, the Fast R-CNN uses selective search to generate its region proposals. === Faster R-CNN === While Fast R-CNN used selective search to generate ROIs, Faster R-CNN integrates the ROI generation into the neural network itself. === Mask R-CNN === While previous versions of R-CNN focused on object detections, Mask R-CNN adds instance segmentation. Mask R-CNN also replaced ROIPooling with a new method called ROIAlign, which can represent fractions of a pixel.

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  World Programming System

  The World Programming System, also known as WPS Analytics or WPS, is a software product developed by a company called World Programming (acquired by Altair Engineering). WPS Analytics supports users of mixed ability to access and process data and to perform data science tasks. It has interactive visual programming tools using data workflows, and it has coding tools supporting the use of the SAS language mixed with Python, R and SQL. == About == WPS can use programs written in the language of SAS without the need for translating them into any other language. In this regard WPS is compatible with the SAS system. WPS has a built-in language interpreter able to process the language of SAS and produce similar results. WPS is available to run on z/OS, Windows, macOS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX. On all supported platforms, programs written in the language of SAS can be executed from a WPS command line interface, often referred to as running in batch mode. WPS can also be used from a graphical user interface known as the WPS Workbench for managing, editing and running programs written in the language of SAS. The WPS Workbench user interface is based on Eclipse. WPS version 4 (released in March 2018) introduced a drag-and-drop workflow canvas providing interactive blocks for data retrieval, blending and preparation, data discovery and profiling, predictive modelling powered by machine learning algorithms, model performance validation and scorecards. WPS version 3 (released in February 2012) provided a new client/server architecture that allows the WPS Workbench GUI to execute SAS programs on remote server installations of WPS in a network or cloud. The resulting output, data sets, logs, etc., can then all be viewed and manipulated from inside the Workbench as if the workloads had been executed locally. SAS programs do not require any special language statements to use this feature. == Summary of main features == Runs on Windows, macOS, z/OS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX An integrated development environment based on Eclipse for Linux, macOS and Windows. Support for language of SAS elements. Support for the language of SAS Macros. Matrix Programming support using PROC IML. Support for generating band plots, bar charts, box plots, bubble plots, contour plots, dendrogram plots, ellipse plots, fringe plots, heat maps, high-low plots, histograms, loess plots, needle plots, pie charts, penalised b-spline, radar charts, reference lines, scatter plots, series plots, step plots, regression plots and vector plots. Support for statistical procedures ACECLUS, ASSOCRULES, ANOVA, BIN, BOXPLOT, CANCORR, CANDISC, CLUSTER, CORRESP, DISCRIM, DISTANCE, FACTOR, FASTCLUS, FREQ, GAM, GANNO, GENMOD, GLIMMIX, GLM, GLMMOD, GLMSELECT, ICLIFETEST, KDE, LIFEREG, LIFETEST, LOESS, LOGISTIC, MDS, MEANS, MI, MIANALYSE, MIXED, MODECLUS, NESTED, NLIN, NPAR1WAY, PHREG, PLAN, PLS, POWER, PRINCOMP, PROBIT, QUANTREG, RBF, REG, ROBUSTREG, RSREG, SCORE, SEGMENT, SIMNORMAL, STANDARD, STDSIZE, STDRATE, STEPDISC, SUMMARY, SURVEYMEANS, SURVEYSELECT, TPSPLINE, TRANSREG, TREE, TTEST, UNIVARIATE, VARCLUS, VARCOMP Support for time series procedures ARIMA, AUTOREG, ESM, EXPAND, FORECAST, LOAN, SEVERITY, SPECTRA, TIMESERIES, X12 Support for machine learning procedures DECISIONFOREST, DECISIONTREE, GMM, MLP, OPTIMALBIN, SEGMENT, SVM Support for ODS. Reads and writes SAS datasets (compressed or uncompressed). Access: Actian Matrix (previously known as ParAccel), DASD, DB2, Excel, Greenplum, Hadoop, Informix, Kognitio Archived 2012-08-24 at the Wayback Machine, MariaDB, MySQL, Netezza, ODBC, OLEDB, Oracle, PostgreSQL, SAND, Snowflake, SPSS/PSPP, SQL Server, Sybase, Sybase IQ, Teradata, VSAM, Vertica and XML. Support for SAS Tape Format. Direct output of reports to CSV, PDF and HTML. Support to connect WPS systems programmatically, remote submit parts of a program to execute on connected remote servers, upload and download data between the connected systems. Support for Hadoop Support for R Support for Python == Industry recognition == Gartner recognized World Programming in their Cool Vendors in Data Science, 2014 Report. == Lawsuit == In 2010 World Programming defended its use of the language of SAS in the High Court of England and Wales in SAS Institute Inc. v World Programming Ltd. The software was the subject of a lawsuit by SAS Institute. The EU Court of Justice ruled in favor of World Programming, stating that the copyright protection does not extend to the software functionality, the programming language used and the format of the data files used by the program. It stated that there is no copyright infringement when a company which does not have access to the source code of a program studies, observes and tests that program to create another program with the same functionality.

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  Time-aware long short-term memory

  Time-aware LSTM (T-LSTM) is a long short-term memory (LSTM) unit capable of handling irregular time intervals in longitudinal patient records. T-LSTM was developed by researchers from Michigan State University, IBM Research, and Cornell University and was first presented in the Knowledge Discovery and Data Mining (KDD) conference. Experiments using real and synthetic data proved that T-LSTM auto-encoder outperformed widely used frameworks including LSTM and MF1-LSTM auto-encoders.

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