An informationist (or information specialist in context) provides research and knowledge management services in the context of clinical care or biomedical research. Although there is no one educational pathway or formalized set of skills or knowledge for informationists, one way to think of the informationist is as one who possesses the knowledge and skill of a medical librarian with extensive research specialization and some formal clinical or public health education that goes beyond on-the-job osmosis. Medical librarians and other biomedical professional organizations have been exploring the possibilities for evaluating how informationists are being used and whether their activities supplement or replace medical library activity. More generally, an informationist is a professional who works with information within a particular business, analytic or scientific context to drive toward outcomes based on evidence, analysis, prediction and execution. For example, an extension of the term is increasingly emerging in financial services, life sciences and health care industries. Though still nascently in use, its adoption applies to individuals with extensive industry expertise, acute familiarity with organizational structures and processes, deep domain level information mastery and information systems technical savvy. Informationists in this context support transformational initiatives within and across functional areas of an enterprise as architects, governance experts, continuous improvement advocates and strategists. == Background == The term was proposed in 2000 by Davidoff & Florance. Their editorial suggested that physicians should be delegating their information needs to informationists, just as they currently order CT scans from radiologists or cardiac catheterizations from cardiologists. They conceived of an information professional who was embedded in (and indeed, supported by) the clinical departments. Supporters of the concept see it as a means for librarians to reinvigorate connections with the faculty/clinicians, as well as provide superior service by dint of informationists' biomedical training. Critics complained that the idea is nothing new; librarians already provide in-depth, high quality information services and clinical medical librarians have been working alongside physicians, nurses and other clinicians for years. Large informationist programs in the U.S. exist at the National Institutes of Health and at Vanderbilt University. Welch Medical Library at Johns Hopkins University (JHU) is developing an informationist service model in which its 10 clinical and public health librarians are moving from serving as liaison librarians for assigned departments toward becoming embedded informationists within their departments. To prepare for the embedded informationist role, librarians are undertaking education as needed to supplement their backgrounds. For example, librarians bring experience in clinical behavior counseling, public health, nursing, and more. Informationist training can then focus upon filling gaps in research methods knowledge more so than on gaining additional knowledge in the librarian's area of expertise. Courses, seminars and workshops being undertaken include those covering systematic reviews, evidence-based medicine, critical appraisal, medical language, anatomy and physiology, biostatistics, and clinical research. The term informationist is related to that of informatician—also informaticist—and many informationists do possess skills in clinical topics, bioinformatics, and biomedical informatics. Harvard University, the University of Pittsburgh, and Washington University in St. Louis are examples of institutional libraries which have hired PhD-level scientists (who may or may not have library degrees) to provide informatics support for biomedical research.
Cloud computing
Cloud computing is defined by the International Organization for Standardization (ISO) as "a paradigm for enabling network access to a scalable and elastic pool of shareable physical or virtual resources with self-service provisioning and administration on demand". It is commonly referred to as "the cloud". == Characteristics == In 2011, the National Institute of Standards and Technology (NIST) identified five "essential characteristics" for cloud systems. Below are the exact definitions according to NIST: On-demand self-service: "A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automatically without requiring human interaction with each service provider." Broad network access: "Capabilities are available over the network and accessed through standard mechanisms that promote use by heterogeneous thin or thick client platforms (e.g., mobile phones, tablets, laptops, and workstations)." Resource pooling: " The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and virtual resources dynamically assigned and reassigned according to consumer demand." Rapid elasticity: "Capabilities can be elastically provisioned and released, in some cases automatically, to scale rapidly outward and inward commensurate with demand. To the consumer, the capabilities available for provisioning often appear unlimited and can be appropriated in any quantity at any time." Measured service: "Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency for both the provider and consumer of the utilized service. By 2023, the International Organization for Standardization (ISO) had expanded and refined the list. == History == The history of cloud computing extends to the 1960s, with the initial concepts of time-sharing becoming popularized via remote job entry (RJE). The "data center" model, where users submitted jobs to operators to run on mainframes, was predominantly used during this era. This period saw broad experimentation with making large-scale computing power more accessible through time-sharing, while optimizing infrastructure, platforms, and applications to improve efficiency for end users. The "cloud" metaphor for virtualized services dates to 1994, when it was used by General Magic for the universe of "places" that mobile agents in the Telescript environment could "go". The metaphor is credited to David Hoffman, a General Magic communications specialist, based on its long-standing use in networking and telecom. The expression cloud computing became more widely known in 1996 when Compaq Computer Corporation drew up a business plan for future computing and the Internet. The company's ambition was to supercharge sales with "cloud computing-enabled applications". The business plan foresaw that online consumer file storage would likely be commercially successful. As a result, Compaq decided to sell server hardware to internet service providers. In the 2000s, the application of cloud computing began to take shape with the establishment of Amazon Web Services (AWS) in 2002, which allowed developers to build applications independently. In 2006 Amazon Simple Storage Service, known as Amazon S3, and the Amazon Elastic Compute Cloud (EC2) were released. In 2008 NASA's development of the first open-source software for deploying private and hybrid clouds. The following decade saw the launch of various cloud services. In 2010, Microsoft launched Microsoft Azure, and Rackspace Hosting and NASA initiated an open-source cloud-software project, OpenStack. IBM introduced the IBM SmartCloud framework in 2011, and Oracle announced the Oracle Cloud in 2012. In December 2019, Amazon launched AWS Outposts, a service that extends AWS infrastructure, services, APIs, and tools to customer data centers, co-location spaces, or on-premises facilities. == Value proposition == Cloud computing can shorten time to market by offering pre-configured tools, scalable resources, and managed services, allowing users to focus on core business value rather than maintaining infrastructure. Cloud platforms can enable organizations and individuals to reduce upfront capital expenditures on physical infrastructure by shifting to an operational expenditure model, where costs scale with usage. Cloud platforms also offer managed services and tools, such as artificial intelligence, data analytics, and machine learning, which might otherwise require significant in-house expertise and infrastructure investment. While cloud computing can offer cost advantages through effective resource optimization, organizations often face challenges such as unused resources, inefficient configurations, and hidden costs without proper oversight and governance. Many cloud platforms provide cost management tools, such as AWS Cost Explorer and Azure Cost Management, and frameworks like FinOps have emerged to standardize financial operations in the cloud. Cloud computing also facilitates collaboration, remote work, and global service delivery by enabling secure access to data and applications from any location with an internet connection. Cloud providers offer various redundancy options for core services, such as managed storage and managed databases, though redundancy configurations often vary by service tier. Advanced redundancy strategies, such as cross-region replication or failover systems, typically require explicit configuration and may incur additional costs or licensing fees. Cloud environments operate under a shared responsibility model, where providers are typically responsible for infrastructure security, physical hardware, and software updates, while customers are accountable for data encryption, identity and access management (IAM), and application-level security. These responsibilities vary depending on the cloud service model—Infrastructure as a Service (IaaS), Platform as a Service (PaaS), or Software as a Service (SaaS)—with customers typically having more control and responsibility in IaaS environments and progressively less in PaaS and SaaS models, often trading control for convenience and managed services. == Adoption and suitability == The decision to adopt cloud computing or maintain on-premises infrastructure depends on factors such as scalability, cost structure, latency requirements, regulatory constraints, and infrastructure customization. Organizations with variable or unpredictable workloads, limited capital for upfront investments, or a focus on rapid scalability benefit from cloud adoption. Startups, SaaS companies, and e-commerce platforms often prefer the pay-as-you-go operational expenditure (OpEx) model of cloud infrastructure. Additionally, companies prioritizing global accessibility, remote workforce enablement, disaster recovery, and leveraging advanced services such as AI/ML and analytics are well-suited for the cloud. In recent years, some cloud providers have started offering specialized services for high-performance computing and low-latency applications, addressing some use cases previously exclusive to on-premises setups. On the other hand, organizations with strict regulatory requirements, highly predictable workloads, or reliance on deeply integrated legacy systems may find cloud infrastructure less suitable. Businesses in industries like defense, government, or those handling highly sensitive data often favor on-premises setups for greater control and data sovereignty. Additionally, companies with ultra-low latency requirements, such as high-frequency trading (HFT) firms, rely on custom hardware (e.g., FPGAs) and physical proximity to exchanges, which most cloud providers cannot fully replicate despite recent advancements. Similarly, tech giants like Google, Meta, and Amazon build their own data centers due to economies of scale, predictable workloads, and the ability to customize hardware and network infrastructure for optimal efficiency. However, these companies also use cloud services selectively for certain workloads and applications where it aligns with their operational needs. In practice, many organizations are increasingly adopting hybrid cloud architectures, combining on-premises infrastructure with cloud services. This approach allows businesses to balance scalability, cost-effectiveness, and control, offering the benefits of both deployment models while mitigating their respective limitations. == Challenges and limitations == One of the primary challenges of cloud computing, compared with traditional on-premises systems, is maintaining data security and privacy. Cloud users entrust their sensitive data to third-party providers, who may not have adequate measures to protect it from unau
Fitness function
A fitness function is a particular type of objective or cost function that is used to summarize, as a single figure of merit, how close a given candidate solution is to achieving the set aims. It is an important component of evolutionary algorithms (EA), such as genetic programming, evolution strategies or genetic algorithms. 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. For this purpose, many candidate solutions are generated, which are evaluated using a fitness function in order to guide the evolutionary development towards the desired goal. Similar quality functions are also used in other metaheuristics, such as ant colony optimization or particle swarm optimization. In the field of EAs, each candidate solution, also called an individual, is commonly represented as a string of numbers (referred to as a chromosome). After each round of testing or simulation the idea is to delete the n worst individuals, and to breed n new ones from the best solutions. Each individual must therefore to be assigned a quality number indicating how close it has come to the overall specification, and this is generated by applying the fitness function to the test or simulation results obtained from that candidate solution. Two main classes of fitness functions exist: one where the fitness function does not change, as in optimizing a fixed function or testing with a fixed set of test cases; and one where the fitness function is mutable, as in niche differentiation or co-evolving the set of test cases. Another way of looking at fitness functions is in terms of a fitness landscape, which shows the fitness for each possible chromosome. In the following, it is assumed that the fitness is determined based on an evaluation that remains unchanged during an optimization run. A fitness function does not necessarily have to be able to calculate an absolute value, as it is sometimes sufficient to compare candidates in order to select the better one. A relative indication of fitness (candidate a is better than b) is sufficient in some cases, such as tournament selection or Pareto optimization. == Requirements of evaluation and fitness function == The quality of the evaluation and calculation of a fitness function is fundamental to the success of an EA optimisation. It implements Darwin's principle of "survival of the fittest". Without fitness-based selection mechanisms for mate selection and offspring acceptance, EA search would be blind and hardly distinguishable from the Monte Carlo method. When setting up a fitness function, one must always be aware that it is about more than just describing the desired target state. Rather, the evolutionary search on the way to the optimum should also be supported as much as possible (see also section on auxiliary objectives), if and insofar as this is not already done by the fitness function alone. If the fitness function is designed badly, the algorithm will either converge on an inappropriate solution, or will have difficulty converging at all. Definition of the fitness function is not straightforward in many cases and often is performed iteratively if the fittest solutions produced by an EA is not what is desired. Interactive genetic algorithms address this difficulty by outsourcing evaluation to external agents which are normally humans. == Computational efficiency == The fitness function should not only closely align with the designer's goal, but also be computationally efficient. Execution speed is crucial, as a typical evolutionary algorithm must be iterated many times in order to produce a usable result for a non-trivial problem. Fitness approximation may be appropriate, especially in the following cases: Fitness computation time of a single solution is extremely high Precise model for fitness computation is missing The fitness function is uncertain or noisy. Alternatively or also in addition to the fitness approximation, the fitness calculations can also be distributed to a parallel computer in order to reduce the execution times. Depending on the population model of the EA used, both the EA itself and the fitness calculations of all offspring of one generation can be executed in parallel. == Multi-objective optimization == Practical applications usually aim at optimizing multiple and at least partially conflicting objectives. Two fundamentally different approaches are often used for this purpose, Pareto optimization and optimization based on fitness calculated using the weighted sum. === Weighted sum and penalty functions === When optimizing with the weighted sum, the single values of the O {\displaystyle O} objectives are first normalized so that they can be compared. This can be done with the help of costs or by specifying target values and determining the current value as the degree of fulfillment. Costs or degrees of fulfillment can then be compared with each other and, if required, can also be mapped to a uniform fitness scale. Without loss of generality, fitness is assumed to represent a value to be maximized. Each objective o i {\displaystyle o_{i}} is assigned a weight w i {\displaystyle w_{i}} in the form of a percentage value so that the overall raw fitness f r a w {\displaystyle f_{raw}} can be calculated as a weighted sum: f r a w = ∑ i = 1 O o i ⋅ w i w i t h ∑ i = 1 O w i = 1 {\displaystyle f_{raw}=\sum _{i=1}^{O}{o_{i}\cdot w_{i}}\quad {\mathsf {with}}\quad \sum _{i=1}^{O}{w_{i}}=1} A violation of R {\displaystyle R} restrictions r j {\displaystyle r_{j}} can be included in the fitness determined in this way in the form of penalty functions. For this purpose, a function p f j ( r j ) {\displaystyle pf_{j}(r_{j})} can be defined for each restriction which returns a value between 0 {\displaystyle 0} and 1 {\displaystyle 1} depending on the degree of violation, with the result being 1 {\displaystyle 1} if there is no violation. The previously determined raw fitness is multiplied by the penalty function(s) and the result is then the final fitness f f i n a l {\displaystyle f_{final}} : f f i n a l = f r a w ⋅ ∏ j = 1 R p f j ( r j ) = ∑ i = 1 O ( o i ⋅ w i ) ⋅ ∏ j = 1 R p f j ( r j ) {\displaystyle f_{final}=f_{raw}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}=\sum _{i=1}^{O}{(o_{i}\cdot w_{i})}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}} This approach is simple and has the advantage of being able to combine any number of objectives and restrictions. The disadvantage is that different objectives can compensate each other and that the weights have to be defined before the optimization. This means that the compromise lines must be defined before optimization, which is why optimization with the weighted sum is also referred to as the a priori method. In addition, certain solutions may not be obtained, see the section on the comparison of both types of optimization. === Pareto optimization === A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution. Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides. It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well. They are therefore well suited as a-posteriori methods for multi-objective optimization, in which the final decision is made by a human decision maker after optimization and determination of the Pareto front. Besides the SPEA2, the NSGA-II and NSGA-III have established themselves as standard methods. The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives. If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods. === Comparison of both types of assessment === With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is convex
Non-negative matrix factorization
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically. NMF finds applications in such fields as astronomy, computer vision, document clustering, missing data imputation, chemometrics, audio signal processing, recommender systems, and bioinformatics. == History == In chemometrics non-negative matrix factorization has a long history under the name "self modeling curve resolution". In this framework the vectors in the right matrix are continuous curves rather than discrete vectors. Also early work on non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more widely known as non-negative matrix factorization after Lee and Seung investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations. == Background == Let matrix V be the product of the matrices W and H, V = W H . {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} Matrix multiplication can be implemented as computing the column vectors of V as linear combinations of the column vectors in W using coefficients supplied by columns of H. That is, each column of V can be computed as follows: v i = W h i , {\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} where vi is the i-th column vector of the product matrix V and hi is the i-th column vector of the matrix H. When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it is this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. For example, if V is an m × n matrix, W is an m × p matrix, and H is a p × n matrix then p can be significantly less than both m and n. Here is an example based on a text-mining application: Let the input matrix (the matrix to be factored) be V with 10000 rows and 500 columns where words are in rows and documents are in columns. That is, we have 500 documents indexed by 10000 words. It follows that a column vector v in V represents a document. Assume we ask the algorithm to find 10 features in order to generate a features matrix W with 10000 rows and 10 columns and a coefficients matrix H with 10 rows and 500 columns. The product of W and H is a matrix with 10000 rows and 500 columns, the same shape as the input matrix V and, if the factorization worked, it is a reasonable approximation to the input matrix V. From the treatment of matrix multiplication above it follows that each column in the product matrix WH is a linear combination of the 10 column vectors in the features matrix W with coefficients supplied by the coefficients matrix H. This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix W as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix H represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in W) where each feature is weighted by the feature's cell value from the document's column in H. == Clustering property == NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data V = ( v 1 , … , v n ) {\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} . More specifically, the approximation of V {\displaystyle \mathbf {V} } by V ≃ W H {\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } is achieved by finding W {\displaystyle W} and H {\displaystyle H} that minimize the error function (using the Frobenius norm) ‖ V − W H ‖ F , {\displaystyle \left\|V-WH\right\|_{F},} subject to W ≥ 0 , H ≥ 0. {\displaystyle W\geq 0,H\geq 0.} , If we furthermore impose an orthogonality constraint on H {\displaystyle \mathbf {H} } , i.e. H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed H {\displaystyle H} gives the cluster membership, i.e., if H k j > H i j {\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} for all i ≠ k, this suggests that the input data v j {\displaystyle v_{j}} belongs to k {\displaystyle k} -th cluster. The computed W {\displaystyle W} gives the cluster centroids, i.e., the k {\displaystyle k} -th column gives the cluster centroid of k {\displaystyle k} -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document clustering method. == Types == === Approximate non-negative matrix factorization === Usually the number of columns of W and the number of rows of H in NMF are selected so the product WH will become an approximation to V. The full decomposition of V then amounts to the two non-negative matrices W and H as well as a residual U, such that: V = WH + U. The elements of the residual matrix can either be negative or positive. When W and H are smaller than V they become easier to store and manipulate. Another reason for factorizing V into smaller matrices W and H, is that if one's goal is to approximately represent the elements of V by significantly less data, then one has to infer some latent structure in the data. === Convex non-negative matrix factorization === In standard NMF, matrix factor W ∈ R+m × k， i.e., W can be anything in that space. Convex NMF restricts the columns of W to convex combinations of the input data vectors ( v 1 , … , v n ) {\displaystyle (v_{1},\dots ,v_{n})} . This greatly improves the quality of data representation of W. Furthermore, the resulting matrix factor H becomes more sparse and orthogonal. === Nonnegative rank factorization === In case the nonnegative rank of V is equal to its actual rank, V = WH is called a nonnegative rank factorization (NRF). The problem of finding the NRF of V, if it exists, is known to be NP-hard. === Different cost functions and regularizations === There are different types of non-negative matrix factorizations. The different types arise from using different cost functions for measuring the divergence between V and WH and possibly by regularization of the W and/or H matrices. Two simple divergence functions studied by Lee and Seung are the squared error (or Frobenius norm) and an extension of the Kullback–Leibler divergence to positive matrices (the original Kullback–Leibler divergence is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules. The factorization problem in the squared error version of NMF may be stated as: Given a matrix V {\displaystyle \mathbf {V} } find nonnegative matrices W and H that minimize the function F ( W , H ) = ‖ V − W H ‖ F 2 {\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} Another type of NMF for images is based on the total variation norm. When L1 regularization (akin to Lasso) is added to NMF with the mean squared error cost function, the resulting problem may be called non-negative sparse coding due to the similarity to the sparse coding problem, although it may also still be referred to as NMF. === Online NMF === Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in streaming fashion. One such use is for collaborative filtering in recommendation systems, where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for o
Random projection
In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. According to theoretical results, random projection preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing. == Dimensionality reduction == Dimensionality reduction, as the name suggests, is reducing the number of random variables using various mathematical methods from statistics and machine learning. Dimensionality reduction is often used to reduce the problem of managing and manipulating large data sets. Dimensionality reduction techniques generally use linear transformations in determining the intrinsic dimensionality of the manifold as well as extracting its principal directions. For this purpose there are various related techniques, including: principal component analysis, linear discriminant analysis, canonical correlation analysis, discrete cosine transform, random projection, etc. Random projection is a simple and computationally efficient way to reduce the dimensionality of data by trading a controlled amount of error for faster processing times and smaller model sizes. The dimensions and distribution of random projection matrices are controlled so as to approximately preserve the pairwise distances between any two samples of the dataset. == Method == The core idea behind random projection is given in the Johnson-Lindenstrauss lemma, which states that if points in a vector space are of sufficiently high dimension, then they may be projected into a suitable lower-dimensional space in a way which approximately preserves pairwise distances between the points with high probability. In random projection, the original d {\displaystyle d} -dimensional data is projected to a k {\displaystyle k} -dimensional subspace, by multiplying on the left by a random matrix R ∈ R k × d {\displaystyle R\in \mathbb {R} ^{k\times d}} . Using matrix notation: If X d × N {\displaystyle X_{d\times N}} is the original set of N d-dimensional observations, then X k × N R P = R k × d X d × N {\displaystyle X_{k\times N}^{RP}=R_{k\times d}X_{d\times N}} is the projection of the data onto a lower k-dimensional subspace. Random projection is computationally simple: form the random matrix "R" and project the d × N {\displaystyle d\times N} data matrix X onto K dimensions of order O ( d k N ) {\displaystyle O(dkN)} . If the data matrix X is sparse with about c nonzero entries per column, then the complexity of this operation is of order O ( c k N ) {\displaystyle O(ckN)} . === Orthogonal random projection === A unit vector can be orthogonally projected to a random subspace. Let u {\displaystyle u} be the original unit vector, and let v {\displaystyle v} be its projection. The norm-squared ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as projecting a random point, uniformly sampled on the unit sphere, to its first k {\displaystyle k} coordinates. This is equivalent to sampling a random point in the multivariate gaussian distribution x ∼ N ( 0 , I d × d ) {\displaystyle x\sim {\mathcal {N}}(0,I_{d\times d})} , then normalizing it. Therefore, ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as ∑ i = 1 k x i 2 ∑ i = 1 k x i 2 + ∑ i = k + 1 d x i 2 {\displaystyle {\frac {\sum _{i=1}^{k}x_{i}^{2}}{\sum _{i=1}^{k}x_{i}^{2}+\sum _{i=k+1}^{d}x_{i}^{2}}}} , which by the chi-squared construction of the Beta distribution, has distribution Beta ( k / 2 , ( d − k ) / 2 ) {\displaystyle \operatorname {Beta} (k/2,(d-k)/2)} , with mean k / d {\displaystyle k/d} . We have a concentration inequality P r [ | ‖ v ‖ 2 − k d | ≥ ϵ k d ] ≤ 3 exp ( − k ϵ 2 / 64 ) {\displaystyle Pr\left[\left|\|v\|_{2}-{\frac {k}{d}}\right|\geq \epsilon {\sqrt {\frac {k}{d}}}\right]\leq 3\exp \left(-k\epsilon ^{2}/64\right)} for any ϵ ∈ ( 0 , 1 ) {\displaystyle \epsilon \in (0,1)} . === Gaussian random projection === The random matrix R can be generated using a Gaussian distribution. The first row is a random unit vector uniformly chosen from S d − 1 {\displaystyle S^{d-1}} . The second row is a random unit vector from the space orthogonal to the first row, the third row is a random unit vector from the space orthogonal to the first two rows, and so on. In this way of choosing R, and the following properties are satisfied: Spherical symmetry: For any orthogonal matrix A ∈ O ( d ) {\displaystyle A\in O(d)} , RA and R have the same distribution. Orthogonality: The rows of R are orthogonal to each other. Normality: The rows of R are unit-length vectors. === More computationally efficient random projections === Achlioptas has shown that the random matrix can be sampled more efficiently. Either the full matrix can be sampled IID according to R i , j = 3 / k × { + 1 with probability 1 6 0 with probability 2 3 − 1 with probability 1 6 {\displaystyle R_{i,j}={\sqrt {3/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{6}}\\0&{\text{with probability }}{\frac {2}{3}}\\-1&{\text{with probability }}{\frac {1}{6}}\end{cases}}} or the full matrix can be sampled IID according to R i , j = 1 / k × { + 1 with probability 1 2 − 1 with probability 1 2 {\displaystyle R_{i,j}={\sqrt {1/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{2}}\\-1&{\text{with probability }}{\frac {1}{2}}\end{cases}}} Both are efficient for database applications because the computations can be performed using integer arithmetic. More related study is conducted in. It was later shown how to use integer arithmetic while making the distribution even sparser, having very few nonzeroes per column, in work on the Sparse JL Transform. This is advantageous since a sparse embedding matrix means being able to project the data to lower dimension even faster. === Random Projection with Quantization === Random projection can be further condensed by quantization (discretization), with 1-bit (sign random projection) or multi-bits. It is the building block of SimHash, RP tree, and other memory efficient estimation and learning methods. == Large quasiorthogonal bases == The Johnson-Lindenstrauss lemma states that large sets of vectors in a high-dimensional space can be linearly mapped in a space of much lower (but still high) dimension n with approximate preservation of distances. One of the explanations of this effect is the exponentially high quasiorthogonal dimension of n-dimensional Euclidean space. There are exponentially large (in dimension n) sets of almost orthogonal vectors (with small value of inner products) in n–dimensional Euclidean space. This observation is useful in indexing of high-dimensional data. Quasiorthogonality of large random sets is important for methods of random approximation in machine learning. In high dimensions, exponentially large numbers of randomly and independently chosen vectors from equidistribution on a sphere (and from many other distributions) are almost orthogonal with probability close to one. This implies that in order to represent an element of such a high-dimensional space by linear combinations of randomly and independently chosen vectors, it may often be necessary to generate samples of exponentially large length if we use bounded coefficients in linear combinations. On the other hand, if coefficients with arbitrarily large values are allowed, the number of randomly generated elements that are sufficient for approximation is even less than dimension of the data space. == Implementations == RandPro - An R package for random projection sklearn.random_projection - A module for random projection from the scikit-learn Python library Weka implementation [1]
List & Label
List & Label is a professional reporting tool for software developers. It provides comprehensive design, print and export functions. The software component runs on Microsoft Windows and can be implemented in desktop, cloud and web applications. List & Label can be used to create user-defined dashboards, lists, invoices, forms and labels. It supports many development environments, frameworks and programming languages such as Microsoft Visual Studio, Embarcadero RAD Studio, .NET Framework, .NET Core, ASP.NET, C++, Delphi, Java, C Sharp and some more. List & Label either retrieves data from various sources via data binding, or works database independent. Reports are designed and created in the so-called List & Label Designer and then exported into a multitude of formats like PDF, Excel, XHTML and RTF. Since version 27 a web report designer for ASP.NET MVC is available. == History == The product was first released in 1992 by combit. The current version is 30. A new major version of List & Label is released every fall, usually in October. Updates are available several times a year via Service Pack. == Features == === Report Designer === The Designer enables users to graphically layout the report. It offers report objects such as tables, charts, crosstabs, gauges, HTML, conditionally formatted text, barcodes, matrix codes, and graphics, and is extensible using third-party add-ons. User applications can interact with the report via the programmable object model of the report. The real-time preview functionality allows users to view changes instantly. Usability features include layer and appearance management, enabling conditional logic to dynamically control the visibility of objects in reports. The Designer also supports the inclusion of multiple report containers in a single project, accommodating complex layouts such as parallel tables and charts. A formula wizard and support for scripting languages such as C# facilitate advanced calculations and logic. The Designer's object model (DOM) provides developers with the ability to modify layouts and behaviors programmatically. === Web Report Designer === The web report designer works browser-based and independent from printer drivers and spoolers - that makes deployments to the cloud easier. Just like the use of the Visual Studio deployment pipeline. === Data Sources === Depending on the programming language, the product offers automatic support for data sources: Databases such as Microsoft SQL Server, Oracle, MySQL, PostgreSQL, IBM Db2, SQLite, MariaDB, MongoDB, Cosmos DB XML data, CSV Business objects Data sources that can be accessed via OLE DB, ODBC or ADO.NET LINQ data and data from web services GraphQL Additionally, the product offers support for unbound data and can be extended to support other data sources via interfaces. === Output Options === Printer Image Formats (JPEG, BMP, EMF, TIFF, PNG, SVG, HEIF, WebP) Document Formats: PDF, PDF/A, Word (DOCX), Excel (XLS), PowerPoint (PPTX) HTML, XHTML, MHTML Barcodes Plain Text, RTF, CSV, JSON XML, ZIP, Email, JSON List & Label preview file === Target Audience === List & Label can be used in Windows development environments. While it competes most notably on the Microsoft .NET platform with other products such as Crystal Reports, SQL Server Reporting Services, ActiveReports, there are few competing products for other programming languages (e.g. Progress, Alaska Xbase++, Visual DataFlex). == Awards == Reader's Choice Award 2005–2008 Stevie Awards 2021: Best Technology for Data Visualization Top 100 Publisher Award Component Source 2013-2014, 2014-2015,2016, 2018, 2019, 2020, 2021, 2022
Dimensionality reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable. Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Linear approaches can be further divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilitate other analyses. == Feature selection == The process of feature selection aims to find a suitable subset of the input variables (features, or attributes) for the task at hand. The three strategies are: the filter strategy (e.g., information gain), the wrapper strategy (e.g., accuracy-guided search), and the embedded strategy (features are added or removed while building the model based on prediction errors). Data analysis such as regression or classification can be done in the reduced space more accurately than in the original space. == Feature projection == Feature projection (also called feature extraction) transforms the data from the high-dimensional space to a space of fewer dimensions. The data transformation may be linear, as in principal component analysis (PCA), but many nonlinear dimensionality reduction techniques also exist. For multidimensional data, tensor representation can be used in dimensionality reduction through multilinear subspace learning. === Principal component analysis (PCA) === The main linear technique for dimensionality reduction, principal component analysis, performs a linear mapping of the data to a lower-dimensional space in such a way that the variance of the data in the low-dimensional representation is maximized. In practice, the covariance (and sometimes the correlation) matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors that correspond to the largest eigenvalues (the principal components) can now be used to reconstruct a large fraction of the variance of the original data. Moreover, the first few eigenvectors can often be interpreted in terms of the large-scale physical behavior of the system, because they often contribute the vast majority of the system's energy, especially in low-dimensional systems. Still, this must be proved on a case-by-case basis as not all systems exhibit this behavior. The original space (with dimension of the number of points) has been reduced (with data loss, but hopefully retaining the most important variance) to the space spanned by a few eigenvectors. === Non-negative matrix factorization (NMF) === NMF decomposes a non-negative matrix to the product of two non-negative ones, which has been a promising tool in fields where only non-negative signals exist, such as astronomy. NMF is well known since the multiplicative update rule by Lee & Seung, which has been continuously developed: the inclusion of uncertainties, the consideration of missing data and parallel computation, sequential construction which leads to the stability and linearity of NMF, as well as other updates including handling missing data in digital image processing. With a stable component basis during construction, and a linear modeling process, sequential NMF is able to preserve the flux in direct imaging of circumstellar structures in astronomy, as one of the methods of detecting exoplanets, especially for the direct imaging of circumstellar discs. In comparison with PCA, NMF does not remove the mean of the matrices, which leads to physical non-negative fluxes; therefore NMF is able to preserve more information than PCA as demonstrated by Ren et al. === Kernel PCA === Principal component analysis can be employed in a nonlinear way by means of the kernel trick. The resulting technique is capable of constructing nonlinear mappings that maximize the variance in the data. The resulting technique is called kernel PCA. === Graph-based kernel PCA === Other prominent nonlinear techniques include manifold learning techniques such as Isomap, locally linear embedding (LLE), Hessian LLE, Laplacian eigenmaps, and methods based on tangent space analysis. These techniques assume that the high-dimensional input data lies near a low-dimensional manifold embedded in the ambient space, and construct a low-dimensional representation using a cost function that retains local properties of the data; they can be viewed as defining a graph-based kernel for Kernel PCA. More recently, techniques have been proposed that, instead of defining a fixed kernel, try to learn the kernel using semidefinite programming. The most prominent example of such a technique is maximum variance unfolding (MVU). The central idea of MVU is to exactly preserve all pairwise distances between nearest neighbors (in the inner product space) while maximizing the distances between points that are not nearest neighbors. An alternative approach to neighborhood preservation is through the minimization of a cost function that measures differences between distances in the input and output spaces. Important examples of such techniques include: classical multidimensional scaling, which is identical to PCA; Isomap, which uses geodesic distances in the data space; diffusion maps, which use diffusion distances in the data space; t-distributed stochastic neighbor embedding (t-SNE), which minimizes the divergence between distributions over pairs of points; and curvilinear component analysis. A different approach to nonlinear dimensionality reduction is through the use of autoencoders, a special kind of feedforward neural networks with a bottleneck hidden layer. The training of deep encoders is typically performed using a greedy layer-wise pre-training (e.g., using a stack of restricted Boltzmann machines) that is followed by a finetuning stage based on backpropagation. === Linear discriminant analysis (LDA) === Linear discriminant analysis (LDA) is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition, and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events. === Generalized discriminant analysis (GDA) === GDA deals with nonlinear discriminant analysis using kernel function operator. The underlying theory is close to the support-vector machines (SVM) insofar as the GDA method provides a mapping of the input vectors into high-dimensional feature space. Similar to LDA, the objective of GDA is to find a projection for the features into a lower dimensional space by maximizing the ratio of between-class scatter to within-class scatter. === Autoencoder === Autoencoders can be used to learn nonlinear dimension reduction functions and codings together with an inverse function from the coding to the original representation. === t-SNE === T-distributed Stochastic Neighbor Embedding (t-SNE) is a nonlinear dimensionality reduction technique useful for the visualization of high-dimensional datasets. It is not recommended for use in analysis such as clustering or outlier detection since it does not necessarily preserve densities or distances well. === UMAP === Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction technique. Visually, it is similar to t-SNE, but it assumes that the data is uniformly distributed on a locally connected Riemannian manifold and that the Riemannian metric is locally constant or approximately locally constant. == Dimension reduction == For high-dimensional datasets, dimension reduction is usually performed prior to applying a k-nearest neighbors (k-NN) algorithm in order to mitigate the curse of dimensionality. Feature extraction and dimension reduction can be combined in one step, using principal component analysis (PCA), linear discriminant analysis (LDA), canonical correlation analysis (CCA), or non-negative matrix factorization (NMF) techniques to pre-process the data, followed by clustering via k-NN on feature vectors in a reduced-dimension space. In machine learning, this process is also called low-dimensional embedding. For high-dimensional datasets (e.g., when performing similarity search on live video streams, DNA data, or high-dimensional time series), running a fast approximate k-NN search using locality-sensitive hashing, random projection, "sketches", or other high-dimensional similarity search techniques from the VLDB conference toolbox may be the only fe