AI Assistant Job Description

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Deep Learning Anti-Aliasing

Deep Learning Anti-Aliasing (DLAA) is a form of spatial anti-aliasing developed by Nvidia. DLAA depends on and requires Tensor Cores available in Nvidia RTX cards. DLAA is similar to Deep Learning Super Sampling (DLSS) in its anti-aliasing method, with one important differentiation being that the goal of DLSS is to increase performance at the cost of image quality, whereas the main priority of DLAA is improving image quality at the cost of performance (irrelevant of resolution upscaling or downscaling). DLAA is similar to temporal anti-aliasing (TAA) in that they are both spatial anti-aliasing solutions relying on past frame data. Compared to TAA, DLAA is substantially better when it comes to shimmering, flickering, and handling small meshes like wires. == Technical overview == DLAA collects game rendering data including raw low-resolution input, motion vectors, depth buffers, and exposure information. This information feeds into a convolutional neural network that processes the image to reduce aliasing while preserving fine detail. The neural network architecture employs an auto-encoder design trained on high-quality reference images. The training dataset includes diverse scenarios focusing on challenging cases like sub-pixel details, high-contrast edges, and transparent surfaces. The network then processes frames in real-time. Unlike traditional anti-aliasing solutions that rely on manually written heuristics, such as TAA, DLAA uses its neural network to preserve fine details while eliminating unwanted visual artifacts. == History == DLAA was initially called and marketed by Nvidia as DLSS 2x. The first game that added support for DLAA was The Elder Scrolls Online, which implemented the feature in 2021. By June 2022, DLAA was only available in six games. This number rose to 17 by February 2023. In June 2023, TechPowerUp reported that "DLAA is seeing sluggish adoption among game developers", and that Nvidia was working on adding DLAA to the quality presets of DLSS to boost adoption. By December 2023, DLAA was supported in 41 games. In early 2025, an update for the Nvidia App added a driver-based DLSS override feature that enables users to activate DLAA even in games that do not support it natively. == Differences between TAA and DLAA == TAA is used in many modern video games and game engines; however, all previous implementations have used some form of manually written heuristics to prevent temporal artifacts such as ghosting and flickering. One example of this is neighborhood clamping which forcefully prevents samples collected in previous frames from deviating too much compared to nearby pixels in newer frames. This helps to identify and fix many temporal artifacts, but deliberately removing fine details in this way is analogous to applying a blur filter, and thus the final image can appear blurry when using this method. DLAA uses an auto-encoder convolutional neural network trained to identify and fix temporal artifacts, instead of manually programmed heuristics as mentioned above. Because of this, DLAA can generally resolve detail better than other TAA and TAAU implementations, while also removing most temporal artifacts. == Differences between DLSS and DLAA == While DLSS handles upscaling with a focus on performance, DLAA handles anti-aliasing with a focus on visual quality. DLAA runs at the given screen resolution with no upscaling or downscaling functionality provided by DLAA. DLSS and DLAA share the same AI-driven anti-aliasing method. As such, DLAA functions like DLSS without the upscaling part. Both are made by Nvidia and require Tensor Cores. However, DLSS and DLAA cannot be enabled at the same time, only one can be selected depending on whether performance or image quality is prioritized. == Reception == TechPowerUp found that "[c]ompared to TAA and DLSS, DLAA is clearly producing the best image quality, especially at lower resolutions", arguing that, while "DLSS was already doing a better job than TAA at reconstructing small objects", "DLAA does an even better job". In a Cyberpunk 2077 performance test, IGN stated that "DLAA provided somewhat similar results [FPS wise] to the normal raster mode in most cases but got significant performance boost with the help of frame generation", a feature not available when using native resolution. Rock Paper Shotgun noted that, while DLAA is "not a completely perfect form of anti-aliasing, as the occasional jaggies are present", it "looks a lot sharper overall [than TAA], and especially in motion." According to PC World, "DLAA offers very good anti-aliasing without losing visual information — alternatives like TAA tend to struggle during motion-filled scenes, where DLAA doesn’t. Furthermore, DLAA’s loss of performance is lower than with conventional anti-aliasing methods."
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Discrete diffusion model

In machine learning, discrete diffusion models are a class of diffusion models, which themselves are a class of latent variable generative models. Each discrete diffusion model consists of two major components: the forward jump diffusion process, and the reverse jump diffusion process. The goal of diffusion modeling is, given a given dataset and a forward process, to learn a model for the reverse process, such that the reverse process can generate new elements that are distributed similarly as the original dataset. A trained discrete diffusion model can be sampled in many ways, which trades off computational efficiency and sample quality. In general, higher quality data can be obtained, but at the price of higher computational cost. In standard diffusion modeling, the diffusion process takes place over a state space that is continuous space of R n {\displaystyle \mathbb {R} ^{n}} , but over a discrete set S {\displaystyle S} . A discrete set is simply a set where one cannot speak of "infinitesimally close" points. Points can be more or less separated from each other, but the separation is always a finite number. This in particular means the standard framework of continuous diffusion does not apply, since it uses gaussian noise, which is continuous. Nevertheless, an analogous theory can be produced. Discrete diffusion is usually used for language modeling. In practice, the state space S {\displaystyle S} is not only discrete, but finite, so this is what we will assume from now on. == Continuous time Markov process == In the case of continuous state space, during the forward discrete diffusion process, at each step t → t + d t {\displaystyle t\to t+dt} , we mix in an infinitesimal amount of gaussian noise d x t = − 1 2 β ( t ) x t d t + β ( t ) d W t {\displaystyle dx_{t}=-{\frac {1}{2}}\beta (t)x_{t}dt+{\sqrt {\beta (t)}}dW_{t}} . This changes the probability density function, by first a convolution with the density of a gaussian, followed by a scaling. In the case of discrete state space, the gaussian noise must be replaced by a noise that takes values over a finite set. For example, if the noise is the uniform distribution over S {\displaystyle S} , then the probability distribution at time t + d t {\displaystyle t+dt} satisfies q t + d t ( x ) = ( 1 − d t ) q t ( x ) + d t ( 1 | S | ∑ y ∈ S q t ( y ) ) {\displaystyle q_{t+dt}(x)=(1-dt)q_{t}(x)+dt\left({\frac {1}{|S|}}\sum _{y\in S}q_{t}(y)\right)} More succinctly, ∂ t q t ( x ) = − ( 1 − 1 | S | ) q t ( x ) + ∑ y ∈ S , y ≠ x 1 | S | q t ( y ) {\displaystyle \partial _{t}q_{t}(x)=-\left(1-{\frac {1}{|S|}}\right)q_{t}(x)+\sum _{y\in S,y\neq x}{\frac {1}{|S|}}q_{t}(y)} In general, we do not need to convolve with a uniformly distributed noise, but with an arbitrary noise process. That is, we use an arbitrary matrix Q t {\displaystyle Q_{t}} such that ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} where Q t {\displaystyle Q_{t}} is called the rate matrix. Any matrix may be used as a rate matrix if it has non-negative off-diagonals, and each column sums to 0: Q t ( y , x ) ≥ 0 ∀ y ≠ x , ∑ y ∈ S Q t ( y , x ) = 0 ∀ x {\displaystyle Q_{t}(y,x)\geq 0\quad \forall y\neq x,\quad \sum _{y\in S}Q_{t}(y,x)=0\quad \forall x} A continuous time Markov chain (CTMC) is defined by a continuous function Q {\displaystyle Q} that maps any time t ∈ [ 0 , T ) {\displaystyle t\in [0,T)} to a rate matrix Q t {\displaystyle Q_{t}} . Given the function Q {\displaystyle Q} , time-evolution under the CTMC is done as follows: Given state x t {\displaystyle x_{t}} at time t {\displaystyle t} , and given an infinitesimal d t {\displaystyle dt} , the state at t + d t {\displaystyle t+dt} is x t + d t {\displaystyle x_{t+dt}} , such that Pr ( x t + d t | x t ) = { 1 + Q t ( x t + d t , x t ) d t if x t + d t = x t Q t ( x t + d t , x t ) d t else {\displaystyle \Pr(x_{t+dt}|x_{t})={\begin{cases}1+Q_{t}(x_{t+dt},x_{t})dt&{\text{if }}x_{t+dt}=x_{t}\\Q_{t}(x_{t+dt},x_{t})dt&{\text{else}}\end{cases}}} This implies that the probability distribution function evolves according to ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} which is what we previously specified. === Backward process === Similarly to the case of continuous diffusion, in discrete diffusion, there exists a backward diffusion process Q ¯ t {\displaystyle {\bar {Q}}_{t}} : s ( x , t ) y := q t ( y ) q t ( x ) , Q ¯ t ( y , x ) := { s ( x , t ) y Q t ( x , y ) if y ≠ x − ∑ y : y ≠ x Q ¯ t ( y , x ) if y = x {\displaystyle s(x,t)_{y}:={\frac {q_{t}(y)}{q_{t}(x)}},\quad {\bar {Q}}_{t}(y,x):={\begin{cases}s(x,t)_{y}Q_{t}(x,y)&{\text{if }}y\neq x\\-\sum _{y:y\neq x}{\bar {Q}}_{t}(y,x)&{\text{if }}y=x\end{cases}}} where s ( x , t ) y {\displaystyle s(x,t)_{y}} should be interpreted as the discrete score or concrete score, since, abusing notation a bit, the score function is ∇ ln ⁡ ρ t ( x ) = 1 d x ( ρ t ( x + d x ) ρ t ( x ) − 1 ) {\displaystyle \nabla \ln \rho _{t}(x)={\frac {1}{dx}}\left({\frac {\rho _{t}(x+dx)}{\rho _{t}(x)}}-1\right)} . If we picture the distribution q t {\displaystyle q_{t}} as a bunch of point-masses, one per state x ∈ S {\displaystyle x\in S} , then the forward diffusion from time t {\displaystyle t} to t + d t {\displaystyle t+dt} is performed by removing Q t ( x , y ) q t ( y ) d t {\displaystyle Q_{t}(x,y)q_{t}(y)dt} from the mass at y {\displaystyle y} and moving it to the mass at x {\displaystyle x} , for each pair x ≠ y {\displaystyle x\neq y} . Thus, the process is reversed in detail by the CTMC defined by Q ¯ {\displaystyle {\bar {Q}}} , since Q ¯ t ( y , x ) q t ( x ) = Q t ( x , y ) q t ( y ) {\displaystyle {\bar {Q}}_{t}(y,x)q_{t}(x)=Q_{t}(x,y)q_{t}(y)} . Given Q ¯ t {\displaystyle {\bar {Q}}_{t}} , if we have a way to sample from q t {\displaystyle q_{t}} , then we can sample from q t − d t {\displaystyle q_{t-dt}} by first sampling x t ∼ q t {\displaystyle x_{t}\sim q_{t}} , then sampling x t − d t {\displaystyle x_{t-dt}} according to Pr ( x t − d t | x t ) = { 1 + Q ¯ t ( x t − d t , x t ) d t if x t − d t = x t Q ¯ t ( x t − d t , x t ) d t else {\displaystyle \Pr(x_{t-dt}|x_{t})={\begin{cases}1+{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{if }}x_{t-dt}=x_{t}\\{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{else}}\end{cases}}} === Overall plan of score-matching discrete diffusion modeling === Similar to score-matching continuous diffusion, score-matching discrete diffusion is a method to sample an initial distribution. If we have a certain function s θ {\displaystyle s_{\theta }} that approximates the true score function s θ ( x , t ) y ≈ s ( x , t ) y {\displaystyle s_{\theta }(x,t)_{y}\approx s(x,t)_{y}} , then it allows a corresponding Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} to be defined in the same way. If we also have a base distribution q base {\displaystyle q_{\text{base}}} such that it is easy to sample from, and approximately equal to the true terminal distribution q base ≈ q T {\displaystyle q_{\text{base}}\approx q_{T}} , then we can perform the backward CTMC with Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} and q T θ := q terminal {\displaystyle q_{T}^{\theta }:=q_{\text{terminal}}} . When both approximations are good, the backward CTMC would give q 0 θ ≈ q 0 {\displaystyle q_{0}^{\theta }\approx q_{0}} . This is the idea of score-matching discrete diffusion modeling. If q data {\displaystyle q_{\text{data}}} is sharp, in the sense that for some x , x ′ {\displaystyle x,x'} , we have q data ( x ) ≫ q data ( x ′ ) {\displaystyle q_{\text{data}}(x)\gg q_{\text{data}}(x')} , then the score function would diverge as 1 / t {\displaystyle 1/t} at the t → 0 {\displaystyle t\to 0} limit. To avoid this in practice, it is common to use early stopping, which is to stop the backward process at some time δ > 0 {\displaystyle \delta >0} , and sample from q δ θ {\displaystyle q_{\delta }^{\theta }} instead of q 0 θ {\displaystyle q_{0}^{\theta }} . === Tractable forward processes === The theory of CTMC works for any continuous choice of rate matrices Q {\displaystyle Q} . However, most choices are computationally expensive and cannot be used in practice. In the case of continuous diffusion, the gaussian noise is used for the simple reason that the sum of any number of gaussians is still a gaussian. This allows one to sample any x t ∼ ρ t {\displaystyle x_{t}\sim \rho _{t}} by sampling a single x 0 ∼ ρ 0 {\displaystyle x_{0}\sim \rho _{0}} , followed by a single gaussian noise z ∼ N ( 0 , I ) {\displaystyle z\sim {\mathcal {N}}(0,I)} , and let x t = α ¯ t x 0 + σ t z {\displaystyle x_{t}={\sqrt {{\bar {\alpha }}_{t}}}x_{0}+\sigma _{t}z} , without needing any x s {\displaystyle x_{s}} for any 0 < s < t {\displaystyle 0 Read more →
Oja's rule

Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}
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State–action–reward–state–action

State–action–reward–state–action (SARSA) is an algorithm for learning a Markov decision process policy, used in the reinforcement learning area of machine learning. It was proposed by Rummery and Niranjan in a technical note with the name "Modified Connectionist Q-Learning" (MCQ-L). The alternative name SARSA, proposed by Rich Sutton, was only mentioned as a footnote. This name reflects the fact that the main function for updating the Q-value depends on the current state of the agent "S1", the action the agent chooses "A1", the reward "R2" the agent gets for choosing this action, the state "S2" that the agent enters after taking that action, and finally the next action "A2" the agent chooses in its new state. The acronym for the quintuple (St, At, Rt+1, St+1, At+1) is SARSA. Some authors use a slightly different convention and write the quintuple (St, At, Rt, St+1, At+1), depending on which time step the reward is formally assigned. The rest of the article uses the former convention. == Algorithm == Q new ( S t , A t ) ← ( 1 − α ) Q ( S t , A t ) + α [ R t + 1 + γ Q ( S t + 1 , A t + 1 ) ] {\displaystyle Q^{\textrm {new}}(S_{t},A_{t})\leftarrow (1-\alpha )Q(S_{t},A_{t})+\alpha \,[R_{t+1}+\gamma \,Q(S_{t+1},A_{t+1})]} A SARSA agent interacts with the environment and updates the policy based on actions taken, hence this is known as an on-policy learning algorithm. The Q value for a state-action is updated by an error, adjusted by the learning rate α. Q values represent the possible reward received in the next time step for taking action a in state s, plus the discounted future reward received from the next state-action observation. Watkin's Q-learning updates an estimate of the optimal state-action value function Q ∗ {\displaystyle Q^{}} based on the maximum reward of available actions. While SARSA learns the Q values associated with taking the policy it follows itself, Watkin's Q-learning learns the Q values associated with taking the optimal policy while following an exploration/exploitation policy. Some optimizations of Watkin's Q-learning may be applied to SARSA. == Hyperparameters == === Learning rate (alpha) === The learning rate determines to what extent newly acquired information overrides old information. A factor of 0 will make the agent not learn anything, while a factor of 1 would make the agent consider only the most recent information. === Discount factor (gamma) === The discount factor determines the importance of future rewards. A discount factor of 0 makes the agent "opportunistic", or "myopic", e.g., by only considering current rewards, while a factor approaching 1 will make it strive for a long-term high reward. If the discount factor meets or exceeds 1, the Q {\displaystyle Q} values may diverge. === Initial conditions (Q(S0, A0)) === Since SARSA is an iterative algorithm, it implicitly assumes an initial condition before the first update occurs. A high (infinite) initial value, also known as "optimistic initial conditions", can encourage exploration: no matter what action takes place, the update rule causes it to have higher values than the other alternative, thus increasing their choice probability. In 2013 it was suggested that the first reward r {\displaystyle r} could be used to reset the initial conditions. According to this idea, the first time an action is taken the reward is used to set the value of Q {\displaystyle Q} . This allows immediate learning in case of fixed deterministic rewards. This resetting-of-initial-conditions (RIC) approach seems to be consistent with human behavior in repeated binary choice experiments.
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Moj

Moj is an Indian short-form video-sharing social networking service owned by Mohalla Tech Pvt Ltd, the parent company of ShareChat. Launched on 29 June 2020, shortly after the Government of India banned TikTok and several other Chinese apps, Moj quickly gained popularity as one of the leading domestic alternatives for short-form video content in India. == History == Moj was introduced by Mohalla Tech, the Bengaluru-based parent company of ShareChat, within days of the TikTok ban in India in June 2020. The app targeted the growing demand for short-form video platforms in the country. By early 2021, Moj had amassed over 100 million downloads on the Google Play Store. In February 2021, Mohalla Tech raised significant funding from investors like Tiger Global, Snapchat, and others, which supported both Moj and ShareChat’s growth. In 2022, Moj partnered with several music labels to expand its licensed music library, competing directly with global platforms such as Instagram Reels and YouTube Shorts. == Features == Short Videos: Users can create and watch videos up to 15–60 seconds. Filters & Effects: The platform provides AR filters, editing tools, stickers, and music integration. Regional Language Support: Moj supports more than 15 Indian languages including Hindi, Bengali, Tamil, Telugu, Kannada, and Marathi. Music Integration: Users can add music tracks to their videos from licensed Indian and international music libraries. Creator Program: Moj launched initiatives to support influencers and creators, offering training, monetization, and promotional opportunities. == Popularity == By mid-2021, Moj reported over 160 million monthly active users. According to reports, Moj consistently ranked among the top social media apps in India in terms of downloads. The app gained traction in Tier-2 and Tier-3 cities due to its multilingual support and focus on local content. == Competitors == Moj competes with several other short video platforms in India, including: Instagram Reels (Meta) YouTube Shorts (Google) Josh (Dailyhunt/VerSe Innovation) Roposo (InMobi) MX TakaTak (later merged with Moj in 2022) RedPost (an emerging Indian social networking platform) == Merger with MX TakaTak == In February 2022, Mohalla Tech announced that Moj would merge with MX TakaTak, another leading short video app owned by Times Internet. The merger created one of the largest short-video ecosystems in India, with a combined user base of over 300 million monthly active users.
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Quickprop

Quickprop is an iterative method for determining the minimum of the loss function of an artificial neural network, following an algorithm inspired by the Newton's method. Sometimes, the algorithm is classified to the group of the second order learning methods. It follows a quadratic approximation of the previous gradient step and the current gradient, which is expected to be close to the minimum of the loss function, under the assumption that the loss function is locally approximately square, trying to describe it by means of an upwardly open parabola. The minimum is sought in the vertex of the parabola. The procedure requires only local information of the artificial neuron to which it is applied. The k {\displaystyle k} -th approximation step is given by: Δ ( k ) w i j = Δ ( k − 1 ) w i j ( ∇ i j E ( k ) ∇ i j E ( k − 1 ) − ∇ i j E ( k ) ) {\displaystyle \Delta ^{(k)}\,w_{ij}=\Delta ^{(k-1)}\,w_{ij}\left({\frac {\nabla _{ij}\,E^{(k)}}{\nabla _{ij}\,E^{(k-1)}-\nabla _{ij}\,E^{(k)}}}\right)} Where w i j {\displaystyle w_{ij}} is the weight of input i {\displaystyle i} of neuron j {\displaystyle j} , and E {\displaystyle E} is the loss function. The Quickprop algorithm is an implementation of the error backpropagation algorithm, but the network can behave chaotically during the learning phase due to large step sizes.
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Types of artificial neural networks

Types of neural networks (NN) include a family of techniques. The simplest types have static components, including number of units, number of layers, unit weights and topology. Dynamic NNs evolve via learning. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. The main types are: Transformers: these use attention to analyze every token in the input stream against every other token in the stream. That technique has enabled neural networks to reach the general public via chatbots, code generators and many other forms. Convolutional neural networks (CNN): a FNN that uses kernels and regularization to evade problems in prior generations of NNs. They are typically used to analyze visual and other two-dimensional data. Generative adversarial networks set networks (of varying structure) against each other, each trying to push the other(s) to produce better results such as winning a game or to deceive the opponent about the authenticity of an input. == Feedforward == In feedforward neural networks the information moves from the input to output directly in every layer. There can be hidden layers with or without cycles/loops to sequence inputs. Feedforward networks can be constructed with various types of units, such as binary McCulloch–Pitts neurons, the simplest of which is the perceptron. Continuous neurons, frequently with sigmoidal activation, are used in the context of backpropagation. == Group method of data handling == The Group Method of Data Handling (GMDH) features fully automatic structural and parametric model optimization. The node activation functions are Kolmogorov–Gabor polynomials that permit additions and multiplications. It uses a deep multilayer perceptron with eight layers. It is a supervised learning network that grows layer by layer, where each layer is trained by regression analysis. Useless items are detected using a validation set, and pruned through regularization. The size and depth of the resulting network depends on the task. == Autoencoder == An autoencoder, autoassociator or Diabolo network is similar to the multilayer perceptron (MLP) – with an input layer, an output layer and one or more hidden layers connecting them. However, the output layer has the same number of units as the input layer. Its purpose is to reconstruct its own inputs (instead of emitting a target value). Therefore, autoencoders are unsupervised learning models. An autoencoder is used for unsupervised learning of efficient codings, typically for the purpose of dimensionality reduction and for learning generative models of data. == Probabilistic == A probabilistic neural network (PNN) is a four-layer feedforward neural network. The layers are Input, hidden pattern, hidden summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each class, the class probability of a new input is estimated and Bayes’ rule is employed to allocate it to the class with the highest posterior probability. It was derived from the Bayesian network and a statistical algorithm called Kernel Fisher discriminant analysis. It is used for classification and pattern recognition. == Time delay == A time delay neural network (TDNN) is a feedforward architecture for sequential data that recognizes features independent of sequence position. In order to achieve time-shift invariance, delays are added to the input so that multiple data points (points in time) are analyzed together. It usually forms part of a larger pattern recognition system. It has been implemented using a perceptron network whose connection weights were trained with back propagation (supervised learning). == Convolutional == A convolutional neural network (CNN, or ConvNet or shift invariant or space invariant) is a class of deep network, composed of one or more convolutional layers with fully connected layers (matching those in typical ANNs) on top. It uses tied weights and pooling layers. In particular, max-pooling. It is often structured via Fukushima's convolutional architecture. They are variations of multilayer perceptrons that use minimal preprocessing. This architecture allows CNNs to take advantage of the 2D structure of input data. Its unit connectivity pattern is inspired by the organization of the visual cortex. Units respond to stimuli in a restricted region of space known as the receptive field. Receptive fields partially overlap, over-covering the entire visual field. Unit response can be approximated mathematically by a convolution operation. CNNs are suitable for processing visual and other two-dimensional data. They have shown superior results in both image and speech applications. They can be trained with standard backpropagation. CNNs are easier to train than other regular, deep, feed-forward neural networks and have many fewer parameters to estimate. Capsule Neural Networks (CapsNet) add structures called capsules to a CNN and reuse output from several capsules to form more stable (with respect to various perturbations) representations. Examples of applications in computer vision include DeepDream and robot navigation. They have wide applications in image and video recognition, recommender systems and natural language processing. == Deep stacking network == A deep stacking network (DSN) (deep convex network) is based on a hierarchy of blocks of simplified neural network modules. It was introduced in 2011 by Deng and Yu. It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization. Each DSN block is a simple module that is easy to train by itself in a supervised fashion without backpropagation for the entire blocks. Each block consists of a simplified multi-layer perceptron (MLP) with a single hidden layer. The hidden layer h has logistic sigmoidal units, and the output layer has linear units. Connections between these layers are represented by weight matrix U; input-to-hidden-layer connections have weight matrix W. Target vectors t form the columns of matrix T, and the input data vectors x form the columns of matrix X. The matrix of hidden units is H = σ ( W T X ) {\displaystyle {\boldsymbol {H}}=\sigma ({\boldsymbol {W}}^{T}{\boldsymbol {X}})} . Modules are trained in order, so lower-layer weights W are known at each stage. The function performs the element-wise logistic sigmoid operation. Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block. Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks. Then learning the upper-layer weight matrix U given other weights in the network can be formulated as a convex optimization problem: min U T f = ‖ U T H − T ‖ F 2 , {\displaystyle \min _{U^{T}}f=\|{\boldsymbol {U}}^{T}{\boldsymbol {H}}-{\boldsymbol {T}}\|_{F}^{2},} which has a closed-form solution. Unlike other deep architectures, such as DBNs, the goal is not to discover the transformed feature representation. The structure of the hierarchy of this kind of architecture makes parallel learning straightforward, as a batch-mode optimization problem. In purely discriminative tasks, DSNs outperform conventional DBNs. === Tensor deep stacking networks === This architecture is a DSN extension. It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer. TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor. While parallelization and scalability are not considered seriously in conventional DNNs, all learning for DSNs and TDSNs is done in batch mode, to allow parallelization. Parallelization allows scaling the design to larger (deeper) architectures and data sets. The basic architecture is suitable for diverse tasks such as classification and regression. == Physics-informed == Such a neural network is designed for the numerical solution of mathematical equations, such as differential, integral, delay, fractional and others. As input parameters, PINN accepts variables (spatial, temporal, and others), transmits them through the network block. At the output, it produces an approximate solution and substitutes it into the mathematical model, considering the initial and boundary conditions. If the solution does not satisfy the required accuracy, one uses the backpropagation and rectify the solution. Besides PINN, other architectures have been developed to produce surrogate models for scientific comput
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BrownBoost

BrownBoost is a boosting algorithm that may be robust to noisy datasets. BrownBoost is an adaptive version of the boost by majority algorithm. As is the case for all boosting algorithms, BrownBoost is used in conjunction with other machine learning methods. BrownBoost was introduced by Yoav Freund in 2001. == Motivation == AdaBoost performs well on a variety of datasets; however, it can be shown that AdaBoost does not perform well on noisy data sets. This is a result of AdaBoost's focus on examples that are repeatedly misclassified. In contrast, BrownBoost effectively "gives up" on examples that are repeatedly misclassified. The core assumption of BrownBoost is that noisy examples will be repeatedly mislabeled by the weak hypotheses and non-noisy examples will be correctly labeled frequently enough to not be "given up on." Thus only noisy examples will be "given up on," whereas non-noisy examples will contribute to the final classifier. In turn, if the final classifier is learned from the non-noisy examples, the generalization error of the final classifier may be much better than if learned from noisy and non-noisy examples. The user of the algorithm can set the amount of error to be tolerated in the training set. Thus, if the training set is noisy (say 10% of all examples are assumed to be mislabeled), the booster can be told to accept a 10% error rate. Since the noisy examples may be ignored, only the true examples will contribute to the learning process. == Algorithm description == BrownBoost uses a non-convex potential loss function, thus it does not fit into the AdaBoost framework. The non-convex optimization provides a method to avoid overfitting noisy data sets. However, in contrast to boosting algorithms that analytically minimize a convex loss function (e.g. AdaBoost and LogitBoost), BrownBoost solves a system of two equations and two unknowns using standard numerical methods. The only parameter of BrownBoost ( c {\displaystyle c} in the algorithm) is the "time" the algorithm runs. The theory of BrownBoost states that each hypothesis takes a variable amount of time ( t {\displaystyle t} in the algorithm) which is directly related to the weight given to the hypothesis α {\displaystyle \alpha } . The time parameter in BrownBoost is analogous to the number of iterations T {\displaystyle T} in AdaBoost. A larger value of c {\displaystyle c} means that BrownBoost will treat the data as if it were less noisy and therefore will give up on fewer examples. Conversely, a smaller value of c {\displaystyle c} means that BrownBoost will treat the data as more noisy and give up on more examples. During each iteration of the algorithm, a hypothesis is selected with some advantage over random guessing. The weight of this hypothesis α {\displaystyle \alpha } and the "amount of time passed" t {\displaystyle t} during the iteration are simultaneously solved in a system of two non-linear equations ( 1. uncorrelated hypothesis w.r.t example weights and 2. hold the potential constant) with two unknowns (weight of hypothesis α {\displaystyle \alpha } and time passed t {\displaystyle t} ). This can be solved by bisection (as implemented in the JBoost software package) or Newton's method (as described in the original paper by Freund). Once these equations are solved, the margins of each example ( r i ( x j ) {\displaystyle r_{i}(x_{j})} in the algorithm) and the amount of time remaining s {\displaystyle s} are updated appropriately. This process is repeated until there is no time remaining. The initial potential is defined to be 1 m ∑ j = 1 m 1 − erf ( c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}({\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Since a constraint of each iteration is that the potential be held constant, the final potential is 1 m ∑ j = 1 m 1 − erf ( r i ( x j ) / c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}(r_{i}(x_{j})/{\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Thus the final error is likely to be near 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} . However, the final potential function is not the 0–1 loss error function. For the final error to be exactly 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} , the variance of the loss function must decrease linearly w.r.t. time to form the 0–1 loss function at the end of boosting iterations. This is not yet discussed in the literature and is not in the definition of the algorithm below. The final classifier is a linear combination of weak hypotheses and is evaluated in the same manner as most other boosting algorithms. == BrownBoost learning algorithm definition == Input: m {\displaystyle m} training examples ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x j ∈ X , y j ∈ Y = { − 1 , + 1 } {\displaystyle x_{j}\in X,\,y_{j}\in Y=\{-1,+1\}} The parameter c {\displaystyle c} Initialise: s = c {\displaystyle s=c} . (The value of s {\displaystyle s} is the amount of time remaining in the game) r i ( x j ) = 0 {\displaystyle r_{i}(x_{j})=0} ∀ j {\displaystyle \forall j} . The value of r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin at iteration i {\displaystyle i} for example x j {\displaystyle x_{j}} . While s > 0 {\displaystyle s>0} : Set the weights of each example: W i ( x j ) = e − ( r i ( x j ) + s ) 2 c {\displaystyle W_{i}(x_{j})=e^{-{\frac {(r_{i}(x_{j})+s)^{2}}{c}}}} , where r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin of example x j {\displaystyle x_{j}} Find a classifier h i : X → { − 1 , + 1 } {\displaystyle h_{i}:X\to \{-1,+1\}} such that ∑ j W i ( x j ) h i ( x j ) y j > 0 {\displaystyle \sum _{j}W_{i}(x_{j})h_{i}(x_{j})y_{j}>0} Find values α , t {\displaystyle \alpha ,t} that satisfy the equation: ∑ j h i ( x j ) y j e − ( r i ( x j ) + α h i ( x j ) y j + s − t ) 2 c = 0 {\displaystyle \sum _{j}h_{i}(x_{j})y_{j}e^{-{\frac {(r_{i}(x_{j})+\alpha h_{i}(x_{j})y_{j}+s-t)^{2}}{c}}}=0} . (Note this is similar to the condition E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} set forth by Schapire and Singer. In this setting, we are numerically finding the W i + 1 = exp ⁡ ( ⋯ ⋯ ) {\displaystyle W_{i+1}=\exp \left({\frac {\cdots }{\cdots }}\right)} such that E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} .) This update is subject to the constraint ∑ ( Φ ( r i ( x j ) + α h ( x j ) y j + s − t ) − Φ ( r i ( x j ) + s ) ) = 0 {\displaystyle \sum \left(\Phi \left(r_{i}(x_{j})+\alpha h(x_{j})y_{j}+s-t\right)-\Phi \left(r_{i}(x_{j})+s\right)\right)=0} , where Φ ( z ) = 1 − erf ( z / c ) {\displaystyle \Phi (z)=1-{\mbox{erf}}(z/{\sqrt {c}})} is the potential loss for a point with margin r i ( x j ) {\displaystyle r_{i}(x_{j})} Update the margins for each example: r i + 1 ( x j ) = r i ( x j ) + α h ( x j ) y j {\displaystyle r_{i+1}(x_{j})=r_{i}(x_{j})+\alpha h(x_{j})y_{j}} Update the time remaining: s = s − t {\displaystyle s=s-t} Output: H ( x ) = sign ( ∑ i α i h i ( x ) ) {\displaystyle H(x)={\textrm {sign}}\left(\sum _{i}\alpha _{i}h_{i}(x)\right)} == Empirical results == In preliminary experimental results with noisy datasets, BrownBoost outperformed AdaBoost's generalization error; however, LogitBoost performed as well as BrownBoost. An implementation of BrownBoost can be found in the open source software JBoost.
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Piranesi (software)

Piranesi is an interactive paint system that enables the user to create artistic images from 3D scenes created using conventional modeling applications. == Image format == Piranesi uses the proprietary EPix file format. For every pixel, additional information is stored, such as distance from the viewer and material settings. EPix files can be rendered from 3D scenes using a fixed viewpoint by Piranesi's companion software, Vedute.
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Prototype methods

Prototype methods are machine learning methods that use data prototypes. A data prototype is a data value that reflects other values in its class, e.g., the centroid in a K-means clustering problem. == Methods == The following are some prototype methods K-means clustering Learning vector quantization (LVQ) Gaussian mixtures == Related Methods == While K-nearest neighbor's does not use prototypes, it is similar to prototype methods like K-means clustering.
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Elastic map

Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. == Energy of elastic map == Let S {\displaystyle {\mathcal {S}}} be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes w j {\displaystyle {\bf {w}}_{j}} in the same space. Each datapoint s ∈ S {\displaystyle s\in {\mathcal {S}}} has a host node, namely the closest node w j {\displaystyle {\bf {w}}_{j}} (if there are several closest nodes then one takes the node with the smallest number). The data set S {\displaystyle {\mathcal {S}}} is divided into classes K j = { s | w j is a host of s } {\displaystyle K_{j}=\{s\ |\ {\bf {w}}_{j}{\mbox{ is a host of }}s\}} . The approximation energy D is the distortion D = 1 2 ∑ j = 1 k ∑ s ∈ K j ‖ s − w j ‖ 2 {\displaystyle D={\frac {1}{2}}\sum _{j=1}^{k}\sum _{s\in K_{j}}\|s-{\bf {w}}_{j}\|^{2}} , which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points { s i } {\displaystyle \{s_{i}\}} . On the set of nodes an additional structure is defined. Some pairs of nodes, ( w i , w j ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j})} , are connected by elastic edges. Call this set of pairs E {\displaystyle E} . Some triplets of nodes, ( w i , w j , w k ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})} , form bending ribs. Call this set of triplets G {\displaystyle G} . The stretching energy is U E = 1 2 λ ∑ ( w i , w j ) ∈ E ‖ w i − w j ‖ 2 {\displaystyle U_{E}={\frac {1}{2}}\lambda \sum _{({\bf {w}}_{i},{\bf {w}}_{j})\in E}\|{\bf {w}}_{i}-{\bf {w}}_{j}\|^{2}} , The bending energy is U G = 1 2 μ ∑ ( w i , w j , w k ) ∈ G ‖ w i − 2 w j + w k ‖ 2 {\displaystyle U_{G}={\frac {1}{2}}\mu \sum _{({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})\in G}\|{\bf {w}}_{i}-2{\bf {w}}_{j}+{\bf {w}}_{k}\|^{2}} , where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U = D + U E + U G . {\displaystyle U=D+U_{E}+U_{G}.} The position of the nodes { w j } {\displaystyle \{{\bf {w}}_{j}\}} is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U {\displaystyle U} . == Expectation-maximization algorithm == For a given splitting of dataset S {\displaystyle {\mathcal {S}}} in classes K j {\displaystyle K_{j}} , minimization of the quadratic functional U {\displaystyle U} is a linear problem with the sparse matrix of coefficients. Therefore, similar to principal component analysis or k-means, a splitting method is used: For given { w j } {\displaystyle \{{\bf {w}}_{j}\}} find { K j } {\displaystyle \{K_{j}\}} ; For given { K j } {\displaystyle \{K_{j}\}} minimize U {\displaystyle U} and find { w j } {\displaystyle \{{\bf {w}}_{j}\}} ; If no change, terminate. This expectation-maximization algorithm guarantees a local minimum of U {\displaystyle U} . For improving the approximation various additional methods are proposed. For example, the softening strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules λ {\displaystyle \lambda } and μ {\displaystyle \mu } coefficients) and finishes with soft grids (small λ {\displaystyle \lambda } and μ {\displaystyle \mu } ). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is growing net: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes. == Applications == Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and for visualisation of data of various nature. The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes. Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios. The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure. Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN. The textbook provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making.
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Diffusion model

In machine learning, diffusion models, also known as diffusion-based generative models or score-based generative models, are a class of latent variable generative models. A diffusion model consists of two major components: the forward diffusion process, and the reverse sampling process. The goal of diffusion models is to learn a diffusion process for a given dataset, such that the process can generate new elements that are distributed similarly as the original dataset. A diffusion model models data as generated by a diffusion process, whereby a new datum performs a random walk with drift through the space of all possible data. A trained diffusion model can be sampled in many ways, with different efficiency and quality. There are various equivalent formalisms, including Markov chains, denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. They are typically trained using variational inference. The model responsible for denoising is typically called its "backbone". The backbone may be of any kind, but they are typically U-nets or transformers. As of 2024, diffusion models are mainly used for computer vision tasks, including image denoising, inpainting, super-resolution, image generation, and video generation. These typically involve training a neural network to sequentially denoise images blurred with Gaussian noise. The model is trained to reverse the process of adding noise to an image. After training to convergence, it can be used for image generation by starting with an image composed of random noise, and applying the network iteratively to denoise the image. Diffusion-based image generators have seen widespread commercial interest, such as Stable Diffusion and DALL-E. These models typically combine diffusion models with other models, such as text-encoders and cross-attention modules to allow text-conditioned generation. Other than computer vision, diffusion models have also found applications in natural language processing such as text generation and summarization, sound generation, and reinforcement learning. == Denoising diffusion model == === Non-equilibrium thermodynamics === Diffusion models were introduced in 2015 as a method to train a model that can sample from a highly complex probability distribution. They used techniques from non-equilibrium thermodynamics, especially diffusion. Consider, for example, how one might model the distribution of all naturally occurring photos. Each image is a point in the space of all images, and the distribution of naturally occurring photos is a "cloud" in space, which, by repeatedly adding noise to the images, diffuses out to the rest of the image space, until the cloud becomes all but indistinguishable from a Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . A model that can approximately undo the diffusion can then be used to sample from the original distribution. This is studied in "non-equilibrium" thermodynamics, as the starting distribution is not in equilibrium, unlike the final distribution. The equilibrium distribution is the Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with pdf ρ ( x ) ∝ e − 1 2 ‖ x ‖ 2 {\displaystyle \rho (x)\propto e^{-{\frac {1}{2}}\|x\|^{2}}} . This is just the Maxwell–Boltzmann distribution of particles in a potential well V ( x ) = 1 2 ‖ x ‖ 2 {\displaystyle V(x)={\frac {1}{2}}\|x\|^{2}} at temperature 1. The initial distribution, being very much out of equilibrium, would diffuse towards the equilibrium distribution, making biased random steps that are a sum of pure randomness (like a Brownian walker) and gradient descent down the potential well. The randomness is necessary: if the particles were to undergo only gradient descent, then they will all fall to the origin, collapsing the distribution. === Denoising Diffusion Probabilistic Model (DDPM) === The 2020 paper proposed the Denoising Diffusion Probabilistic Model (DDPM), which improves upon the previous method by variational inference. ==== Forward diffusion ==== To present the model, some notation is required. β 1 , . . . , β T ∈ ( 0 , 1 ) {\displaystyle \beta _{1},...,\beta _{T}\in (0,1)} are fixed constants. α t := 1 − β t {\displaystyle \alpha _{t}:=1-\beta _{t}} α ¯ t := α 1 ⋯ α t {\displaystyle {\bar {\alpha }}_{t}:=\alpha _{1}\cdots \alpha _{t}} σ t := 1 − α ¯ t {\displaystyle \sigma _{t}:={\sqrt {1-{\bar {\alpha }}_{t}}}} σ ~ t := σ t − 1 σ t β t {\displaystyle {\tilde {\sigma }}_{t}:={\frac {\sigma _{t-1}}{\sigma _{t}}}{\sqrt {\beta _{t}}}} μ ~ t ( x t , x 0 ) := α t ( 1 − α ¯ t − 1 ) x t + α ¯ t − 1 ( 1 − α t ) x 0 σ t 2 {\displaystyle {\tilde {\mu }}_{t}(x_{t},x_{0}):={\frac {{\sqrt {\alpha _{t}}}(1-{\bar {\alpha }}_{t-1})x_{t}+{\sqrt {{\bar {\alpha }}_{t-1}}}(1-\alpha _{t})x_{0}}{\sigma _{t}^{2}}}} N ( μ , Σ ) {\displaystyle {\mathcal {N}}(\mu ,\Sigma )} is the normal distribution with mean μ {\displaystyle \mu } and variance Σ {\displaystyle \Sigma } , and N ( x | μ , Σ ) {\displaystyle {\mathcal {N}}(x|\mu ,\Sigma )} is the probability density at x {\displaystyle x} . A vertical bar denotes conditioning. A forward diffusion process starts at some starting point x 0 ∼ q {\displaystyle x_{0}\sim q} , where q {\displaystyle q} is the probability distribution to be learned, then repeatedly adds noise to it by x t = 1 − β t x t − 1 + β t z t {\displaystyle x_{t}={\sqrt {1-\beta _{t}}}x_{t-1}+{\sqrt {\beta _{t}}}z_{t}} where z 1 , . . . , z T {\displaystyle z_{1},...,z_{T}} are IID (Independent and identically distributed random variables) samples from N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The coefficients 1 − β t {\displaystyle {\sqrt {1-\beta _{t}}}} and β t {\displaystyle {\sqrt {\beta _{t}}}} ensure that Var ( X t ) = I {\displaystyle {\mbox{Var}}(X_{t})=I} assuming that Var ( X 0 ) = I {\displaystyle {\mbox{Var}}(X_{0})=I} . The values of β t {\displaystyle \beta _{t}} are chosen such that for any starting distribution of x 0 {\displaystyle x_{0}} , if it has finite second moment, then lim t → ∞ x t | x 0 {\displaystyle \lim _{t\to \infty }x_{t}|x_{0}} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The entire diffusion process then satisfies q ( x 0 : T ) = q ( x 0 ) q ( x 1 | x 0 ) ⋯ q ( x T | x T − 1 ) = q ( x 0 ) N ( x 1 | α 1 x 0 , β 1 I ) ⋯ N ( x T | α T x T − 1 , β T I ) {\displaystyle q(x_{0:T})=q(x_{0})q(x_{1}|x_{0})\cdots q(x_{T}|x_{T-1})=q(x_{0}){\mathcal {N}}(x_{1}|{\sqrt {\alpha _{1}}}x_{0},\beta _{1}I)\cdots {\mathcal {N}}(x_{T}|{\sqrt {\alpha _{T}}}x_{T-1},\beta _{T}I)} or ln ⁡ q ( x 0 : T ) = ln ⁡ q ( x 0 ) − ∑ t = 1 T 1 2 β t ‖ x t − 1 − β t x t − 1 ‖ 2 + C {\displaystyle \ln q(x_{0:T})=\ln q(x_{0})-\sum _{t=1}^{T}{\frac {1}{2\beta _{t}}}\|x_{t}-{\sqrt {1-\beta _{t}}}x_{t-1}\|^{2}+C} where C {\displaystyle C} is a normalization constant and often omitted. In particular, we note that x 1 : T | x 0 {\displaystyle x_{1:T}|x_{0}} is a Gaussian process, which affords us considerable freedom in reparameterization. For example, by standard manipulation with Gaussian process, x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} x t − 1 | x t , x 0 ∼ N ( μ ~ t ( x t , x 0 ) , σ ~ t 2 I ) {\displaystyle x_{t-1}|x_{t},x_{0}\sim {\mathcal {N}}({\tilde {\mu }}_{t}(x_{t},x_{0}),{\tilde {\sigma }}_{t}^{2}I)} In particular, notice that for large t {\displaystyle t} , the variable x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . That is, after a long enough diffusion process, we end up with some x T {\displaystyle x_{T}} that is very close to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with all traces of the original x 0 ∼ q {\displaystyle x_{0}\sim q} gone. For example, since x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} we can sample x t | x 0 {\displaystyle x_{t}|x_{0}} directly "in one step", instead of going through all the intermediate steps x 1 , x 2 , . . . , x t − 1 {\displaystyle x_{1},x_{2},...,x_{t-1}} . ==== Backward diffusion ==== The key idea of DDPM is to use a neural network parametrized by θ {\displaystyle \theta } . The network takes in two arguments x t , t {\displaystyle x_{t},t} , and outputs a vector μ θ ( x t , t ) {\displaystyle \mu _{\theta }(x_{t},t)} and a matrix Σ θ ( x t , t ) {\displaystyle \Sigma _{\theta }(x_{t},t)} , such that each step in the forward diffusion process can be approximately undone by x t − 1 ∼ N ( μ θ ( x t , t ) , Σ θ ( x t , t ) ) {\displaystyle x_{t-1}\sim {\mathcal {N}}(\mu _{\theta }(x_{t},t),\Sigma _{\theta }(x_{t},t))} . This then gives us a backward diffusion process p θ {\displaystyle p_{\theta }} defined by p θ ( x T ) = N ( x T | 0 , I ) {\displaystyle p_{\theta }(x
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Teamwork (project management)

Teamwork.com is an Irish, privately owned, web-based software company headquartered in Cork, Ireland. Teamwork creates task management and team collaboration software. Founded in 2007, as of 2016 the company stated that its software was in use by over 370,000 organisations worldwide (including Disney, Spotify and HP), and that it had over 2.4m users. == History == Peter Coppinger and Dan Mackey founded a company, Digital Crew, in 2007. This company built websites, intranets and custom web-based solutions for clients in Cork, Ireland. Frustrated by whiteboards and software management tools, Coppinger wanted a software system that would help manage client projects and which would be easy to use and generic enough to be used by different types of companies. Originally 37signals Basecamp users themselves, Coppinger and Mackey were frustrated by the limited feature set, and by Basecamp's apparent inaction on their feedback. In October 2007, Coppinger and Mackey launched Teamwork Project Manager, nicknamed TeamworkPM. In March 2015, this was renamed as Teamwork Projects. In 2014, after two years of negotiations, TeamworkPM bought the domain name 'Teamwork.com' for US$675,000 (€500,000). At the time this was one of the most expensive domain name purchases by an Irish company, and involved the transfer of a domain name which had been dormant since it was first acquired by the original owner in 1999. In 2015, Teamwork.com was named by Gartner to be one of their "Cool Vendors" in the Program and Portfolio Management Category. This was followed by the launch of a new real-time messaging product, Teamwork Chat, in January 2015. In June 2015, the company announced a drive to recruit for 40 positions by the end of the year. This was followed by the announcement that the company was investing more than €1 million in a new office, and had leased office space in Park House, Blackpool. In June 2016, Teamwork.com undertook a further recruitment drive to entice developers to Cork. In July 2021, the company announced that it had raised an investment of $70 million (€59.1 million) from venture capital firm Bregal Milestone to fund further growth. == Products == Teamwork markets a number of cloud-based applications, including Teamwork, Teamwork Desk, Teamwork Spaces, Teamwork CRM and Teamwork Chat. Teamwork was launched on 4 October 2007, at which time it had time management, milestone management, file sharing, time tracking, and messaging features. Teamwork's platform reportedly integrates with martech software like HubSpot, as well as other productivity tools like Slack, G Suite, MS Teams, Zapier, Dropbox and QuickBooks. == Awards == In 2016, Teamwork was awarded Cork's Best SME in the Cork Chamber of Commerce "Company of the Year" awards. In 2016, Teamwork was named number 7 in Deloitte's Fast 50 tech companies hit €1.6bn turnover. In 2015, Teamwork was identified as a Gartner "Cool Vendor" in the Program and Portfolio Management Category.
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Iris flower data set

The Iris flower data set or Fisher's Iris data set is a multivariate data set used and made famous by the British statistician and biologist Ronald Fisher in his 1936 paper The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. It is sometimes called Anderson's Iris data set because Edgar Anderson collected the data to quantify the morphologic variation of Iris flowers of three related species. Two of the three species were collected in the Gaspé Peninsula "all from the same pasture, and picked on the same day and measured at the same time by the same person with the same apparatus". The data set consists of 50 samples from each of three species of Iris (Iris setosa, Iris virginica and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimeters. Based on the combination of these four features, Fisher developed a linear discriminant model to distinguish each species. Fisher's paper was published in the Annals of Eugenics (today the Annals of Human Genetics). == Use of the data set == Originally used as an example data set on which Fisher's linear discriminant analysis was applied, it became a typical test case for many statistical classification techniques in machine learning such as support vector machines. The use of this data set in cluster analysis however is not common, since the data set only contains two clusters with rather obvious separation. One of the clusters contains Iris setosa, while the other cluster contains both Iris virginica and Iris versicolor and is not separable without the species information Fisher used. This makes the data set a good example to explain the difference between supervised and unsupervised techniques in data mining: Fisher's linear discriminant model can only be obtained when the object species are known: class labels and clusters are not necessarily the same. Nevertheless, all three species of Iris are separable in the projection on the nonlinear and branching principal component. The data set is approximated by the closest tree with some penalty for the excessive number of nodes, bending and stretching. Then the so-called "metro map" is constructed. The data points are projected into the closest node. For each node the pie diagram of the projected points is prepared. The area of the pie is proportional to the number of the projected points. It is clear from the diagram (left) that the absolute majority of the samples of the different Iris species belong to the different nodes. Only a small fraction of Iris-virginica is mixed with Iris-versicolor (the mixed blue-green nodes in the diagram). Therefore, the three species of Iris (Iris setosa, Iris virginica and Iris versicolor) are separable by the unsupervising procedures of nonlinear principal component analysis. To discriminate them, it is sufficient just to select the corresponding nodes on the principal tree. == Data set == The data set contains a set of 150 records under five attributes: sepal length, sepal width, petal length, petal width and species. The iris data set is widely used as a beginner's data set for machine learning purposes. The data set is included in R base and Python in the machine learning library scikit-learn, so that users can access it without having to find a source for it. Several versions of the data set have been published. === R code illustrating usage === The example R code shown below reproduce the scatterplot displayed at the top of this article: === Python code illustrating usage === This code gives:
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Computational learning theory

In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms. == Overview == Theoretical results in machine learning often focus on a type of inductive learning known as supervised learning. In supervised learning, an algorithm is provided with labeled samples. For instance, the samples might be descriptions of mushrooms, with labels indicating whether they are edible or not. The algorithm uses these labeled samples to create a classifier. This classifier assigns labels to new samples, including those it has not previously encountered. The goal of the supervised learning algorithm is to optimize performance metrics, such as minimizing errors on new samples. In addition to performance bounds, computational learning theory studies the time complexity and feasibility of learning . In computational learning theory, a computation is considered feasible if it can be done in polynomial time . There are two kinds of time complexity results: Positive results – Showing that a certain class of functions is learnable in polynomial time. Negative results – Showing that certain classes cannot be learned in polynomial time. Negative results often rely on commonly believed, but yet unproven assumptions, such as: Computational complexity – P ≠ NP (the P versus NP problem); Cryptographic – One-way functions exist. There are several different approaches to computational learning theory based on making different assumptions about the inference principles used to generalise from limited data. This includes different definitions of probability (see frequency probability, Bayesian probability) and different assumptions on the generation of samples. The different approaches include: Exact learning, proposed by Dana Angluin; Probably approximately correct learning (PAC learning), proposed by Leslie Valiant; VC theory, proposed by Vladimir Vapnik and Alexey Chervonenkis; Inductive inference as developed by Ray Solomonoff; Algorithmic learning theory, from the work of E. Mark Gold; Online machine learning, from the work of Nick Littlestone. While its primary goal is to understand learning abstractly, computational learning theory has led to the development of practical algorithms. For example, PAC theory inspired boosting, VC theory led to support vector machines, and Bayesian inference led to belief networks.
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