Data Refuge is a public and collaborative project designed to address concerns about federal climate and environmental data that is in danger of being lost. In particular, the initiative addresses five main concerns: What are the best ways to safeguard data? How do federal agencies play a crucial role in collecting, managing, and distributing data? How do government priorities impact data's accessibility? Which projects and research fields depend on federal data? Which data sets are of value to research and local communities, and why? Data Refuge began as a grassroots organization in opposition to government data on climate change and the environment not being archived systemically. Data Refuge's main goal is to collect and allocate data in multiple safe locations to create a sustainable way of archiving old and new data. Data Refuge was initiated in 2016 to protect federal climate and environmental data that is vulnerable under an administration that denies climate change. The system aims to make public research-quality copies of federal climate and environmental data. Data Refuge is supported by the National Geographic Foundation, private donors, Libraries+ Network, Preserving Electronic Governance Initiative (PEGI), the Union of Concerned Scientists (USC), and the Penn Program in Environmental Humanities (PPEH). == Types of data == Data Refuge collects public federal data on the climate and environment in the form of satellite imagery, PDFs, and stories. The data are stored in multiple trusted locations as they are less vulnerable if in only one location, and to ensure accessibility for researchers. Through the Data Rescue events, Data Refuge has accumulated 4 terabytes of data, 30,000 URLs, and 800 participants. === Storytelling === Data Refuge collects stories on vulnerable federal climate and environmental data through: surveys, oral history, photo essays, maps, video shorts, and animations. The stories are archived in a public bank that showcase how federal environmental data support health and safety in communities. Data Stories are collected at Data Rescue events, which are partnered with universities, city and town halls, and advocacy groups. Data stories are collected and used to emphasize the importance of Data Refuge, in how the data on climate change and the environment are being used by people in the United States and across the world for meaningful practices.
DevOps toolchain
A DevOps toolchain is a set or combination of tools that aid in the delivery, development, and management of software applications throughout the systems development life cycle, as coordinated by an organization that uses DevOps practices. Generally, DevOps tools fit into one or more activities, which supports specific DevOps initiatives: Plan, Create, Verify, Package, Release, Configure, Monitor, and Version Control. == Toolchains == In software, a toolchain is the set of programming tools that is used to perform a complex software development task or to create a software product, which is typically another computer program or a set of related programs. In general, the tools forming a toolchain are executed consecutively so the output or resulting environment state of each tool becomes the input or starting environment for the next one, but the term is also used when referring to a set of related tools that are not necessarily executed consecutively. As DevOps is a set of practices that emphasizes the collaboration and communication of both software developers and other information technology (IT) professionals, while automating the process of software delivery and infrastructure changes, its implementation can include the definition of the series of tools used at various stages of the lifecycle; because DevOps is a cultural shift and collaboration between development and operations, there is no one product that can be considered a single DevOps tool. Instead a collection of tools, potentially from a variety of vendors, are used in one or more stages of the lifecycle. == Stages of DevOps == === Plan === Plan consists of two elements: "define" and "plan". This activity refers to the business value and application requirements. Specifically "Plan" activities include: Production metrics, objects and feedback Requirements Business metrics Update release metrics Release plan, timing and business case Security policy and requirement A combination of the IT personnel will be involved in these activities: business application owners, software development, software architects, continual release management, security officers and the organization responsible for managing the production of IT infrastructure. === Create === Create consists of the building, coding, and configuring of the software development process. The specific activities are: Design of the software and configuration Coding including code quality and performance Software build and build performance Release candidate Tools and vendors in this category often overlap with other categories. Because DevOps is about breaking down silos, this is reflective in the activities and product solutions. === Verify === Verify is directly associated with ensuring the quality of the software release; activities designed to ensure code quality is maintained and the highest quality is deployed to production. The main activities in this are: Acceptance testing Regression testing Security and vulnerability analysis Performance Configuration testing Solutions for verify-related activities generally fall under four main categories: Test automation, Static analysis, Test Lab, and Security. === Package === Package refers to the activities involved once the release is ready for deployment, often also referred to as staging or Preproduction / "preprod". This often includes tasks and activities such as: Approval/preapprovals Package configuration Triggered releases Release staging and holding === Release === Release related activities include schedule, orchestration, provisioning and deploying software into production and targeted environment. The specific Release activities include: Release coordination Deploying and promoting applications Fallbacks and recovery Scheduled/timed releases Solutions that cover this aspect of the toolchain include application release automation, deployment automation and release management. === Configure === Configure activities fall under the operation side of DevOps. Once software is deployed, there may be additional IT infrastructure provisioning and configuration activities required. Specific activities including: Infrastructure storage, database and network provisioning and configuring Application provision and configuration. The main types of solutions that facilitate these activities are continuous configuration automation, configuration management, and infrastructure as code tools. === Monitor === Monitoring is an important link in a DevOps toolchain. It allows IT organization to identify specific issues of specific releases and to understand the impact on end-users. A summary of Monitor related activities are: Performance of IT infrastructure End-user response and experience Production metrics and statistics Information from monitoring activities often impacts Plan activities required for changes and for new release cycles. === Version Control === Version Control is an important link in a DevOps toolchain and a component of software configuration management. Version Control is the management of changes to documents, computer programs, large web sites, and other collections of information. A summary of Version Control related activities are: Non-linear development Distributed development Compatibility with existent systems and protocols Toolkit-based design Information from Version Control often supports Release activities required for changes and for new release cycles.
Memetic algorithm
In computer science and operations research, a memetic algorithm (MA) is an extension of an evolutionary algorithm (EA) that aims to accelerate the evolutionary search for the optimum. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. An MA uses one or more suitable heuristics or local search techniques to improve the quality of solutions generated by the EA and to speed up the search. The effects on the reliability of finding the global optimum depend on both the use case and the design of the MA. Memetic algorithms represent one of the recent growing areas of research in evolutionary computation. The term MA is now widely used as a synergy of evolutionary or any population-based approach with separate individual learning or local improvement procedures for problem search. Quite often, MAs are also referred to in the literature as Baldwinian evolutionary algorithms, Lamarckian EAs, cultural algorithms, or genetic local search. == Introduction == Inspired by both Darwinian principles of natural evolution and Dawkins' notion of a meme, the term memetic algorithm (MA) was introduced by Pablo Moscato in his technical report in 1989 where he viewed MA as being close to a form of population-based hybrid genetic algorithm (GA) coupled with an individual learning procedure capable of performing local refinements. The metaphorical parallels, on the one hand, to Darwinian evolution and, on the other hand, between memes and domain specific (local search) heuristics are captured within memetic algorithms thus rendering a methodology that balances well between generality and problem specificity. This two-stage nature makes them a special case of dual-phase evolution. The basic idea behind an MA is to combine the advantages of a global search performed by an EA (or another global search method) with the local refinement provided by one or more local search techniques, while avoiding their drawbacks. The main disadvantage of EAs is that, when searching in the vicinity of an optimum, they perform poorly in determining the exact position of that optimum. The downside of local search methods lies simply in the locality of their search relative to the chosen starting point. The combination of these two classes of methods aims to merge global and local search so that the advantages of both approaches can be leveraged. The idea of this approach can be illustrated by the search for the highest mountain in the Alps. A local search method would climb one of the mountains near the starting point, ignoring Mont Blanc as long as the starting point is not in its vicinity. An EA, on the other hand, will likely only find Mont Blanc after examining many other mountains, valleys, and hills, and then it will have difficulty identifying the summit cross. From the perspective of an MA’s global search procedure, however, only the summits of hills and mountains are seen, and its search is limited to finding the best summit. The open question is whether the additional effort required for the local search is worthwhile. This depends not only on the design of the MA but also on the specific application and the local search methods used. In the context of complex optimization, many different instantiations of memetic algorithms have been reported across a wide range of application domains, in general, converging to high-quality solutions more efficiently than their conventional evolutionary counterparts. In general, using the ideas of memetics within a computational framework is called memetic computing or memetic computation (MC). With MC, the traits of universal Darwinism are more appropriately captured. Viewed in this perspective, MA is a more constrained notion of MC. More specifically, MA covers one area of MC, in particular dealing with areas of evolutionary algorithms that marry other deterministic refinement techniques for solving optimization problems. MC extends the notion of memes to cover conceptual entities of knowledge-enhanced procedures or representations. == Theoretical Background == The no-free-lunch theorems of optimization and search state that all optimization strategies are equally effective with respect to the set of all optimization problems. Conversely, this means that one can expect the following: The more efficiently an algorithm solves a problem or class of problems, the less general it is and the more problem-specific knowledge it builds on. This insight leads directly to the recommendation to complement generally applicable metaheuristics with application-specific methods or heuristics, which fits well with the concept of MAs. == The development of MAs == === 1st generation === Pablo Moscato characterized an MA as follows: "Memetic algorithms are a marriage between a population-based global search and the heuristic local search made by each of the individuals. ... The mechanisms to do local search can be to reach a local optimum or to improve (regarding the objective cost function) up to a predetermined level." And he emphasizes "I am not constraining an MA to a genetic representation.". This original definition of MA although encompasses characteristics of cultural evolution (in the form of local refinement) in the search cycle, it may not qualify as a true evolving system according to universal Darwinism, since all the core principles of inheritance/memetic transmission, variation, and selection are missing. This suggests why the term MA stirred up criticisms and controversies among researchers when first introduced. The following pseudo code would correspond to this general definition of an MA: Pseudo code Procedure Memetic Algorithm Initialize: Generate an initial population, evaluate the individuals and assign a quality value to them; while Stopping conditions are not satisfied do Evolve a new population using stochastic search operators. Evaluate all individuals in the population and assign a quality value to them. Select the subset of individuals, Ω i l {\displaystyle \Omega _{il}} , that should undergo the individual improvement procedure. for each individual in Ω i l {\displaystyle \Omega _{il}} do Perform individual learning using meme(s) with frequency or probability of f i l {\displaystyle f_{il}} , with an intensity of t i l {\displaystyle t_{il}} . Proceed with Lamarckian or Baldwinian learning. end for end while Lamarckian learning in this context means to update the chromosome according to the improved solution found by the individual learning step, while Baldwinian learning leaves the chromosome unchanged and uses only the improved fitness. This pseudo code leaves open which steps are based on the fitness of the individuals and which are not. In question are the evolving of the new population and the selection of Ω i l {\displaystyle \Omega _{il}} . Since most MA implementations are based on EAs, the pseudo code of a corresponding representative of the first generation is also given here, following Krasnogor: Pseudo code Procedure Memetic Algorithm Based on an EA Initialization: t = 0 {\displaystyle t=0} ; // Initialization of the generation counter Randomly generate an initial population P ( t ) {\displaystyle P(t)} ; Compute the fitness f ( p ) ∀ p ∈ P ( t ) {\displaystyle f(p)\ \ \forall p\in P(t)} ; while Stopping conditions are not satisfied do Selection: Accordingly to f ( p ) {\displaystyle f(p)} choose a subset of P ( t ) {\displaystyle P(t)} and store it in M ( t ) {\displaystyle M(t)} ; Offspring: Recombine and mutate individuals p ∈ M ( t ) {\displaystyle p\in M(t)} and store them in M ′ ( t ) {\displaystyle M'(t)} ; Learning: Improve p ′ {\displaystyle p'} by local search or heuristic ∀ p ′ ∈ M ′ ( t ) {\displaystyle \forall p'\in M'(t)} ; Evaluation: Compute the fitness f ( p ′ ) ∀ p ′ ∈ M ′ ( t ) {\displaystyle f(p')\ \ \forall p'\in M'(t)} ; if Lamarckian learning then Update chromosome of p ′ {\displaystyle p'} according to improvement ∀ p ′ ∈ M ′ ( t ) {\displaystyle \forall p'\in M'(t)} ; fi New generation: Generate P ( t + 1 ) {\displaystyle P(t+1)} by selecting some individuals from P ( t ) {\displaystyle P(t)} and M ′ ( t ) {\displaystyle M'(t)} ; t = t + 1 {\displaystyle t=t+1} ; // Increment the generation counter end while Return best individual p ∈ P ( t − 1 ) {\displaystyle p\in P(t-1)} as result; There are some alternatives for this MA scheme. For example: All or some of the initial individuals may be improved by the meme(s). The parents may be locally improved instead of the offspring. Instead of all offspring, only a randomly selected or fitness-dependent fraction may undergo local improvement. The latter requires the evaluation of the offspring in M ′ ( t ) {\displaystyle M'(t)} prior to the Learning step. === 2nd generation === Multi-meme, hyper-heuristic and meta-Lamarckian MA are referred to as second generation MA exhibiting the principles of me
Rectified linear unit
In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU ( x ) = x + = max ( 0 , x ) = x + | x | 2 = { x if x > 0 , 0 x ≤ 0 {\displaystyle \operatorname {ReLU} (x)=x^{+}=\max(0,x)={\frac {x+|x|}{2}}={\begin{cases}x&{\text{if }}x>0,\\0&x\leq 0\end{cases}}} where x {\displaystyle x} is the input to a neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognition using deep neural nets and computational neuroscience. == History == The ReLU was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used ReLU in the context of visual feature extraction in hierarchical neural networks. In 1998, Gregory Woodbury demonstrated that the rectified linear function could account for a broad range of emergent properties in the visual cortex. His work showed that a single unified model could drive the joint development of refined retinotopic maps, ocular dominance columns, and orientation selectivity. By utilizing the rectifier's "cutoff" property, Woodbury achieved a close quantitative fit to biological data, matching the spatial periodicities and topographic refinement patterns observed in macaque and cat cortical maps. Furthermore, he extended this framework to adult plasticity, accurately replicating the spatial and temporal dynamics of lesion-induced cortical reorganization. This research established that the rectified linear response was a necessary mechanism for the stable self-organisation and maintenance of complex, multi-feature neural maps. In 2000, Hahnloser et al. argued that ReLU approximates the biological relationship between neural firing rates and input current, in addition to enabling recurrent neural network dynamics to stabilise under weaker criteria. Prior to 2010, most activation functions used were the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more numerically efficient counterpart, the hyperbolic tangent. Around 2010, the use of ReLU became common again. Jarrett et al. (2009) noted that rectification by either absolute or ReLU (which they called "positive part") was critical for object recognition in convolutional neural networks (CNNs), specifically because it allows average pooling without neighboring filter outputs cancelling each other out. They hypothesized that the use of sigmoid or tanh was responsible for poor performance in previous CNNs. Nair and Hinton (2010) made a theoretical argument that the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by considering a single binary neuron in a Boltzmann machine that takes x {\displaystyle x} as input, and produces 1 as output with probability σ ( x ) = 1 1 + e − x {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}} . They then considered extending its range of output by making infinitely many copies of it X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } , that all take the same input, offset by an amount 0.5 , 1.5 , 2.5 , … {\displaystyle 0.5,1.5,2.5,\dots } , then their outputs are added together as ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} . They then demonstrated that ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} is approximately equal to N ( log ( 1 + e x ) , σ ( x ) ) {\displaystyle {\mathcal {N}}(\log(1+e^{x}),\sigma (x))} , which is also approximately equal to ReLU ( N ( x , σ ( x ) ) ) {\displaystyle \operatorname {ReLU} ({\mathcal {N}}(x,\sigma (x)))} , where N {\displaystyle {\mathcal {N}}} stands for the gaussian distribution. They also argued for another reason for using ReLU: that it allows "intensity equivariance" in image recognition. That is, multiplying input image by a constant k {\displaystyle k} multiplies the output also. In contrast, this is false for other activation functions like sigmoid or tanh. They found that ReLU activation allowed good empirical performance in restricted Boltzmann machines. Glorot et al (2011) argued that ReLU has the following advantages over sigmoid or tanh: ReLU is more similar to biological neurons' responses in their main operating regime. ReLU avoids vanishing gradients. ReLU is cheaper to compute. ReLU creates sparse representation naturally, because many hidden units output exactly zero for a given input. They also found empirically that deep networks trained with ReLU can achieve strong performance without unsupervised pre-training, especially on large, purely supervised tasks. In 2017, the rectified linear function became a central component of the transformer architecture introduced in the Vaswani et al paper "Attention Is All You Need". Within every transformer layer, ReLU is utilized in the position-wise feed-forward networks (FFN), defined by Equation 2 of their paper: FFN ( x ) = max ( 0 , x W 1 + b 1 ) W 2 + b 2 {\displaystyle \operatorname {FFN} (x)=\max(0,xW_{1}+b_{1})W_{2}+b_{2}} This equation is foundational to the model's capacity; while the attention mechanism determines the relationships between tokens, the ReLU-based FFN performs the majority of the numerical computation and houses the bulk of the model's parameters. The efficiency and scalability of this rectified framework triggered a global technological revolution, enabling the development of Large Language Models that have had a profound economic impact. The industrial response to this architecture—including the massive expansion of AI-specific hardware and the birth of the generative AI sector—has positioned the Transformer as a cornerstone of 21st-century infrastructure. During the post 2017 period of rapid AI advancement, the rectified linear unit function has been key to achieving increased model performance and scaling due to the fact that it zeros out responses that are immaterial for a given stimuli, preventing them from accumulating in massive scale models. It is the complete silencing of the parts of the model found to be stimuli-irrelevant during learning that allows for scaling. As the stimuli-irrelevant proportion of the model becomes more massive, these highly numerous connections within the model would inevitably accumulate during scaling no matter how small each individual response is. Therefore, the rectified linear unit function, with its absolute zeroing property, enabled the scaling to hundred billion parameter models and beyond. Early Transformer scaling giants like GPT-3 (2020) and Falcon-180B (2023) relied on the rectified linear unit function explicitly, while successors such as GPT-4 (2023) and Llama 3 (2024) utilized smoother variants like GELU or SwiGLU. These variants were used to improve training stability while fundamentally preserving the rectified principle of zeroing low responses. At the centre of modern artificial intelligence ReLU and its variants maintain absolute zero response across the bulk of the model at any one time, while maintaining approximately linear reponses for stimuli-relevant connections enabling high performance on each specific cognitive task. This feature of activation sparsity has been critical for massive scaling and performance gains of AI models right up to the present day. == Advantages == Advantages of ReLU include: Sparse activation: for example, in a randomly initialized network, only about 50% of hidden units are activated (i.e. have a non-zero output). Better gradient propagation: fewer vanishing gradient problems compared to sigmoidal activation functions that saturate in both directions. Efficiency: only requires comparison and addition. Scale-invariant (homogeneous, or "intensity equivariance"): max ( 0 , a x ) = a max ( 0 , x ) for a ≥ 0 {\displaystyle \max(0,ax)=a\max(0,x){\text{ for }}a\geq 0} . == Potential problems == Possible downsides can include: Non-differentiability at zero (however, it is differentiable anywhere else, and the value of the derivative at zero can be chosen to be 0 or 1 arbitrarily). Not zero-centered: ReLU outputs are always non-negative. This can make it harder for the network to learn during backpropagation, because gradient updates tend to push weights in one direction (positive or negative). Batch normalization can help address this. ReLU is unbounded. Redundancy of the parametrization: Because ReLU is scale-invariant, the network computes the exact same function by scaling the weights and biases in front of a ReLU activation by k {\displaystyle k} , and the weights after by 1 / k {\displaystyle 1/k} . Dying ReLU: ReLU neurons can sometimes be pushed into states
Transkribus
Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.
Graphics address remapping table
The graphics address remapping table (GART), also known as the graphics aperture remapping table, or graphics translation table (GTT), is an I/O memory management unit (IOMMU) used by Accelerated Graphics Port (AGP) and PCI Express (PCIe) graphics cards. The GART allows the graphics card direct memory access (DMA) to the host system memory, through which buffers of textures, polygon meshes and other data are loaded. AMD later reused the same mechanism for I/O virtualization with other peripherals including disk controllers and network adapters. A GART is used as a means of data exchange between the main memory and video memory through which buffers (i.e. paging/swapping) of textures, polygon meshes and other data are loaded, but can also be used to expand the amount of video memory available for systems with only integrated or shared graphics (i.e. no discrete or inbuilt graphics processor), such as Intel HD Graphics processors. However, this type of memory (expansion) remapping has a caveat that affects the entire system: specifically, any GART, pre-allocated memory becomes pooled and cannot be utilised for any other purposes but graphics memory and display rendering. Since PCI Express, the GART is extended to the GTT (Graphics Translation Table), which act as a buffer or cache between system memory and graphics card, and in PCI Express, the GTT buffer size is changeable by the GPU driver. == Operating system support == === Windows === Support for AGP GART was added since Windows 95 OSR2. Later, support for GTT was added since Windows XP SP2 and Windows Vista. === Linux === Jeff Hartmann served as the primary maintainer of the Linux kernel's agpgart driver, which began as part of Brian Paul's Utah GLX accelerated Mesa 3D driver project. The developers primarily targeted Linux 2.4.x kernels, but made patches available against older 2.2.x kernels. Dave Jones heavily reworked agpgart for the Linux 2.6.x kernels, along with more contributions from Jeff Hartmann. === FreeBSD === In FreeBSD, the agpgart driver appeared in its 4.1 release. === Solaris === AGPgart support was introduced into Solaris Express Developer Edition as of its 7/05 release.
Large margin nearest neighbor
Large margin nearest neighbor (LMNN) classification is a statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based on semidefinite programming, a sub-class of convex optimization. The goal of supervised learning (more specifically classification) is to learn a decision rule that can categorize data instances into pre-defined classes. The k-nearest neighbor rule assumes a training data set of labeled instances (i.e. the classes are known). It classifies a new data instance with the class obtained from the majority vote of the k closest (labeled) training instances. Closeness is measured with a pre-defined metric. Large margin nearest neighbors is an algorithm that learns this global (pseudo-)metric in a supervised fashion to improve the classification accuracy of the k-nearest neighbor rule. == Setup == The main intuition behind LMNN is to learn a pseudometric under which all data instances in the training set are surrounded by at least k instances that share the same class label. If this is achieved, the leave-one-out error (a special case of cross validation) is minimized. Let the training data consist of a data set D = { ( x → 1 , y 1 ) , … , ( x → n , y n ) } ⊂ R d × C {\displaystyle D=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}\subset R^{d}\times C} , where the set of possible class categories is C = { 1 , … , c } {\displaystyle C=\{1,\dots ,c\}} . The algorithm learns a pseudometric of the type d ( x → i , x → j ) = ( x → i − x → j ) ⊤ M ( x → i − x → j ) {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})=({\vec {x}}_{i}-{\vec {x}}_{j})^{\top }\mathbf {M} ({\vec {x}}_{i}-{\vec {x}}_{j})} . For d ( ⋅ , ⋅ ) {\displaystyle d(\cdot ,\cdot )} to be well defined, the matrix M {\displaystyle \mathbf {M} } needs to be positive semi-definite. The Euclidean metric is a special case, where M {\displaystyle \mathbf {M} } is the identity matrix. This generalization is often (falsely) referred to as Mahalanobis metric. Figure 1 illustrates the effect of the metric under varying M {\displaystyle \mathbf {M} } . The two circles show the set of points with equal distance to the center x → i {\displaystyle {\vec {x}}_{i}} . In the Euclidean case this set is a circle, whereas under the modified (Mahalanobis) metric it becomes an ellipsoid. The algorithm distinguishes between two types of special data points: target neighbors and impostors. === Target neighbors === Target neighbors are selected before learning. Each instance x → i {\displaystyle {\vec {x}}_{i}} has exactly k {\displaystyle k} different target neighbors within D {\displaystyle D} , which all share the same class label y i {\displaystyle y_{i}} . The target neighbors are the data points that should become nearest neighbors under the learned metric. Let us denote the set of target neighbors for a data point x → i {\displaystyle {\vec {x}}_{i}} as N i {\displaystyle N_{i}} . === Impostors === An impostor of a data point x → i {\displaystyle {\vec {x}}_{i}} is another data point x → j {\displaystyle {\vec {x}}_{j}} with a different class label (i.e. y i ≠ y j {\displaystyle y_{i}\neq y_{j}} ) which is one of the nearest neighbors of x → i {\displaystyle {\vec {x}}_{i}} . During learning the algorithm tries to minimize the number of impostors for all data instances in the training set. == Algorithm == Large margin nearest neighbors optimizes the matrix M {\displaystyle \mathbf {M} } with the help of semidefinite programming. The objective is twofold: For every data point x → i {\displaystyle {\vec {x}}_{i}} , the target neighbors should be close and the impostors should be far away. Figure 1 shows the effect of such an optimization on an illustrative example. The learned metric causes the input vector x → i {\displaystyle {\vec {x}}_{i}} to be surrounded by training instances of the same class. If it was a test point, it would be classified correctly under the k = 3 {\displaystyle k=3} nearest neighbor rule. The first optimization goal is achieved by minimizing the average distance between instances and their target neighbors ∑ i , j ∈ N i d ( x → i , x → j ) {\displaystyle \sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})} . The second goal is achieved by penalizing distances to impostors x → l {\displaystyle {\vec {x}}_{l}} that are less than one unit further away than target neighbors x → j {\displaystyle {\vec {x}}_{j}} (and therefore pushing them out of the local neighborhood of x → i {\displaystyle {\vec {x}}_{i}} ). The resulting value to be minimized can be stated as: ∑ i , j ∈ N i , l , y l ≠ y i [ d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ] + {\displaystyle \sum _{i,j\in N_{i},l,y_{l}\neq y_{i}}[d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})]_{+}} With a hinge loss function [ ⋅ ] + = max ( ⋅ , 0 ) {\textstyle [\cdot ]_{+}=\max(\cdot ,0)} , which ensures that impostor proximity is not penalized when outside the margin. The margin of exactly one unit fixes the scale of the matrix M {\displaystyle M} . Any alternative choice c > 0 {\displaystyle c>0} would result in a rescaling of M {\displaystyle M} by a factor of 1 / c {\displaystyle 1/c} . The final optimization problem becomes: min M ∑ i , j ∈ N i d ( x → i , x → j ) + λ ∑ i , j , l ξ i j l {\displaystyle \min _{\mathbf {M} }\sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})+\lambda \sum _{i,j,l}\xi _{ijl}} ∀ i , j ∈ N i , l , y l ≠ y i {\displaystyle \forall _{i,j\in N_{i},l,y_{l}\neq y_{i}}} d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ≤ ξ i j l {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})\leq \xi _{ijl}} ξ i j l ≥ 0 {\displaystyle \xi _{ijl}\geq 0} M ⪰ 0 {\displaystyle \mathbf {M} \succeq 0} The hyperparameter λ > 0 {\textstyle \lambda >0} is some positive constant (typically set through cross-validation). Here the variables ξ i j l {\displaystyle \xi _{ijl}} (together with two types of constraints) replace the term in the cost function. They play a role similar to slack variables to absorb the extent of violations of the impostor constraints. The last constraint ensures that M {\displaystyle \mathbf {M} } is positive semi-definite. The optimization problem is an instance of semidefinite programming (SDP). Although SDPs tend to suffer from high computational complexity, this particular SDP instance can be solved very efficiently due to the underlying geometric properties of the problem. In particular, most impostor constraints are naturally satisfied and do not need to be enforced during runtime (i.e. the set of variables ξ i j l {\displaystyle \xi _{ijl}} is sparse). A particularly well suited solver technique is the working set method, which keeps a small set of constraints that are actively enforced and monitors the remaining (likely satisfied) constraints only occasionally to ensure correctness. == Extensions and efficient solvers == LMNN was extended to multiple local metrics in the 2008 paper. This extension significantly improves the classification error, but involves a more expensive optimization problem. In their 2009 publication in the Journal of Machine Learning Research, Weinberger and Saul derive an efficient solver for the semi-definite program. It can learn a metric for the MNIST handwritten digit data set in several hours, involving billions of pairwise constraints. An open source Matlab implementation is freely available at the authors web page. Kumal et al. extended the algorithm to incorporate local invariances to multivariate polynomial transformations and improved regularization.