2024–present global memory supply shortage

A global computer memory supply shortage started in 2024 due to supply constraints and rapid price escalation in the semiconductor memory market, particularly affecting DRAM and NAND flash memory. This shortage is sometimes labelled by tech media outlets as "RAMmageddon" or the "RAMpocalypse". Unlike the 2020–2023 global chip shortage, which stemmed primarily from pandemic-related supply chain disruptions from COVID-19, this shortage is driven by a structural reallocation of manufacturing capacity toward high-margin products for artificial intelligence infrastructure, creating scarcity of computer memory in consumer and enterprise PC markets. According to a 2026 Kearney's PERLab analysis, the shortage is expected to last at least until 2030, with CEOs agreeing with the timelines. == Background == Following a severe market downturn in 2022–2023, major memory manufacturers—Samsung Electronics, SK Hynix, and Micron Technology—implemented strategic production cuts to stabilize pricing. By mid-2024, the rapid expansion of generative AI services triggered unprecedented demand for specialized memory products, particularly High Bandwidth Memory (HBM) used in AI accelerators and data center GPUs. Specialized components of semiconductor technology are also experiencing supply constraints due to high demand in AI application. For example, glass cloth, a high-performance glass fiber substrate used for power efficient high speed data transfer and a crucial component of semiconductor manufacturing, is experiencing a supply crisis. Nitto Boseki, a Japanese firm having overwhelming monopoly in its production, is not able to meet increased demands, making chip-makers such as Qualcomm, Apple, Nvidia and AMD compete for securing supply. There are also reports of smaller electronics companies struggling to find suppliers for components such as NAND flash. Memory suppliers are adapting to increased demands and market unpredictability by requiring prepayment or shorter time-frame of payment, which makes it more difficult for smaller firms to acquire capital to survive. By 2026, due to steadily increased demand on resources, CPUs are also experiencing shortage issues due to low fabrication capacity, prioritisation of server CPUs, and increased demand, with CPU prices also being forecast to increase by as much as 15%. The demand on memory has also increased strain on other electronic components such as hard disk devices, with reports such as Western Digital's hard disk supply for 2026 being booked for enterprise applications before February 2026. A 2024 McKinsey analysis projected that global demand for AI-ready data center capacity would grow at approximately 33% annually through 2030, with AI workloads consuming roughly 70% of total data center capacity by the decade's end. In addition, according to Kearney's State of Semiconductor 2025 Report, executives were already expecting a shortage in the <8nm wafer size with memory chips being mentioned as an acute source of concern. Multiple companies mentioned being prepared for it through long-term agreements with RAM suppliers or amassing additional inventory. On 24 March 2026, Google announced TurboQuant, a memory compression technology focused on large language models (LLM) and vector search engines, which it claimed achieves 6x lower memory consumption in tested local LLMs and 8x performance enhancement in tests running on H100 accelerators. The technology is also a drop in enhancement for existing inference pipeline. Amid speculation about memory demand trends, memory manufacturers, SanDisk, Micron, Western Digital and Seagate, among other companies involved in memory manufacture experienced stock price declines. Prices of memory kits also reduced in the following months, although still at inflated prices. == Causes == === HBM production displacement === HBM manufacturing requires significantly more wafer capacity per bit than standard DRAM modules. Industry sources reported that as manufacturers allocated increasing wafer capacity to HBM production to meet contracts with AI infrastructure providers, the supply of conventional DDR4 and DDR5 modules for consumer PCs and smartphones contracted sharply. By September 2025, Samsung Electronics had reportedly expanded its 1c DRAM capacity to target 60,000 wafers per month specifically for HBM4 production, further diverting resources from consumer memory lines. === Geopolitical and trade barriers === The supply chain was further constrained by escalating trade tensions between the United States and China. Throughout 2025, fears of U.S. regulatory backlash and new tariff structures led major manufacturers like Samsung and SK Hynix to halt sales of older semiconductor manufacturing equipment to Chinese entities, effectively capping production capacity in the region. Additionally, proposed tariff policies by the U.S. administration in late 2025 prompted supply chain realignments, with Apple reportedly accelerating plans to source all U.S.-bound iPhones from India to avoid potential levies. === NAND flash capacity constraints === In the NAND flash segment, manufacturers prioritized higher-margin enterprise SSDs for data center applications while phasing out older process nodes more rapidly than anticipated. In November 2025, contract prices for NAND wafers increased by more than 60% month-over-month for certain product categories, with 512GB TLC experiencing the steepest rise as legacy manufacturing capacity was retired. == Impact on industry and consumers == === Manufacturer responses === Major PC manufacturers responded to component cost increases with significant price adjustments and supply chain strategies. Dell Technologies Chief Operating Officer Jeff Clarke stated during a November 2025 analyst call that the company had "never witnessed costs escalating at the current pace," describing tighter availability across DRAM, hard drives, and NAND flash memory. Analysts at Morgan Stanley downgraded Dell Technologies stock from "Overweight" to "Underweight" in late 2025, citing the company's heavy exposure to rising server memory costs. The firm warned that skyrocketing memory prices could significantly erode margins for server and PC OEMs. Conversely, Apple Inc. was reportedly less affected than its competitors, having secured long-term supply agreements for DRAM through the first quarter of 2026. Lenovo Chief Financial Officer Winston Cheng described the cost surge as "unprecedented" and disclosed that the company's memory inventories were approximately 50% above normal levels in anticipation of further price increases. === Consumer electronics sector === The shortage particularly affected smartphone manufacturers and other consumer electronics producers. DRAM prices reportedly rose by 172% throughout 2025, leading manufacturers like Samsung to halt new orders for DDR5 modules to reassess pricing structures and Micron to exit its 'Crucial' brand of consumer products. In Tokyo's Akihabara electronics district, retailers began limiting purchases of memory products to prevent hoarding, with prices for popular DDR5 memory modules more than doubling in some cases. Despite the broad trend of rising hardware costs, some companies engaged in aggressive pricing strategies to maintain market share; for example, Sony reduced the price of the PlayStation 5 by $100 for Black Friday 2025, potentially absorbing increased component costs to stimulate software ecosystem growth. Due to memory prices more than doubling in a single quarter, HP revealed in its Q1 2026 earnings call that memory costs account for 35% of PC build materials up from 15-18% previous quarter. Despite showing strong Q1 2026 earning driven by Windows 11 upgrade cycle and AI PC adoption, HP warned investors of low operating margins and up to double digit percentage decline for coming quarter. Trendforce, an IT analytics company, updated its forecast from 1.7% year-over-year growth in PC market to 2.6% year-over-year decline for 2026, amid backdrop of steadily increasing prices and supply crisis. Research and analytics firms, Gartner and IDC expect worldwide PC market to decline 10-11% and smartphone market to decline 8-9% in 2026. Gartner also projects that rising memory prices will make low-margin entry level laptops under 500 USD financially unviable in two years. The RAM shortage has delayed the release of Valve's second Steam Machine due to increased memory prices. The device was originally set to launch in early 2026. === AI infrastructure competition === Technology companies including Google, Amazon, Microsoft, and Meta Platforms placed open-ended orders with memory suppliers, indicating they would accept as much supply as available regardless of cost, according to Reuters sources. The limited supply of AI chips has been cited as a reason for the slow down in compute growth. In October 2025, OpenAI formally announced a strategic partnership using letters of intent with Samsung Electronics and SK Hynix

Learning curve (machine learning)

In machine learning (ML), a learning curve (or training curve) is a graphical representation that shows how a model's performance on a training set (and usually a validation set) changes with the number of training iterations (epochs) or the amount of training data. Typically, the number of training epochs or training set size is plotted on the x-axis, and the value of the loss function (and possibly some other metric such as the cross-validation score) on the y-axis. Synonyms include error curve, experience curve, improvement curve and generalization curve. More abstractly, learning curves plot the difference between learning effort and predictive performance, where "learning effort" usually means the number of training samples, and "predictive performance" means accuracy on testing samples. Learning curves have many useful purposes in ML, including: choosing model parameters during design, adjusting optimization to improve convergence, and diagnosing problems such as overfitting (or underfitting). Learning curves can also be tools for determining how much a model benefits from adding more training data, and whether the model suffers more from a variance error or a bias error. If both the validation score and the training score converge to a certain value, then the model will no longer significantly benefit from more training data. == Formal definition == When creating a function to approximate the distribution of some data, it is necessary to define a loss function L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} to measure how good the model output is (e.g., accuracy for classification tasks or mean squared error for regression). We then define an optimization process which finds model parameters θ {\displaystyle \theta } such that L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} is minimized, referred to as θ ∗ {\displaystyle \theta ^{}} . === Training curve for amount of data === If the training data is { x 1 , x 2 , … , x n } , { y 1 , y 2 , … y n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\},\{y_{1},y_{2},\dots y_{n}\}} and the validation data is { x 1 ′ , x 2 ′ , … x m ′ } , { y 1 ′ , y 2 ′ , … y m ′ } {\displaystyle \{x_{1}',x_{2}',\dots x_{m}'\},\{y_{1}',y_{2}',\dots y_{m}'\}} , a learning curve is the plot of the two curves i ↦ L ( f θ ∗ ( X i , Y i ) ( X i ) , Y i ) {\displaystyle i\mapsto L(f_{\theta ^{}(X_{i},Y_{i})}(X_{i}),Y_{i})} i ↦ L ( f θ ∗ ( X i , Y i ) ( X i ′ ) , Y i ′ ) {\displaystyle i\mapsto L(f_{\theta ^{}(X_{i},Y_{i})}(X_{i}'),Y_{i}')} where X i = { x 1 , x 2 , … x i } {\displaystyle X_{i}=\{x_{1},x_{2},\dots x_{i}\}} === Training curve for number of iterations === Many optimization algorithms are iterative, repeating the same step (such as backpropagation) until the process converges to an optimal value. Gradient descent is one such algorithm. If θ i ∗ {\displaystyle \theta _{i}^{}} is the approximation of the optimal θ {\displaystyle \theta } after i {\displaystyle i} steps, a learning curve is the plot of i ↦ L ( f θ i ∗ ( X , Y ) ( X ) , Y ) {\displaystyle i\mapsto L(f_{\theta _{i}^{}(X,Y)}(X),Y)} i ↦ L ( f θ i ∗ ( X , Y ) ( X ′ ) , Y ′ ) {\displaystyle i\mapsto L(f_{\theta _{i}^{}(X,Y)}(X'),Y')}

Lingua Libre

Lingua Libre is an online collaborative project and tool by the Wikimédia France association, which aims to build a collaborative, multilingual, audiovisual speech corpus under a free license. It mostly consists of a rapid recording online service which allows the user to chain hundreds of recordings. Contributors have produced content in 310+ languages. == Description == Lingua Libre enables the recording of words, phrases or sentences of any language, oral (audio recording) or signed (video recording). Words are presented to the speaker in the form of a list, created on the spot, in advance, or by reusing an existing Wikimedia category. The speaker simply reads the word displayed on the screen, and the software moves on to the next word when it detects a silence after the read word. This principle, borrowed from the open source software Shtooka recorder with the help of its creator, Nicolas Vion, makes it possible to record several hundreds of words per hour. The recordings are then uploaded automatically from the web client to the Wikimedia Commons media library. In spring 2021, Lingua Libre was offline due to a fire in Strasbourg, but no audio recordings were lost. === Use of the recordings === The recordings can be consulted either on Lingua Libre or on Commons. They are mainly used on other Wikimedia projects, for example to illustrate entries on Wiktionaries or proper nouns in Wikipedia articles. The re-use of the recordings in a language teaching context is envisaged. Language learners can freely download pronunciations and use them on GoldenDict, a popular dictionary software. Thus, audio recordings can be used as “Pronunciation Dictionaries” on GoldenDict without needing internet connection. The recordings are also reused in Natural Language Processing projects, for example to drive Mozilla's DeepSpeech speech recognition engines. == Versions == Lingua Libre was initiated on January 23, 2015 and has had three successive versions: === Lingua Libre v.1 (2016) === As part of the Languages of France project, which aims to document and promote the regional languages of France on Wikimedia and Internet projects in general, the conception of Lingua Libre started in November 2015, partly funded by the DGLFLF (General Delegation for the French language and the languages of France). The first version of the project was launched in August 2016. Only suitable for audio recording, Lingua Libre was shown during a workshop on Occitan language in December 2016, and then presented to the online Wikimedia community and at international events in 2017. === Lingua Libre v.2 (2018) === A complete rebuilding was launched at the end of 2017. The new version of Lingua Libre is based on MediaWiki, uses Wikibase and OAuth to better integrate into the Wikimedia environment. The interface is translated via Translatewiki.net so that the project can be used by a large number of communities. The new version of the site was ready in June 2018 and opened to the public in August 2018. === Lingua Libre v.2.2 (2020) === In 2020, important changes were made to the platform; a new look was developed especially for the site, the .org domain replaced the .fr domain used until then, and added support for sign languages through video recording. == Statistics == In the first two years of the project's launch, approximately 10,000 recordings were made. The transition to v.2 was accompanied by a sharp increase in the contributions. The number of recordings multiplied by 10 in less than a year, exceeding the 100,000 threshold in May 2019. These recordings were made by 127 speakers in almost 50 languages. By September 2020, the platform had more than 300,000 recordings in 90 languages with more than 350 speakers. The 500,000 recordings milestone was reached in June 2021, thanks to 540 speakers of 120 languages.

DaVinci (software)

DaVinci was a development tool produced by Incross, which aimed at creating HTML5 mobile applications and media content. It included a jQuery framework and a JavaScript library that enabled developers and designers to craft web applications designed for mobile devices with a user experience similar to native applications. Business applications, games, rich media content, such as HTML5 multi-media magazines, advertisements, and animation, may be produced with the tool. DaVinci was based on standard web technology – including HTML5, CSS3, and JavaScript. == Features == DaVinci comprised DaVinci Studio and DaVinci Animator, which handled application programming and UI design. The tool had a WYSIWYG authoring environment. Open-source libraries, such as KnockOut, JsRender/JsViews, Impress.js, and turn.js, were included in the tool. Other open-source frameworks could also be integrated. The Model View Controller (MVC) and Data Binding in JavaScript could be handled through DaVinci's Data-Set Editor. In this mode, view components and model data could be visually bound, which allowed users to create web applications with server-integrated UI components without coding. Additionally, DaVinci included an N-Screen editor, which automatically adjusted designs and functionalities to fit the screen sizes of various devices, including smartphones, tablet PCs, and TVs. == DaVinci and jQuery == In collaboration with the jQuery Foundation, DaVinci played a significant role in hosting the first jQuery conference in an Asian district, which took place on November 12, 2012, in Seoul, South Korea. The conference showcased how DaVinci could be utilized in application development demonstrations.

Structural synthesis of programs

Structural synthesis of programs (SSP) is a special form of (automatic) program synthesis that is based on propositional calculus. More precisely, it uses intuitionistic logic for describing the structure of a program in such a detail that the program can be automatically composed from pieces like subroutines or even computer commands. It is assumed that these pieces have been implemented correctly, hence no correctness verification of these pieces is needed. SSP is well suited for automatic composition of services for service-oriented architectures and for synthesis of large simulation programs. == History == Automatic program synthesis began in the artificial intelligence field, with software intended for automatic problem solving. The first program synthesizer was developed by Cordell Green in 1969. At about the same time, mathematicians including R. Constable, Z. Manna, and R. Waldinger explained the possible use of formal logic for automatic program synthesis. Practically applicable program synthesizers appeared considerably later. The idea of structural synthesis of programs was introduced at a conference on algorithms in modern mathematics and computer science organized by Andrey Ershov and Donald Knuth in 1979. The idea originated from G. Pólya’s well-known book on problem solving. The method for devising a plan for solving a problem in SSP was presented as a formal system. The inference rules of the system were restructured and justified in logic by G. Mints and E. Tyugu in 1982. A programming tool PRIZ that uses SSP was developed in the 1980s. A recent Integrated development environment that supports SSP is CoCoViLa — a model-based software development platform for implementing domain specific languages and developing large Java programs. == The logic of SSP == Structural synthesis of programs is a method for composing programs from already implemented components (e.g. from computer commands or software object methods) that can be considered as functions. A specification for synthesis is given in intuitionistic propositional logic by writing axioms about the applicability of functions. An axiom about the applicability of a function f is a logical implication X1 ∧ X2 ∧ ... ∧ Xm → Y1 ∧ Y2 ... Yn, where X1, X2, ... Xm are preconditions and Y1, Y2, ... Yn are postconditions of the application of the function f. In intuitionistic logic, the function f is called a realization of this formula. A precondition can be a proposition stating that input data exists, e.g. Xi may have the meaning “variable xi has received a value”, but it may denote also some other condition, e.g. that resources needed for using the function f are available, etc. A precondition may also be an implication of the same form as the axiom given above; then it is called a subtask. A subtask denotes a function that must be available as an input when the function f is applied. This function itself must be synthesized in the process of SSP. In this case, realization of the axiom is a higher order function, i.e., a function that uses another function as an input. For instance, the formula (state → nextState) ∧ initialState → result can specify a higher order function with two inputs and an output result. The first input is a function that has to be synthesized for computing nextState from state, and the second input is initialState. Higher order functions give generality to the SSP – any control structure needed in a synthesized program can be preprogrammed and used then automatically with a respective specification. In particular, the last axiom presented here is a specification of a complex program – a simulation engine for simulating dynamic systems on models where nextState can be computed from state of the system.

Energy-based model

An energy-based model (EBM), also called Canonical Ensemble Learning (CEL) or Learning via Canonical Ensemble (LCE), is an application of canonical ensemble formulation from statistical physics for learning from data. The approach prominently appears in generative artificial intelligence. EBMs provide a unified framework for many probabilistic and non-probabilistic approaches to such learning, particularly for training graphical and other structured models. An EBM learns the characteristics of a target dataset and generates a similar but larger dataset. EBMs detect the latent variables of a dataset and generate new datasets with a similar distribution. Energy-based generative neural networks is a class of generative models, which aim to learn explicit probability distributions of data in the form of energy-based models, the energy functions of which are parameterized by modern deep neural networks. Boltzmann machines are a special form of energy-based models with a specific parametrization of the energy. == Description == For a given input x {\displaystyle x} , the model describes an energy E θ ( x ) {\displaystyle E_{\theta }(x)} such that the Boltzmann distribution P θ ( x ) = e − β E θ ( x ) Z ( θ ) {\displaystyle P_{\theta }(x)={e^{-\beta E_{\theta }(x)} \over Z(\theta )}} is a probability (density), and typically β = 1 {\displaystyle \beta =1} . Since the normalization constant: Z ( θ ) := ∫ x ∈ X e − β E θ ( x ) d x {\displaystyle Z(\theta ):=\int _{x\in X}e^{-\beta E_{\theta }(x)}dx} (also known as the partition function) depends on all the Boltzmann factors of all possible inputs x {\displaystyle x} , it cannot be easily computed or reliably estimated during training simply using standard maximum likelihood estimation. However, for maximizing the likelihood during training, the gradient of the log-likelihood of a single training example x {\displaystyle x} is given by using the chain rule: ∂ θ log ⁡ ( P θ ( x ) ) = E x ′ ∼ P θ [ ∂ θ E θ ( x ′ ) ] − ∂ θ E θ ( x ) ( ∗ ) {\displaystyle \partial _{\theta }\log \left(P_{\theta }(x)\right)=\mathbb {E} _{x'\sim P_{\theta }}[\partial _{\theta }E_{\theta }(x')]-\partial _{\theta }E_{\theta }(x)\,()} The expectation in the above formula for the gradient can be approximately estimated by drawing samples x ′ {\displaystyle x'} from the distribution P θ {\displaystyle P_{\theta }} using Markov chain Monte Carlo (MCMC). Early energy-based models, such as the 2003 Boltzmann machine by Hinton, estimated this expectation via blocked Gibbs sampling. Newer approaches make use of more efficient Stochastic Gradient Langevin Dynamics (LD), drawing samples using: x 0 ′ ∼ P 0 , x i + 1 ′ = x i ′ − α 2 ∂ E θ ( x i ′ ) ∂ x i ′ + ϵ {\displaystyle x_{0}'\sim P_{0},x_{i+1}'=x_{i}'-{\frac {\alpha }{2}}{\frac {\partial E_{\theta }(x_{i}')}{\partial x_{i}'}}+\epsilon } , where ϵ ∼ N ( 0 , α ) {\displaystyle \epsilon \sim {\mathcal {N}}(0,\alpha )} . A replay buffer of past values x i ′ {\displaystyle x_{i}'} is used with LD to initialize the optimization module. The parameters θ {\displaystyle \theta } of the neural network are therefore trained in a generative manner via MCMC-based maximum likelihood estimation: the learning process follows an "analysis by synthesis" scheme, where within each learning iteration, the algorithm samples the synthesized examples from the current model by a gradient-based MCMC method (e.g., Langevin dynamics or Hybrid Monte Carlo), and then updates the parameters θ {\displaystyle \theta } based on the difference between the training examples and the synthesized ones – see equation ( ∗ ) {\displaystyle ()} . This process can be interpreted as an alternating mode seeking and mode shifting process, and also has an adversarial interpretation. Essentially, the model learns a function E θ {\displaystyle E_{\theta }} that associates low energies to correct values, and higher energies to incorrect values. After training, given a converged energy model E θ {\displaystyle E_{\theta }} , the Metropolis–Hastings algorithm can be used to draw new samples. The acceptance probability is given by: P a c c ( x i → x ∗ ) = min ( 1 , P θ ( x ∗ ) P θ ( x i ) ) . {\displaystyle P_{acc}(x_{i}\to x^{})=\min \left(1,{\frac {P_{\theta }(x^{})}{P_{\theta }(x_{i})}}\right).} == History == The term "energy-based models" was first coined in a 2003 JMLR paper where the authors defined a generalisation of independent components analysis to the overcomplete setting using EBMs. Other early work on EBMs proposed models that represented energy as a composition of latent and observable variables. == Characteristics == EBMs demonstrate useful properties: Simplicity and stability. The EBM is the only object that needs to be designed and trained. Separate networks need not be trained to ensure balance. Adaptive computation time. An EBM can generate sharp, diverse samples or (more quickly) coarse, less diverse samples. Given infinite time, this procedure produces true samples. Flexibility. In Variational Autoencoders (VAE) and flow-based models, the generator learns a map from a continuous space to a (possibly) discontinuous space containing different data modes. EBMs can learn to assign low energies to disjoint regions (multiple modes). Adaptive generation. EBM generators are implicitly defined by the probability distribution, and automatically adapt as the distribution changes (without training), allowing EBMs to address domains where generator training is impractical, as well as minimizing mode collapse and avoiding spurious modes from out-of-distribution samples. Compositionality. Individual models are unnormalized probability distributions, allowing models to be combined through product of experts or other hierarchical techniques. == Experimental results == On image datasets such as CIFAR-10 and ImageNet 32x32, an EBM model generated high-quality images relatively quickly. It supported combining features learned from one type of image for generating other types of images. It was able to generalize using out-of-distribution datasets, outperforming flow-based and autoregressive models. EBM was relatively resistant to adversarial perturbations, behaving better than models explicitly trained against them with training for classification. == Applications == Target applications include natural language processing, robotics and computer vision. The first energy-based generative neural network is the generative ConvNet proposed in 2016 for image patterns, where the neural network is a convolutional neural network. The model has been generalized to various domains to learn distributions of videos, and 3D voxels. They are made more effective in their variants. They have proven useful for data generation (e.g., image synthesis, video synthesis, 3D shape synthesis, etc.), data recovery (e.g., recovering videos with missing pixels or image frames, 3D super-resolution, etc), data reconstruction (e.g., image reconstruction and linear interpolation ). == Alternatives == EBMs compete with techniques such as variational autoencoders (VAEs), generative adversarial networks (GANs) or normalizing flows. == Extensions == === Joint energy-based models === Joint energy-based models (JEM), proposed in 2020 by Grathwohl et al., allow any classifier with softmax output to be interpreted as energy-based model. The key observation is that such a classifier is trained to predict the conditional probability p θ ( y | x ) = e f → θ ( x ) [ y ] ∑ j = 1 K e f → θ ( x ) [ j ] for y = 1 , … , K and f → θ = ( f 1 , … , f K ) ∈ R K , {\displaystyle p_{\theta }(y|x)={\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{\sum _{j=1}^{K}e^{{\vec {f}}_{\theta }(x)[j]}}}\ \ {\text{ for }}y=1,\dotsc ,K{\text{ and }}{\vec {f}}_{\theta }=(f_{1},\dotsc ,f_{K})\in \mathbb {R} ^{K},} where f → θ ( x ) [ y ] {\displaystyle {\vec {f}}_{\theta }(x)[y]} is the y-th index of the logits f → {\displaystyle {\vec {f}}} corresponding to class y. Without any change to the logits it was proposed to reinterpret the logits to describe a joint probability density: p θ ( y , x ) = e f → θ ( x ) [ y ] Z ( θ ) , {\displaystyle p_{\theta }(y,x)={\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}},} with unknown partition function Z ( θ ) {\displaystyle Z(\theta )} and energy E θ ( x , y ) = − f θ ( x ) [ y ] {\displaystyle E_{\theta }(x,y)=-f_{\theta }(x)[y]} . By marginalization, we obtain the unnormalized density p θ ( x ) = ∑ y p θ ( y , x ) = ∑ y e f → θ ( x ) [ y ] Z ( θ ) =: e − E θ ( x ) , {\displaystyle p_{\theta }(x)=\sum _{y}p_{\theta }(y,x)=\sum _{y}{\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}}=:e^{-E_{\theta }(x)},} therefore, E θ ( x ) = − log ⁡ ( ∑ y e f → θ ( x ) [ y ] Z ( θ ) ) , {\displaystyle E_{\theta }(x)=-\log \left(\sum _{y}{\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}}\right),} so that any classifier can be used to define an energy function E θ ( x ) {\displaystyle E_{\theta }(x)} .

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: ∫ 0 t φ ( s ) ψ ( t − s ) d s , 0 ≤ t < ∞ , {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,} is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. == Circular c