AI Art History

AI Art History — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • Self-supervised learning

    Self-supervised learning

    Self-supervised learning (SSL) is a paradigm in machine learning where a model is trained on a task using the data itself to generate supervisory signals, rather than relying on externally-provided labels. In the context of neural networks, self-supervised learning aims to leverage inherent structures or relationships within the input data to create meaningful training signals. SSL tasks are designed so that solving them requires capturing essential features or relationships in the data. The input data is typically augmented or transformed in a way that creates pairs of related samples, where one sample serves as the input, and the other is used to formulate the supervisory signal. This augmentation can involve introducing noise, cropping, rotation, or other transformations. Self-supervised learning more closely imitates the way humans learn to classify objects. During SSL, the model learns in two steps. First, the task is solved based on an auxiliary or pretext classification task using pseudo-labels, which help to initialize the model parameters. Next, the actual task is performed with supervised or unsupervised learning. Self-supervised learning has produced promising results in recent years, and has found practical application in fields such as audio processing, and is being used by Facebook and others for speech recognition. == Pseudo-labels == Pseudo-labels are automatically generated labels that a model assigns to unlabeled data based on its own predictions. They are widely used in self-supervised and semi-supervised learning, where ground-truth annotations are limited or unavailable. By treating predicted labels as surrogate ground truth, learning algorithms can make use of large quantities of unlabeled data in the training process. Pseudo-labeling also plays an important role in systems that must adapt to concept drift, where the statistical properties of the data change over time. In these scenarios, the model may detect that an incoming instance deviates from previously learned behavior. The system then generates a classification result for that instance, and this predicted class is used as a pseudo-label for updating or retraining model components that are becoming outdated. This approach enables continuous adaptation in dynamic environments without requiring manual annotation. In many adaptive learning pipelines, pseudo-labels are chosen when the classifier produces sufficiently confident predictions, reducing the risk of propagating errors. These pseudo-labeled instances are then incorporated into training to refresh or evolve the model's understanding of emerging data patterns, particularly when existing components show signs of “aging” due to drift or distributional shifts. This strategy reduces reliance on manual labeling while helping maintain long-term model performance. == Types == === Autoassociative self-supervised learning === Autoassociative self-supervised learning is a specific category of self-supervised learning where a neural network is trained to reproduce or reconstruct its own input data. In other words, the model is tasked with learning a representation of the data that captures its essential features or structure, allowing it to regenerate the original input. The term "autoassociative" comes from the fact that the model is essentially associating the input data with itself. This is often achieved using autoencoders, which are a type of neural network architecture used for representation learning. Autoencoders consist of an encoder network that maps the input data to a lower-dimensional representation (latent space), and a decoder network that reconstructs the input from this representation. The training process involves presenting the model with input data and requiring it to reconstruct the same data as closely as possible. The loss function used during training typically penalizes the difference between the original input and the reconstructed output (e.g. mean squared error). By minimizing this reconstruction error, the autoencoder learns a meaningful representation of the data in its latent space. === Contrastive self-supervised learning === For a binary classification task, training data can be divided into positive examples and negative examples. Positive examples are those that match the target. For example, if training a classifier to identify birds, the positive training data would include images that contain birds. Negative examples would be images that do not. Contrastive self-supervised learning uses both positive and negative examples. The loss function in contrastive learning is used to minimize the distance between positive sample pairs, while maximizing the distance between negative sample pairs. An early example uses a pair of 1-dimensional convolutional neural networks to process a pair of images and maximize their agreement. Contrastive Language-Image Pre-training (CLIP) allows joint pretraining of a text encoder and an image encoder, such that a matching image-text pair have image encoding vector and text encoding vector that span a small angle (having a large cosine similarity). InfoNCE (Noise-Contrastive Estimation) is a method to optimize two models jointly, based on Noise Contrastive Estimation (NCE). Given a set X = { x 1 , … x N } {\displaystyle X=\left\{x_{1},\ldots x_{N}\right\}} of N {\displaystyle N} random samples containing one positive sample from p ( x t + k ∣ c t ) {\displaystyle p\left(x_{t+k}\mid c_{t}\right)} and N − 1 {\displaystyle N-1} negative samples from the 'proposal' distribution p ( x t + k ) {\displaystyle p\left(x_{t+k}\right)} , it minimizes the following loss function: L N = − E X [ log ⁡ f k ( x t + k , c t ) ∑ x j ∈ X f k ( x j , c t ) ] {\displaystyle {\mathcal {L}}_{\mathrm {N} }=-\mathbb {E} _{X}\left[\log {\frac {f_{k}\left(x_{t+k},c_{t}\right)}{\sum _{x_{j}\in X}f_{k}\left(x_{j},c_{t}\right)}}\right]} === Non-contrastive self-supervised learning === Non-contrastive self-supervised learning (NCSSL) uses only positive examples. Counterintuitively, NCSSL converges on a useful local minimum rather than reaching a trivial solution, with zero loss. For the example of binary classification, it would trivially learn to classify each example as positive. Effective NCSSL requires an extra predictor on the online side that does not back-propagate on the target side. === Joint-Embedding and Predictive Architectures === A major class of self-supervised learning moves beyond contrastive pairs, instead maximizing the agreement between views while preventing collapse through statistical constraints. Rooted in Deep Canonical Correlation Analysis (Deep CCA), this approach includes Joint-Embedding Architectures (JEA) like Barlow Twins and VICReg, which enforce covariance constraints to learn invariant representations without negative sampling. Deep Latent Variable Path Modelling (DLVPM) generalizes this to multimodal systems, using path models to enforce correlation and orthogonality across diverse data types. In 2022 Yann LeCun introduced Joint-Embedding Predictive Architectures (JEPA) as a step towards decision making, reasoning, and autonomous human intelligence in machines, including self-improvement through autonomous learning. Founded in representation learning, LeCun included the concept of a “world model” in JEPA which aims to enable machines to replicate human intellect by providing machines with a concept for the world in which they exist. Unlike autoencoders, JEPAs operate entirely in latent space, avoiding pixel-level noise to focus on semantic structure. Rather than just learning invariance, JEPAs learn by predicting masked latent representations from visible context. JEPA has been applied to domains such as image analysis, audio processing, and motion in images and video. == Comparison with other forms of machine learning == SSL belongs to supervised learning methods insofar as the goal is to generate a classified output from the input. At the same time, however, it does not require the explicit use of labeled input-output pairs. Instead, correlations, metadata embedded in the data, or domain knowledge present in the input are implicitly and autonomously extracted from the data. These supervisory signals, extracted from the data, can then be used for training. SSL is similar to unsupervised learning in that it does not require labels in the sample data. Unlike unsupervised learning, however, learning is not done using inherent data structures. Semi-supervised learning combines supervised and unsupervised learning, requiring only a small portion of the learning data be labeled. In transfer learning, a model designed for one task is reused on a different task. Training an autoencoder intrinsically constitutes a self-supervised process, because the output pattern needs to become an optimal reconstruction of the input pattern itself. However, in current jargon, the term 'self-supervised' often refers to tasks based on a pretext-task training setup

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  • International Conference on Computer Vision

    International Conference on Computer Vision

    The International Conference on Computer Vision (ICCV) is a research conference sponsored by the Institute of Electrical and Electronics Engineers (IEEE) held every other year. It is considered to be one of the top conferences in computer vision, alongside CVPR and ECCV, and it is held on years in which ECCV is not. The conference is usually spread over four to five days. Typically, experts in the focus areas give tutorial talks on the first day, then the technical sessions (and poster sessions in parallel) follow. Recent conferences have also had an increasing number of focused workshops and a commercial exhibition. == Awards == === Azriel Rosenfeld Lifetime Achievement Award === The Azriel Rosenfeld Award, or Azriel Rosenfeld Lifetime Achievement Award, recognizes researchers who have made significant contributions to the field of computer vision over their careers. It is named in memory of computer scientist and mathematician Azriel Rosenfeld. The following people have received this award: === Helmholtz Prize === The ICCV Helmholtz Prize, known as the Test of Time Award before 2013, is awarded every other year at the ICCV, recognizing ICCV papers from ten or more years earlier that had a significant impact on computer vision research. Winners are selected by the IEEE Computer Society's Technical Committee on Pattern Analysis and Machine Intelligence. The award is named after the 19th century physician and physicist Hermann von Helmholtz, and the ICCV's award is not related to the various Helmholtz Prizes in physics, or the Hermann von Helmholtz Prize in neuroscience. === Marr Prize === The ICCV best-paper award is the Marr Prize, named after British neuroscientist David Marr. === Mark Everingham Prize === The Mark Everingham Prize is an award given yearly by the Technical Committee on Pattern Analysis and Machine Intelligence of the IEEE Computer Society at the IEEE International Conference on Computer Vision or the European Conference on Computer Vision to commemorate the late Mark Everingham, "one of the rising stars of computer vision", and to encourage others to follow in his footsteps by acting to further progress in the computer vision community as a whole. The prize is given to a researcher, or a team of researchers, who have made a selfless contribution of significant benefit to other members of the computer vision community. The Mark Everingham Prize for Rigorous Evaluation was an award given in 2012 at the British Machine Vision Conference. === PAMI Distinguished Researcher Award === The PAMI Distinguished Researcher Award (until 2013 called Significant Researcher Award) is awarded to candidates whose research projects have significantly contributed to the progress of computer vision. Awards are made based on major research contributions, as well as the role of those contributions in influencing and inspiring other research. Candidates are nominated by the community. The following people have received this award: == Conference list == The conference is usually held in the Spring in various international locations.

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  • Operational taxonomic unit

    Operational taxonomic unit

    An operational taxonomic unit (OTU) is an operational definition used to classify groups of closely related individuals. The term was originally introduced in 1963 by Robert R. Sokal and Peter H. A. Sneath in the context of numerical taxonomy, where an "operational taxonomic unit" is simply the group of organisms currently being studied. In this sense, an OTU is a pragmatic definition to group individuals by similarity, equivalent to but not necessarily in line with classical Linnaean taxonomy or modern evolutionary taxonomy. Nowadays, however, the term is commonly used in a different context and refers to clusters of (uncultivated or unknown) organisms, grouped by DNA sequence similarity of a specific taxonomic marker gene (originally coined as mOTU; molecular OTU). In other words, OTUs are pragmatic proxies for "species" at different taxonomic levels, in the absence of traditional systems of biological classification as are available for macroscopic organisms. For several years, OTUs have been the most commonly used units of diversity, especially when analysing small subunit 16S (for prokaryotes) or 18S rRNA (for eukaryotes) marker gene sequence datasets. == Molecular OTU by clustering of marker gene sequences == In the approach represented by DNA barcoding, a particular locus is chosen to be used as the marker gene for classification. This locus should be universally present in the scope selected, variable enough to be different among close-related species, and be flanked by conservative sequences that allow for easy amplification and detection. There are databases containing sequences for such marker genes from many different species, allowing for comparison. (Sometimes only using one locus does not provide sufficient resolution, so multiple marker genes are used. This is the case for plants, where rbcL+matK is common.) Sequences obtained this way can be clustered according to their similarity to one another, and operational taxonomic units are defined based on the similarity threshold set by the researcher. The exact threshold depends on the taxa in question and the mutational rates of the selected locus in the taxon. 97–99% are commonly used, but "it is now recognized to be somewhat arbitrary as sequence variation within and among species varies across taxa". 100% similarity (fully identical) is also common, also known as single variants. It remains debatable how well this commonly used method recapitulates true microbial species phylogeny or ecology. Although OTUs can be calculated differently when using different algorithms or thresholds, research by Schmidt et al. (2014) demonstrated that 16S-derived microbial OTUs were generally ecologically consistent across habitats and several clustering approaches. The number of OTUs defined may be inflated due to errors in DNA sequencing. === OTU clustering approaches === There are three main approaches to clustering OTUs: De novo, for which the clustering is based on similarities between sequencing reads. Closed-reference, for which the clustering is performed against a reference database of sequences. Open-reference, where clustering is first performed against a reference database of sequences, then any remaining sequences that could not be mapped to the reference are clustered de novo. Using a reference provides taxonomic context for the OTUs found. Alternatively, taxonomic context can be found after the construction of clusters by comparing representative sequences from clusters against a reference database. There are also specialized classifiers for this purpose which are much faster than naive comparison using BLAST. === OTU clustering algorithms === Hierarchical clustering algorithms (HCA): uclust & cd-hit & ESPRIT Bayesian clustering: CROP == Molecular OTU by other methods == In addition to similarity-based grouping, marker gene sequences can be sorted into OTUs using molecular phylogeny, k-mer composition, or hybrid methods combining these methods with similarity. There are also Bayesian tree-less methods and machine learning approaches. Using phylogeny often involves manually assigning terminal clades or single nodes to an OTU, so this is usually only done for refinement. Genome skimming can be used to obtain high-copy DNA without the need to choose marker genes or to design PCR primers for the chosen genes. It can provide fairly good coverage of organelle DNA and repetitive elements such as ribosomal DNA, both of which can be used like marker genes in OTU analysis. Whole-genome sequencing is more expensive and involves the production and processing of more data. By considering the entire genome, many (sometimes over 100) marker genes can be used at the same time, producing highly resolved phylogenies that correctly identify problematic taxa. It is also possible to use entire genomes for OTU assignment. For example, genomes from different bacterial species almost always have an average nucleotide identity lower than 95%, a fact that can be used to define new OTUs (and likely new species).

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  • ImageNets

    ImageNets

    ImageNets is an open source framework for rapid prototyping of machine vision algorithms, developed by the Institute of Automation. == Description == ImageNets is an open source and platform independent (Windows & Linux) framework for rapid prototyping of machine vision algorithms. With the GUI ImageNet Designer, no programming knowledge is required to perform operations on images. A configured ImageNet can be loaded and executed from C++ code without the need for loading the ImageNet Designer GUI to achieve higher execution performance. == History == ImageNets was developed by the Institute of Automation, University of Bremen, Germany. The software was first publicly released in 2010. Originally, ImageNets was developed for the Care-Providing Robot FRIEND but it can be used for a wide range of computer vision applications.

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  • Wadhwani Institute for Artificial Intelligence

    Wadhwani Institute for Artificial Intelligence

    Wadhwani AI, based in Mumbai, Maharashtra, is an independent, non-profit institute. Founded in 2018, it is dedicated to developing Artificial intelligence solutions for social good. Their mission is to build AI-based innovations and solutions for underserved communities in developing countries, for a wide range of domains including agriculture, education, financial inclusion, healthcare, and infrastructure. == History and funding == The institute was founded with a $30 million philanthropic effort by the Wadhwani brothers, Romesh Wadhwani and Sunil Wadhwani. The institute was inaugurated and dedicated to the nation by Narendra Modi, the 14th Prime Minister of India. In 2019, the institute received a $2 million grant from Google.org to create technologies to help reduce crop losses in cotton farming, through integrated pest management. The United States Agency for International Development awarded $2 million to the institute in 2020 to develop tools, using mathematical modeling techniques and digital technologies such as artificial intelligence and machine learning, to forecast COVID-19 disease patterns, estimate resources needed, and plan interventions. == Collaboration == With assistance from Google, the Ministry of Agriculture and Farmers' Welfare and the Wadhwani AI developed Krishi 24/7, the first AI-powered automated agricultural news monitoring and analysis tool. Through better decision-making, Krishi 24/7 will support the identification of valuable news, provide timely notifications, and respond quickly to safeguard farmers' interests and advance sustainable agricultural growth. The application converts news articles into English after scanning them in several languages. It ensures that the ministry is informed in a timely manner about pertinent occurrences that are published online by extracting key information from news items, including the headline, crop name, event type, date, location, severity, summary, and source link. The National Center for Disease Control has effectively implemented a comparable automated surveillance and analysis tool for disease outbreaks.

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  • XGBoost

    XGBoost

    XGBoost (eXtreme Gradient Boosting) is an open-source software library which provides a regularizing gradient boosting framework for C++, Java, Python, R, Julia, Perl, and Scala. It works on Linux, Microsoft Windows, and macOS. From the project description, it aims to provide a "Scalable, Portable and Distributed Gradient Boosting (GBM, GBRT, GBDT) Library". It runs on a single machine, as well as the distributed processing frameworks Apache Hadoop, Apache Spark, Apache Flink, and Dask. XGBoost gained much popularity and attention in the mid-2010s as the algorithm of choice for many winning teams of machine learning competitions. == History == XGBoost initially started as a research project by Tianqi Chen as part of the Distributed (Deep) Machine Learning Community (DMLC) group at the University of Washington. Initially, it began as a terminal application which could be configured using a libsvm configuration file. It became well known in the ML competition circles after its use in the winning solution of the Higgs Machine Learning Challenge. Soon after, the Python and R packages were built, and XGBoost now has package implementations for Java, Scala, Julia, Perl, and other languages. This brought the library to more developers and contributed to its popularity among the Kaggle community, where it has been used for a large number of competitions. It was soon integrated with a number of other packages making it easier to use in their respective communities. It has now been integrated with scikit-learn for Python users and with the caret package for R users. It can also be integrated into Data Flow frameworks like Apache Spark, Apache Hadoop, and Apache Flink using the abstracted Rabit and XGBoost4J. XGBoost is also available on OpenCL for FPGAs. An efficient, scalable implementation of XGBoost has been published by Tianqi Chen and Carlos Guestrin. While the XGBoost model often achieves higher accuracy than a single decision tree, it sacrifices the intrinsic interpretability of decision trees. For example, following the path that a decision tree takes to make its decision is trivial and self-explained, but following the paths of hundreds or thousands of trees is much harder. == Features == Salient features of XGBoost which make it different from other gradient boosting algorithms include: Clever penalization of trees A proportional shrinking of leaf nodes Newton Boosting Extra randomization parameter Implementation on single, distributed systems and out-of-core computation Automatic feature selection Theoretically justified weighted quantile sketching for efficient computation Parallel tree structure boosting with sparsity Efficient cacheable block structure for decision tree training == The algorithm == XGBoost works as Newton–Raphson in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton–Raphson method. A generic unregularized XGBoost algorithm is: == Parameters == XGBoost has parameters which can be specified to affect how it functions and performs. Some parameters include: Learning rate (also known as the "step size" or “"shrinkage"), it is a number between 0 and 1 (default is 0.3), which determines the rate the algorithm learns from each iteration. n_estimators, sets the number of trees to be built in the ensemble, where more trees generally increases the complexity of the model, but can lead to overfitting with too many trees. Gamma (also known as Lagrange multiplier or the minimum loss reduction parameter), controls the minimum amount of loss reduction required to make a further split on a leaf node of the tree. The default in XGBoost is 0 . max_depth, represents how deeply each tree in the boosting process can grow during training, where the default is 6. == Awards == John Chambers Award (2016) High Energy Physics meets Machine Learning award (HEP meets ML) (2016)

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  • Neural gas

    Neural gas

    Neural gas is an artificial neural network, inspired by the self-organizing map and introduced in 1991 by Thomas Martinetz and Klaus Schulten. The neural gas is a simple algorithm for finding optimal data representations based on feature vectors. The algorithm was coined "neural gas" because of the dynamics of the feature vectors during the adaptation process, which distribute themselves like a gas within the data space. It is applied where data compression or vector quantization is an issue, for example speech recognition, image processing or pattern recognition. As a robustly converging alternative to the k-means clustering it is also used for cluster analysis. == Algorithm == Suppose we want to model a probability distribution P ( x ) {\displaystyle P(x)} of data vectors x {\displaystyle x} using a finite number of feature vectors w i {\displaystyle w_{i}} , where i = 1 , ⋯ , N {\displaystyle i=1,\cdots ,N} . For each time step t {\displaystyle t} Sample data vector x {\displaystyle x} from P ( x ) {\displaystyle P(x)} Compute the distance between x {\displaystyle x} and each feature vector. Rank the distances. Let i 0 {\displaystyle i_{0}} be the index of the closest feature vector, i 1 {\displaystyle i_{1}} the index of the second closest feature vector, and so on. Update each feature vector by: w i k t + 1 = w i k t + ε ⋅ e − k / λ ⋅ ( x − w i k t ) , k = 0 , ⋯ , N − 1 {\displaystyle w_{i_{k}}^{t+1}=w_{i_{k}}^{t}+\varepsilon \cdot e^{-k/\lambda }\cdot (x-w_{i_{k}}^{t}),k=0,\cdots ,N-1} In the algorithm, ε {\displaystyle \varepsilon } can be understood as the learning rate, and λ {\displaystyle \lambda } as the neighborhood range. ε {\displaystyle \varepsilon } and λ {\displaystyle \lambda } are reduced with increasing t {\displaystyle t} so that the algorithm converges after many adaptation steps. The adaptation step of the neural gas can be interpreted as gradient descent on a cost function. By adapting not only the closest feature vector but all of them with a step size decreasing with increasing distance order, compared to (online) k-means clustering a much more robust convergence of the algorithm can be achieved. The neural gas model does not delete a node and also does not create new nodes. === Comparison with SOM === Compared to self-organized map, the neural gas model does not assume that some vectors are neighbors. If two vectors happen to be close together, they would tend to move together, and if two vectors happen to be apart, they would tend to not move together. In contrast, in an SOM, if two vectors are neighbors in the underlying graph, then they will always tend to move together, no matter whether the two vectors happen to be neighbors in the Euclidean space. The name "neural gas" is because one can imagine it to be what an SOM would be like if there is no underlying graph, and all points are free to move without the bonds that bind them together. == Variants == A number of variants of the neural gas algorithm exists in the literature so as to mitigate some of its shortcomings. More notable is perhaps Bernd Fritzke's growing neural gas, but also one should mention further elaborations such as the Growing When Required network and also the incremental growing neural gas. A performance-oriented approach that avoids the risk of overfitting is the Plastic Neural gas model. === Growing neural gas === Fritzke describes the growing neural gas (GNG) as an incremental network model that learns topological relations by using a "Hebb-like learning rule", only, unlike the neural gas, it has no parameters that change over time and it is capable of continuous learning, i.e. learning on data streams. GNG has been widely used in several domains, demonstrating its capabilities for clustering data incrementally. The GNG is initialized with two randomly positioned nodes which are initially connected with a zero age edge and whose errors are set to 0. Since in the GNG input data is presented sequentially one by one, the following steps are followed at each iteration: It is calculating the errors (distances) between the two closest nodes to the current input data. The error of the winner node (only the closest one) is respectively accumulated. The winner node and its topological neighbors (connected by an edge) are moving towards the current input by different fractions of their respective errors. The age of all edges connected to the winner node are incremented. If the winner node and the second-winner are connected by an edge, such an edge is set to 0. Else, an edge is created between them. If there are edges with an age larger than a threshold, they are removed. Nodes without connections are eliminated. If the current iteration is an integer multiple of a predefined frequency-creation threshold, a new node is inserted between the node with the largest error (among all) and its topological neighbor presenting the highest error. The link between the former and the latter nodes is eliminated (their errors are decreased by a given factor) and the new node is connected to both of them. The error of the new node is initialized as the updated error of the node which had the largest error (among all). The accumulated error of all nodes is decreased by a given factor. If the stopping criterion is not met, the algorithm takes a following input. The criterion might be a given number of epochs, i.e., a pre-set number of times where all data is presented, or the reach of a maximum number of nodes. === Incremental growing neural gas === Another neural gas variant inspired by the GNG algorithm is the incremental growing neural gas (IGNG). The authors propose the main advantage of this algorithm to be "learning new data (plasticity) without degrading the previously trained network and forgetting the old input data (stability)." === Growing when required === Having a network with a growing set of nodes, like the one implemented by the GNG algorithm was seen as a great advantage, however some limitation on the learning was seen by the introduction of the parameter λ, in which the network would only be able to grow when iterations were a multiple of this parameter. The proposal to mitigate this problem was a new algorithm, the Growing When Required network (GWR), which would have the network grow more quickly, by adding nodes as quickly as possible whenever the network identified that the existing nodes would not describe the input well enough. === Plastic neural gas === The ability to only grow a network may quickly introduce overfitting; on the other hand, removing nodes on the basis of age only, as in the GNG model, does not ensure that the removed nodes are actually useless, because removal depends on a model parameter that should be carefully tuned to the "memory length" of the stream of input data. The "Plastic Neural Gas" model solves this problem by making decisions to add or remove nodes using an unsupervised version of cross-validation, which controls an equivalent notion of "generalization ability" for the unsupervised setting. While growing-only methods only cater for the incremental learning scenario, the ability to grow and shrink is suited to the more general streaming data problem. == Implementations == To find the ranking i 0 , i 1 , … , i N − 1 {\displaystyle i_{0},i_{1},\ldots ,i_{N-1}} of the feature vectors, the neural gas algorithm involves sorting, which is a procedure that does not lend itself easily to parallelization or implementation in analog hardware. However, implementations in both parallel software and analog hardware were actually designed.

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  • Robust principal component analysis

    Robust principal component analysis

    Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to grossly corrupted observations. A number of different approaches exist for Robust PCA, including an idealized version of Robust PCA, which aims to recover a low-rank matrix L0 from highly corrupted measurements M = L0 +S0. This decomposition in low-rank and sparse matrices can be achieved by techniques such as Principal Component Pursuit method (PCP), Stable PCP, Quantized PCP, Block based PCP, and Local PCP. Then, optimization methods are used such as the Augmented Lagrange Multiplier Method (ALM), Alternating Direction Method (ADM), Fast Alternating Minimization (FAM), Iteratively Reweighted Least Squares (IRLS ) or alternating projections (AP). == Algorithms == === Non-convex method === The 2014 guaranteed algorithm for the robust PCA problem (with the input matrix being M = L + S {\displaystyle M=L+S} ) is an alternating minimization type algorithm. The computational complexity is O ( m n r 2 log ⁡ 1 ϵ ) {\displaystyle O\left(mnr^{2}\log {\frac {1}{\epsilon }}\right)} where the input is the superposition of a low-rank (of rank r {\displaystyle r} ) and a sparse matrix of dimension m × n {\displaystyle m\times n} and ϵ {\displaystyle \epsilon } is the desired accuracy of the recovered solution, i.e., ‖ L ^ − L ‖ F ≤ ϵ {\displaystyle \|{\widehat {L}}-L\|_{F}\leq \epsilon } where L {\displaystyle L} is the true low-rank component and L ^ {\displaystyle {\widehat {L}}} is the estimated or recovered low-rank component. Intuitively, this algorithm performs projections of the residual onto the set of low-rank matrices (via the SVD operation) and sparse matrices (via entry-wise hard thresholding) in an alternating manner - that is, low-rank projection of the difference the input matrix and the sparse matrix obtained at a given iteration followed by sparse projection of the difference of the input matrix and the low-rank matrix obtained in the previous step, and iterating the two steps until convergence. This alternating projections algorithm is later improved by an accelerated version, coined AccAltProj. The acceleration is achieved by applying a tangent space projection before projecting the residue onto the set of low-rank matrices. This trick improves the computational complexity to O ( m n r log ⁡ 1 ϵ ) {\displaystyle O\left(mnr\log {\frac {1}{\epsilon }}\right)} with a much smaller constant in front while it maintains the theoretically guaranteed linear convergence. Another fast version of accelerated alternating projections algorithm is IRCUR. It uses the structure of CUR decomposition in alternating projections framework to dramatically reduces the computational complexity of RPCA to O ( max { m , n } r 2 log ⁡ ( m ) log ⁡ ( n ) log ⁡ 1 ϵ ) {\displaystyle O\left(\max\{m,n\}r^{2}\log(m)\log(n)\log {\frac {1}{\epsilon }}\right)} === Convex relaxation === This method consists of relaxing the rank constraint r a n k ( L ) {\displaystyle rank(L)} in the optimization problem to the nuclear norm ‖ L ‖ ∗ {\displaystyle \|L\|_{}} and the sparsity constraint ‖ S ‖ 0 {\displaystyle \|S\|_{0}} to ℓ 1 {\displaystyle \ell _{1}} -norm ‖ S ‖ 1 {\displaystyle \|S\|_{1}} . The resulting program can be solved using methods such as the method of Augmented Lagrange Multipliers. === Deep-learning augmented method === Some recent works propose RPCA algorithms with learnable/training parameters. Such a learnable/trainable algorithm can be unfolded as a deep neural network whose parameters can be learned via machine learning techniques from a given dataset or problem distribution. The learned algorithm will have superior performance on the corresponding problem distribution. == Applications == RPCA has many real life important applications particularly when the data under study can naturally be modeled as a low-rank plus a sparse contribution. Following examples are inspired by contemporary challenges in computer science, and depending on the applications, either the low-rank component or the sparse component could be the object of interest: === Video surveillance === Given a sequence of surveillance video frames, it is often required to identify the activities that stand out from the background. If we stack the video frames as columns of a matrix M, then the low-rank component L0 naturally corresponds to the stationary background and the sparse component S0 captures the moving objects in the foreground. === Face recognition === Images of a convex, Lambertian surface under varying illuminations span a low-dimensional subspace. This is one of the reasons for effectiveness of low-dimensional models for imagery data. In particular, it is easy to approximate images of a human's face by a low-dimensional subspace. To be able to correctly retrieve this subspace is crucial in many applications such as face recognition and alignment. It turns out that RPCA can be applied successfully to this problem to exactly recover the face.

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  • Self-management (computer science)

    Self-management (computer science)

    Self-management is the process by which computer systems manage their own operation without human intervention. Self-management technologies are expected to pervade the next generation of network management systems. The growing complexity of modern networked computer systems is a limiting factor in their expansion. The increasing heterogeneity of corporate computer systems, the inclusion of mobile computing devices, and the combination of different networking technologies like WLAN, cellular phone networks, and mobile ad hoc networks make the conventional, manual management difficult, time-consuming, and error-prone. More recently, self-management has been suggested as a solution to increasing complexity in cloud computing. An industrial initiative towards realizing self-management is the Autonomic Computing Initiative (ACI) started by IBM in 2001. The ACI defines the following four functional areas: Self-configuration Auto-configuration of components Self-healing Automatic discovery, and correction of faults; automatically applying all necessary actions to bring system back to normal operation Self-optimization Automatic monitoring and control of resources to ensure the optimal functioning with respect to the defined requirements Self-protection Proactive identification and protection from arbitrary attacks

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  • Logic learning machine

    Logic learning machine

    Logic learning machine (LLM) is a machine learning method based on the generation of intelligible rules. LLM is an efficient implementation of the Switching Neural Network (SNN) paradigm, developed by Marco Muselli, Senior Researcher at the Italian National Research Council CNR-IEIIT in Genoa. LLM has been employed in many different sectors, including the field of medicine (orthopedic patient classification, DNA micro-array analysis and Clinical Decision Support Systems), financial services and supply chain management. == History == The Switching Neural Network approach was developed in the 1990s to overcome the drawbacks of the most commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but could not provide deep insight into the studied phenomenon. On the other hand, decision trees were able to describe the phenomenon but often lacked accuracy. Switching Neural Networks made use of Boolean algebra to build sets of intelligible rules able to obtain very good performance. In 2014, an efficient version of Switching Neural Network was developed and implemented in the Rulex suite with the name Logic Learning Machine. Also, an LLM version devoted to regression problems was developed. == General == Like other machine learning methods, LLM uses data to build a model able to perform a good forecast about future behaviors. LLM starts from a table including a target variable (output) and some inputs and generates a set of rules that return the output value y {\displaystyle y} corresponding to a given configuration of inputs. A rule is written in the form: if premise then consequence where consequence contains the output value whereas premise includes one or more conditions on the inputs. According to the input type, conditions can have different forms: for categorical variables the input value must be in a given subset: x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} . for ordered variables the condition is written as an inequality or an interval: x 2 ≤ α {\displaystyle x_{2}\leq \alpha } or β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } A possible rule is therefore in the form if x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} AND x 2 ≤ α {\displaystyle x_{2}\leq \alpha } AND β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } then y = y ¯ {\displaystyle y={\bar {y}}} == Types == According to the output type, different versions of the Logic Learning Machine have been developed: Logic Learning Machine for classification, when the output is a categorical variable, which can assume values in a finite set Logic Learning Machine for regression, when the output is an integer or real number.

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  • Multiclass classification

    Multiclass classification

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

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  • Prototype methods

    Prototype methods

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

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  • Tradeshift

    Tradeshift

    Tradeshift is a cloud based business network and platform for purchase-to-pay automation, supply chain payments, marketplaces, virtual cards and supply chain financing. Its 2018 round of funding, led by Goldman Sachs, raised US$250 million at a valuation of $1.1 billion, giving the company unicorn status. Tradeshift is headquartered in San Francisco, California and has offices in London, Copenhagen, Bucharest and Kuala Lumpur. Tradeshift has reprocessed over $1 trillion USD through transactions on its network. == History == Tradeshift was founded in 2010 by Christian Lanng, Mikkel Hippe Brun, and Gert Sylvest. Inspiration for Tradeshift came after they created the world's first large scale peer-to-peer infrastructure for an e-business called NemHandel. The founders also had leading roles (Governing board member, Technical Director) in the European Commission project PEPPOL inside the European Union. In 2010, the Tradeshift platform launched in May in Copenhagen. Tradeshift won the European Startup Awards in the category of "Best Business or Enterprise Startup." In 2011, Tradeshift made its app marketplace available. In 2012, Tradeshift moved their headquarters from Copenhagen to San Francisco. In 2013, Tradeshift opened an R&D center in Suzhou, China. Tradeshift opened an additional office in London. And LATAM e-invoicing capabilities were added through partnership with Invoiceware. In 2014, Tradeshift expanded with offices in Tokyo, Paris, and Munich. The EU Commission officially approved the Universal Business Language (UBL) data format – a format Tradeshift supports – as eligible for referencing in tenders from public administrations. In 2015, Tradeshift won the Circulars "Digital Disruptor" Award at the WEF conference in Davos, Switzerland. Tradeshift also acquired product information management company Merchantry, and launched e-procurement and supplier risk management solutions. In 2016, Tradeshift acquired Hyper Travel and secured a $75 million series-D round funding. In 2017, Tradeshift acquired IBX Business Network and launches Tradeshift Ada. In 2018, Tradeshift secured a $250 million series-E round funding. and launched Blockchain Payments, the latter as part of Tradeshift Pay. In December 2018 Tradeshift acquired Babelway, an online B2B integration platform. The acquisition added three new office locations to Tradeshift (Salt Lake City, Louvain-la-neuve, Belgium, Cairo Egypt). In Q3 2018, Tradeshift reported year-over-year revenue growth of 400%, new bookings growth of 284%, and gross merchandise volume (GMV) growth of 262%. New total contract value also grew by US$47 million. Additionally, it added 27 new customers including Hertz, Shiseido, ECU and multiple Fortune 500 companies. In July 2023, HSBC and Tradeshift announced an agreement to launch a new, jointly owned business focused on the development of embedded finance solutions and financial services apps. As part of the agreement, HSBC made a $35 million investment into Tradeshift and joined its board. The agreement was part of a funding round which is expected to raise a minimum of $70 million from HSBC and other investors. The new joint venture will allow HSBC and Tradeshift to deploy a range of digital solutions across Tradeshift and other platforms. This includes payment and fintech services embedded into trade, e-commerce and marketplace experiences. In September 2023, CEO Lanng was fired for "gross misconduct on multiple grounds," including "allegations of sexual assault and harassment." Tradeshift was alleged to have fired his accuser after she complained to the company's human resources department, its co-founders and members of its board of directors about his abuse. == Financials == The company's valuation as of May 2018 was $1.1 billion. Tradeshift is now considered a unicorn, and, according to Bloomberg, will not need any further funding. Jan 14, 2020, Tradeshift announced that they had raised $240 million in Series F finance. == Acquisitions == In 2015, Tradeshift acquired product information management company Merchantry. Merchantry is a retail product information management (PIM) software for multi-vendor ecommerce retailers. In 2016, Tradeshift acquired Hyper Travel. Hyper Travel is a travel management service that allows customers to access travel agents via its native messaging apps, SMS, and email. In 2017, Tradeshift acquired IBX Group. In 2018, Tradeshift acquired Babelway, an online B2B integration platform.

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  • Medoid

    Medoid

    Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be members of the data set. Medoids are most commonly used on data when a mean or centroid cannot be defined, such as graphs. They are also used in contexts where the centroid is not representative of the dataset like in images, 3-D trajectories and gene expression (where while the data is sparse the medoid need not be). These are also of interest while wanting to find a representative using some distance other than squared euclidean distance (for instance in movie-ratings). For some data sets there may be more than one medoid, as with medians. A common application of the medoid is the k-medoids clustering algorithm, which is similar to the k-means algorithm but works when a mean or centroid is not definable. This algorithm basically works as follows. First, a set of medoids is chosen at random. Second, the distances to the other points are computed. Third, data are clustered according to the medoid they are most similar to. Fourth, the medoid set is optimized via an iterative process. Note that a medoid is not equivalent to a median, a geometric median, or centroid. A median is only defined on 1-dimensional data, and it only minimizes dissimilarity to other points for metrics induced by a norm (such as the Manhattan distance or Euclidean distance). A geometric median is defined in any dimension, but unlike a medoid, it is not necessarily a point from within the original dataset. == Definition == Let X := { x 1 , x 2 , … , x n } {\textstyle {\mathcal {X}}:=\{x_{1},x_{2},\dots ,x_{n}\}} be a set of n {\textstyle n} points in a space with a distance function d. Medoid is defined as x medoid = arg ⁡ min y ∈ X ∑ i = 1 n d ( y , x i ) . {\displaystyle x_{\text{medoid}}=\arg \min _{y\in {\mathcal {X}}}\sum _{i=1}^{n}d(y,x_{i}).} == Clustering with medoids == Medoids are a popular replacement for the cluster mean when the distance function is not (squared) Euclidean distance, or not even a metric (as the medoid does not require the triangle inequality). When partitioning the data set into clusters, the medoid of each cluster can be used as a representative of each cluster. Clustering algorithms based on the idea of medoids include: Partitioning Around Medoids (PAM), the standard k-medoids algorithm Hierarchical Clustering Around Medoids (HACAM), which uses medoids in hierarchical clustering == Algorithms to compute the medoid of a set == From the definition above, it is clear that the medoid of a set X {\displaystyle {\mathcal {X}}} can be computed after computing all pairwise distances between points in the ensemble. This would take O ( n 2 ) {\textstyle O(n^{2})} distance evaluations (with n = | X | {\displaystyle n=|{\mathcal {X}}|} ). In the worst case, one can not compute the medoid with fewer distance evaluations. However, there are many approaches that allow us to compute medoids either exactly or approximately in sub-quadratic time under different statistical models. If the points lie on the real line, computing the medoid reduces to computing the median which can be done in O ( n ) {\textstyle O(n)} by Quick-select algorithm of Hoare. However, in higher dimensional real spaces, no linear-time algorithm is known. RAND is an algorithm that estimates the average distance of each point to all the other points by sampling a random subset of other points. It takes a total of O ( n log ⁡ n ϵ 2 ) {\textstyle O\left({\frac {n\log n}{\epsilon ^{2}}}\right)} distance computations to approximate the medoid within a factor of ( 1 + ϵ Δ ) {\textstyle (1+\epsilon \Delta )} with high probability, where Δ {\textstyle \Delta } is the maximum distance between two points in the ensemble. Note that RAND is an approximation algorithm, and moreover Δ {\textstyle \Delta } may not be known apriori. RAND was leveraged by TOPRANK which uses the estimates obtained by RAND to focus on a small subset of candidate points, evaluates the average distance of these points exactly, and picks the minimum of those. TOPRANK needs O ( n 5 3 log 4 3 ⁡ n ) {\textstyle O(n^{\frac {5}{3}}\log ^{\frac {4}{3}}n)} distance computations to find the exact medoid with high probability under a distributional assumption on the average distances. trimed presents an algorithm to find the medoid with O ( n 3 2 2 Θ ( d ) ) {\textstyle O(n^{\frac {3}{2}}2^{\Theta (d)})} distance evaluations under a distributional assumption on the points. The algorithm uses the triangle inequality to cut down the search space. Meddit leverages a connection of the medoid computation with multi-armed bandits and uses an upper-Confidence-bound type of algorithm to get an algorithm which takes O ( n log ⁡ n ) {\textstyle O(n\log n)} distance evaluations under statistical assumptions on the points. Correlated Sequential Halving also leverages multi-armed bandit techniques, improving upon Meddit. By exploiting the correlation structure in the problem, the algorithm is able to provably yield drastic improvement (usually around 1-2 orders of magnitude) in both number of distance computations needed and wall clock time. == Implementations == An implementation of RAND, TOPRANK, and trimed can be found here. An implementation of Meddit can be found here and here. An implementation of Correlated Sequential Halving can be found here. == Medoids in text and natural language processing (NLP) == Medoids can be applied to various text and NLP tasks to improve the efficiency and accuracy of analyses. By clustering text data based on similarity, medoids can help identify representative examples within the dataset, leading to better understanding and interpretation of the data. === Text clustering === Text clustering is the process of grouping similar text or documents together based on their content. Medoid-based clustering algorithms can be employed to partition large amounts of text into clusters, with each cluster represented by a medoid document. This technique helps in organizing, summarizing, and retrieving information from large collections of documents, such as in search engines, social media analytics and recommendation systems. === Text summarization === Text summarization aims to produce a concise and coherent summary of a larger text by extracting the most important and relevant information. Medoid-based clustering can be used to identify the most representative sentences in a document or a group of documents, which can then be combined to create a summary. This approach is especially useful for extractive summarization tasks, where the goal is to generate a summary by selecting the most relevant sentences from the original text. === Sentiment analysis === Sentiment analysis involves determining the sentiment or emotion expressed in a piece of text, such as positive, negative, or neutral. Medoid-based clustering can be applied to group text data based on similar sentiment patterns. By analyzing the medoid of each cluster, researchers can gain insights into the predominant sentiment of the cluster, helping in tasks such as opinion mining, customer feedback analysis, and social media monitoring. === Topic modeling === Topic modeling is a technique used to discover abstract topics that occur in a collection of documents. Medoid-based clustering can be applied to group documents with similar themes or topics. By analyzing the medoids of these clusters, researchers can gain an understanding of the underlying topics in the text corpus, facilitating tasks such as document categorization, trend analysis, and content recommendation. === Techniques for measuring text similarity in medoid-based clustering === When applying medoid-based clustering to text data, it is essential to choose an appropriate similarity measure to compare documents effectively. Each technique has its advantages and limitations, and the choice of the similarity measure should be based on the specific requirements and characteristics of the text data being analyzed. The following are common techniques for measuring text similarity in medoid-based clustering: ==== Cosine similarity ==== Cosine similarity is a widely used measure to compare the similarity between two pieces of text. It calculates the cosine of the angle between two document vectors in a high-dimensional space. Cosine similarity ranges between -1 and 1, where a value closer to 1 indicates higher similarity, and a value closer to -1 indicates lower similarity. By visualizing two lines originating from the origin and extending to the respective points of interest, and then measuring the angle between these lines, one can determine the similarity between the associated points. Cosine similarity is less affected by document length, so it may be better at producing medoids that are representative of the content of a cluster instead of the lengt

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  • Silhouette (clustering)

    Silhouette (clustering)

    Silhouette is a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette value ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. A clustering with an average silhouette width of over 0.7 is considered to be "strong", a value over 0.5 "reasonable", and over 0.25 "weak". However, with an increasing dimensionality of the data, it becomes difficult to achieve such high values because of the curse of dimensionality, as the distances become more similar. The silhouette score is specialized for measuring cluster quality when the clusters are convex-shaped, and may not perform well if the data clusters have irregular shapes or are of varying sizes. The silhouette value can be calculated with any distance metric, such as Euclidean distance or Manhattan distance. == Definition == Assume the data have been clustered via any technique, such as k-medoids or k-means, into k {\displaystyle k} clusters. For data point i ∈ C i {\displaystyle i\in C_{i}} (data point i {\displaystyle i} in the cluster C i {\displaystyle C_{i}} ), calculate a ( i ) {\displaystyle a(i)} , the average distance that i {\displaystyle i} is from all other points in that cluster: a ( i ) = 1 | C i | − 1 ∑ j ∈ C i , i ≠ j d ( i , j ) {\displaystyle a(i)={\frac {1}{|C_{i}|-1}}\sum _{j\in C_{i},i\neq j}d(i,j)} where | C i | {\displaystyle |C_{i}|} is the number of points belonging to cluster C i {\displaystyle C_{i}} , and d ( i , j ) {\displaystyle d(i,j)} is the distance between data points i {\displaystyle i} and j {\displaystyle j} in the cluster C i {\displaystyle C_{i}} (we divide by | C i | − 1 {\displaystyle |C_{i}|-1} because the distance d ( i , i ) {\displaystyle d(i,i)} is not included in the sum). a ( i ) {\displaystyle a(i)} can be interpreted as a measure of how well i {\displaystyle i} is assigned to its cluster (the smaller the value, the better the assignment). We then define the mean dissimilarity of point i {\displaystyle i} to some cluster C j {\displaystyle C_{j}} as the mean of the distance from i {\displaystyle i} to all points in C j {\displaystyle C_{j}} (where C j ≠ C i {\displaystyle C_{j}\neq C_{i}} ). For each data point i ∈ C i {\displaystyle i\in C_{i}} , we now define b ( i ) {\displaystyle b(i)} as the average distance between i {\displaystyle i} and the points in the closest cluster (hence: "min") that i {\displaystyle i} does not belong to: b ( i ) = min j ≠ i 1 | C j | ∑ l ∈ C j d ( i , l ) {\displaystyle b(i)=\min _{j\neq i}{\frac {1}{|C_{j}|}}\sum _{l\in C_{j}}d(i,l)} The cluster with the smallest mean dissimilarity is said to be the "neighboring cluster" of i {\displaystyle i} because it is the next best fit cluster for point i {\displaystyle i} . We now define a silhouette (value) of one data point i {\displaystyle i} s ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } {\displaystyle s(i)={\frac {b(i)-a(i)}{\max\{a(i),b(i)\}}}} , if | C i | > 1 {\displaystyle |C_{i}|>1} and s ( i ) = 0 {\displaystyle s(i)=0} , if | C i | = 1 {\displaystyle |C_{i}|=1} , which can also be written as s ( i ) = { 1 − a ( i ) b ( i ) , if a ( i ) < b ( i ) 0 , if a ( i ) = b ( i ) b ( i ) a ( i ) − 1 , if a ( i ) > b ( i ) {\displaystyle s(i)={\begin{cases}1-{\frac {a(i)}{b(i)}},&{\mbox{ if }}a(i)b(i)\\\end{cases}}} From the above definition, s ( i ) {\displaystyle s(i)} is bounded to the interval [ − 1 , 1 ] {\displaystyle [-1,1]} , i.e. − 1 ≤ s ( i ) ≤ 1. {\displaystyle -1\leq s(i)\leq 1.} Note that a ( i ) {\displaystyle a(i)} is not clearly defined for clusters with size = 1, in which case we set s ( i ) = 0 {\displaystyle s(i)=0} . This choice is arbitrary, but neutral in the sense that it is at the midpoint of the bounds, -1 and 1. For s ( i ) {\displaystyle s(i)} to be close to 1 we require a ( i ) ≪ b ( i ) {\displaystyle a(i)\ll b(i)} . As a ( i ) {\displaystyle a(i)} is a measure of how dissimilar i {\displaystyle i} is to its own cluster, a small value means it is well matched. Furthermore, a large b ( i ) {\displaystyle b(i)} implies that i {\displaystyle i} is badly matched to its neighbouring cluster. Thus an s ( i ) {\displaystyle s(i)} close to 1 means that the data is appropriately clustered. If s ( i ) {\displaystyle s(i)} is close to -1, then by the same logic we see that i {\displaystyle i} would be more appropriate if it was clustered in its neighbouring cluster. An s ( i ) {\displaystyle s(i)} near zero means that the datum is on the border of two natural clusters. The mean s ( i ) {\displaystyle s(i)} over all points of a cluster is a measure of how tightly grouped all the points in the cluster are. Thus the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset is a measure of how appropriately the data have been clustered. If there are too many or too few clusters, as may occur when a poor choice of k {\displaystyle k} is used in the clustering algorithm (e.g., k-means), some of the clusters will typically display much narrower silhouettes than the rest. Thus silhouette plots and means may be used to determine the natural number of clusters within a dataset. One can also increase the likelihood of the silhouette being maximized at the correct number of clusters by re-scaling the data using feature weights that are cluster specific. Kaufman et al. introduced the term silhouette coefficient for the maximum value of the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset, i.e., S C = max k s ~ ( k ) , {\displaystyle SC=\max _{k}{\tilde {s}}\left(k\right),} where s ~ ( k ) {\displaystyle {\tilde {s}}\left(k\right)} represents the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset for a specific number of clusters k {\displaystyle k} . The silhouette coefficient describes the best possible clustering possible for a given number of clusters, as measured by the highest average silhouette score for all points in the dataset. == Simplified and medoid silhouette == Computing the silhouette coefficient needs all O ( N 2 ) {\displaystyle {\mathcal {O}}(N^{2})} pairwise distances, making this evaluation much more costly than clustering with k-means. For a clustering with centers μ C I {\displaystyle \mu _{C_{I}}} for each cluster C I {\displaystyle C_{I}} , we can use the following simplified Silhouette for each point i ∈ C I {\displaystyle i\in C_{I}} instead, which can be computed using only O ( N k ) {\displaystyle {\mathcal {O}}(Nk)} distances: a ′ ( i ) = d ( i , μ C I ) {\displaystyle a'(i)=d(i,\mu _{C_{I}})} and b ′ ( i ) = min C J ≠ C I d ( i , μ C J ) {\displaystyle b'(i)=\min _{C_{J}\neq C_{I}}d(i,\mu _{C_{J}})} , which has the additional benefit that a ′ ( i ) {\displaystyle a'(i)} is always defined, then define accordingly the simplified silhouette and simplified silhouette coefficient s ′ ( i ) = b ′ ( i ) − a ′ ( i ) max { a ′ ( i ) , b ′ ( i ) } {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{\max\{a'(i),b'(i)\}}}} S C ′ = max k 1 N ∑ i s ′ ( i ) {\displaystyle SC'=\max _{k}{\frac {1}{N}}\sum _{i}s'\left(i\right)} . If the cluster centers are medoids (as in k-medoids clustering) instead of arithmetic means (as in k-means clustering), this is also called the medoid-based silhouette or medoid silhouette. If every object is assigned to the nearest medoid (as in k-medoids clustering), we know that a ′ ( i ) ≤ b ′ ( i ) {\displaystyle a'(i)\leq b'(i)} , and hence s ′ ( i ) = b ′ ( i ) − a ′ ( i ) b ′ ( i ) = 1 − a ′ ( i ) b ′ ( i ) {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{b'(i)}}=1-{\frac {a'(i)}{b'(i)}}} . == Silhouette clustering == Instead of using the average silhouette to evaluate a clustering obtained from, e.g., k-medoids or k-means, we can try to directly find a solution that maximizes the Silhouette. We do not have a closed form solution to maximize this, but it will usually be best to assign points to the nearest cluster as done by these methods. Van der Laan et al. proposed to adapt the standard algorithm for k-medoids, PAM, for this purpose and call this algorithm PAMSIL: Choose initial medoids by using PAM Compute the average silhouette of this initial solution For each pair of a medoid m and a non-medoid x swap m and x compute the average silhouette of the resulting solution remember the best swap un-swap m and x for the next iteration Perform the best swap and return to

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