AI App Developer Free

AI App Developer Free — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • Transfer learning

    Transfer learning

    Transfer learning (TL) is a technique in machine learning (ML) in which knowledge learned from a task is re-used in order to boost performance on a related task. For example, for image classification, knowledge gained while learning to recognize cars could be applied when trying to recognize trucks. This topic is related to the psychological literature on transfer of learning, although practical ties between the two fields are limited. Reusing or transferring information from previously learned tasks to new tasks has the potential to significantly improve learning efficiency. Since transfer learning makes use of training with multiple objective functions it is related to cost-sensitive machine learning and multi-objective optimization. == History == In 1976, Bozinovski and Fulgosi published a paper addressing transfer learning in neural network training. The paper gives a mathematical and geometrical model of the topic. In 1981, a report considered the application of transfer learning to a dataset of images representing letters of computer terminals, experimentally demonstrating positive and negative transfer learning. In 1992, Lorien Pratt formulated the discriminability-based transfer (DBT) algorithm. By 1998, the field had advanced to include multi-task learning, along with more formal theoretical foundations. Influential publications on transfer learning include the book Learning to Learn in 1998, a 2009 survey and a 2019 survey. Ng said in his NIPS 2016 tutorial that TL would become the next driver of machine learning commercial success after supervised learning. In the 2020 paper, "Rethinking Pre-Training and self-training", Zoph et al. reported that pre-training can hurt accuracy, and advocate self-training instead. == Definition == The definition of transfer learning is given in terms of domains and tasks. A domain D {\displaystyle {\mathcal {D}}} consists of: a feature space X {\displaystyle {\mathcal {X}}} and a marginal probability distribution P ( X ) {\displaystyle P(X)} , where X = { x 1 , . . . , x n } ∈ X {\displaystyle X=\{x_{1},...,x_{n}\}\in {\mathcal {X}}} . Given a specific domain, D = { X , P ( X ) } {\displaystyle {\mathcal {D}}=\{{\mathcal {X}},P(X)\}} , a task consists of two components: a label space Y {\displaystyle {\mathcal {Y}}} and an objective predictive function f : X → Y {\displaystyle f:{\mathcal {X}}\rightarrow {\mathcal {Y}}} . The function f {\displaystyle f} is used to predict the corresponding label f ( x ) {\displaystyle f(x)} of a new instance x {\displaystyle x} . This task, denoted by T = { Y , f ( x ) } {\displaystyle {\mathcal {T}}=\{{\mathcal {Y}},f(x)\}} , is learned from the training data consisting of pairs { x i , y i } {\displaystyle \{x_{i},y_{i}\}} , where x i ∈ X {\displaystyle x_{i}\in {\mathcal {X}}} and y i ∈ Y {\displaystyle y_{i}\in {\mathcal {Y}}} . Given a source domain D S {\displaystyle {\mathcal {D}}_{S}} and learning task T S {\displaystyle {\mathcal {T}}_{S}} , a target domain D T {\displaystyle {\mathcal {D}}_{T}} and learning task T T {\displaystyle {\mathcal {T}}_{T}} , where D S ≠ D T {\displaystyle {\mathcal {D}}_{S}\neq {\mathcal {D}}_{T}} , or T S ≠ T T {\displaystyle {\mathcal {T}}_{S}\neq {\mathcal {T}}_{T}} , transfer learning aims to help improve the learning of the target predictive function f T ( ⋅ ) {\displaystyle f_{T}(\cdot )} in D T {\displaystyle {\mathcal {D}}_{T}} using the knowledge in D S {\displaystyle {\mathcal {D}}_{S}} and T S {\displaystyle {\mathcal {T}}_{S}} . == Applications == Algorithms for transfer learning are available in Markov logic networks and Bayesian networks. Transfer learning has been applied to cancer subtype discovery, building utilization, general game playing, text classification, digit recognition, medical imaging and spam filtering. In 2020, it was discovered that, due to their similar physical natures, transfer learning is possible between electromyographic (EMG) signals from the muscles and classifying the behaviors of electroencephalographic (EEG) brainwaves, from the gesture recognition domain to the mental state recognition domain. It was noted that this relationship worked in both directions, showing that electroencephalographic can likewise be used to classify EMG. The experiments noted that the accuracy of neural networks and convolutional neural networks were improved through transfer learning both prior to any learning (compared to standard random weight distribution) and at the end of the learning process (asymptote). That is, results are improved by exposure to another domain. Moreover, the end-user of a pre-trained model can change the structure of fully-connected layers to improve performance.

    Read more →
  • Mutation (evolutionary algorithm)

    Mutation (evolutionary algorithm)

    Mutation is a genetic operator used to maintain genetic diversity of the chromosomes of a population of an evolutionary algorithm (EA), including genetic algorithms in particular. It is analogous to biological mutation. The classic example of a mutation operator of a binary coded genetic algorithm (GA) involves a probability that an arbitrary bit in a genetic sequence will be flipped from its original state. A common method of implementing the mutation operator involves generating a random variable for each bit in a sequence. This random variable tells whether or not a particular bit will be flipped. This mutation procedure, based on the biological point mutation, is called single point mutation. Other types of mutation operators are commonly used for representations other than binary, such as floating-point encodings or representations for combinatorial problems. The purpose of mutation in EAs is to introduce diversity into the sampled population. Mutation operators are used in an attempt to avoid local minima by preventing the population of chromosomes from becoming too similar to each other, thus slowing or even stopping convergence to the global optimum. This reasoning also leads most EAs to avoid only taking the fittest of the population in generating the next generation, but rather selecting a random (or semi-random) set with a weighting toward those that are fitter. The following requirements apply to all mutation operators used in an EA: every point in the search space must be reachable by one or more mutations. there must be no preference for parts or directions in the search space (no drift). small mutations should be more probable than large ones. For different genome types, different mutation types are suitable. Some mutations are Gaussian, Uniform, Zigzag, Scramble, Insertion, Inversion, Swap, and so on. An overview and more operators than those presented below can be found in the introductory book by Eiben and Smith or in. == Bit string mutation == The mutation of bit strings ensue through bit flips at random positions. Example: The probability of a mutation of a bit is 1 l {\displaystyle {\frac {1}{l}}} , where l {\displaystyle l} is the length of the binary vector. Thus, a mutation rate of 1 {\displaystyle 1} per mutation and individual selected for mutation is reached. == Mutation of real numbers == Many EAs, such as the evolution strategy or the real-coded genetic algorithms, work with real numbers instead of bit strings. This is due to the good experiences that have been made with this type of coding. The value of a real-valued gene can either be changed or redetermined. A mutation that implements the latter should only ever be used in conjunction with the value-changing mutations and then only with comparatively low probability, as it can lead to large changes. In practical applications, the respective value range of the decision variables to be changed of the optimisation problem to be solved is usually limited. Accordingly, the values of the associated genes are each restricted to an interval [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} . Mutations may or may not take these restrictions into account. In the latter case, suitable post-treatment is then required as described below. === Mutation without consideration of restrictions === A real number x {\displaystyle x} can be mutated using normal distribution N ( 0 , σ ) {\displaystyle {\mathcal {N}}(0,\sigma )} by adding the generated random value to the old value of the gene, resulting in the mutated value x ′ {\displaystyle x'} : x ′ = x + N ( 0 , σ ) {\displaystyle x'=x+{\mathcal {N}}(0,\sigma )} In the case of genes with a restricted range of values, it is a good idea to choose the step size of the mutation σ {\displaystyle \sigma } so that it reasonably fits the range [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} of the gene to be changed, e.g.: σ = x max − x min 6 {\displaystyle \sigma ={\frac {x_{\text{max}}-x_{\text{min}}}{6}}} The step size can also be adjusted to the smaller permissible change range depending on the current value. In any case, however, it is likely that the new value x ′ {\displaystyle x'} of the gene will be outside the permissible range of values. Such a case must be considered a lethal mutation, since the obvious repair by using the respective violated limit as the new value of the gene would lead to a drift. This is because the limit value would then be selected with the entire probability of the values beyond the limit of the value range. The evolution strategy works with real numbers and mutation based on normal distribution. The step sizes are part of the chromosome and are subject to evolution together with the actual decision variables. === Mutation with consideration of restrictions === One possible form of changing the value of a gene while taking its value range [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} into account is the mutation relative parameter change of the evolutionary algorithm GLEAM (General Learning Evolutionary Algorithm and Method), in which, as with the mutation presented earlier, small changes are more likely than large ones. First, an equally distributed decision is made as to whether the current value x {\displaystyle x} should be increased or decreased and then the corresponding total change interval is determined. Without loss of generality, an increase is assumed for the explanation and the total change interval is then [ x , x max ] {\displaystyle [x,x_{\max }]} . It is divided into k {\displaystyle k} sub-areas of equal size with the width δ {\displaystyle \delta } , from which k {\displaystyle k} sub-change intervals of different size are formed: i {\displaystyle i} -th sub-change interval: [ x , x + δ ⋅ i ] {\displaystyle [x,x+\delta \cdot i]} with δ = ( x max − x ) k {\displaystyle \delta ={\frac {(x_{\text{max}}-x)}{k}}} and i = 1 , … , k {\displaystyle i=1,\dots ,k} Subsequently, one of the k {\displaystyle k} sub-change intervals is selected in equal distribution and a random number, also equally distributed, is drawn from it as the new value x ′ {\displaystyle x'} of the gene. The resulting summed probabilities of the sub-change intervals result in the probability distribution of the k {\displaystyle k} sub-areas shown in the adjacent figure for the exemplary case of k = 10 {\displaystyle k=10} . This is not a normal distribution as before, but this distribution also clearly favours small changes over larger ones. This mutation for larger values of k {\displaystyle k} , such as 10, is less well suited for tasks where the optimum lies on one of the value range boundaries. This can be remedied by significantly reducing k {\displaystyle k} when a gene value approaches its limits very closely. === Common properties === For both mutation operators for real-valued numbers, the probability of an increase and decrease is independent of the current value and is 50% in each case. In addition, small changes are considerably more likely than large ones. For mixed-integer optimization problems, rounding is usually used. == Mutation of permutations == Mutations of permutations are specially designed for genomes that are themselves permutations of a set. These are often used to solve combinatorial tasks. In the two mutations presented, parts of the genome are rotated or inverted. === Rotation to the right === The presentation of the procedure is illustrated by an example on the right: === Inversion === The presentation of the procedure is illustrated by an example on the right: === Variants with preference for smaller changes === The requirement raised at the beginning for mutations, according to which small changes should be more probable than large ones, is only inadequately fulfilled by the two permutation mutations presented, since the lengths of the partial lists and the number of shift positions are determined in an equally distributed manner. However, the longer the partial list and the shift, the greater the change in gene order. This can be remedied by the following modifications. The end index j {\displaystyle j} of the partial lists is determined as the distance d {\displaystyle d} to the start index i {\displaystyle i} : j = ( i + d ) mod | P 0 | {\displaystyle j=(i+d){\bmod {\left|P_{0}\right|}}} where d {\displaystyle d} is determined randomly according to one of the two procedures for the mutation of real numbers from the interval [ 0 , | P 0 | − 1 ] {\displaystyle \left[0,\left|P_{0}\right|-1\right]} and rounded. For the rotation, k {\displaystyle k} is determined similarly to the distance d {\displaystyle d} , but the value 0 {\displaystyle 0} is forbidden. For the inversion, note that i ≠ j {\displaystyle i\neq j} must hold, so for d {\displaystyle d} the value 0 {\displaystyle 0} must be excluded.

    Read more →
  • NETtalk (artificial neural network)

    NETtalk (artificial neural network)

    NETtalk is an artificial neural network that learns to pronounce written English text by supervised learning. It takes English text as input, and produces a matching phonetic transcriptions as output. It is the result of research carried out in the mid-1980s by Terrence Sejnowski and Charles Rosenberg. The intent behind NETtalk was to construct simplified models that might shed light on the complexity of learning human level cognitive tasks, and their implementation as a connectionist model that could also learn to perform a comparable task. The authors trained it by backpropagation. The network was trained on a large amount of English words and their corresponding pronunciations, and is able to generate pronunciations for unseen words with a high level of accuracy. The output of the network was a stream of phonemes, which fed into DECtalk to produce audible speech. It achieved popular success, appearing on the Today show. From the point of view of modeling human cognition, NETtalk does not specifically model the image processing stages and letter recognition of the visual cortex. Rather, it assumes that the letters have been pre-classified and recognized. It is NETtalk's task to learn proper associations between the correct pronunciation with a given sequence of letters based on the context in which the letters appear. A similar architecture was subsequently used for the opposite task, that of converting continuous speech signal to a phoneme sequence. == Training == The training dataset was a 20,008-word subset of the Brown Corpus, with manually annotated phoneme and stress for each letter. The development process was described in a 1993 interview. It took three months -- 250 person-hours -- to create the training dataset, but only a few days to train the network. After it was run successfully on this, the authors tried it on a phonological transcription of an interview with a young Latino boy from a barrio in Los Angeles. This resulted in a network that reproduced his Spanish accent. The original NETtalk was implemented on a Ridge 32, which took 0.275 seconds per learning step (one forward and one backward pass). Training NETtalk became a benchmark to test for the efficiency of backpropagation programs. For example, an implementation on Connection Machine-1 (with 16384 processors) ran at 52x speedup. An implementation on a 10-cell Warp ran at 340x speedup. The following table compiles the benchmark scores as of 1988. Speed is measured in "millions of connections per second" (MCPS). For example, the original NETtalk on Ridge 32 took 0.275 seconds per forward-backward pass, giving 18629 / 10 6 0.275 = 0.068 {\displaystyle {\frac {18629/10^{6}}{0.275}}=0.068} MCPS. Relative times are normalized to the MicroVax. == Architecture == The network had three layers and 18,629 adjustable weights, large by the standards of 1986. There were worries that it would overfit the dataset, but it was trained successfully. The input of the network has 203 units, divided into 7 groups of 29 units each. Each group is a one-hot encoding of one character. There are 29 possible characters: 26 letters, comma, period, and word boundary (whitespace). To produce the pronunciation of a single character, the network takes the character itself, as well as 3 characters before and 3 characters after it. The hidden layer has 80 units. The output has 26 units. 21 units encode for articulatory features (point of articulation, voicing, vowel height, etc.) of phonemes, and 5 units encode for stress and syllable boundaries. Sejnowski studied the learned representation in the network, and found that phonemes that sound similar are clustered together in representation space. The output of the network degrades, but remains understandable, when some hidden neurons are removed.

    Read more →
  • Julia (programming language)

    Julia (programming language)

    Julia is a dynamic general-purpose programming language. As a high-level language, distinctive aspects of Julia's design include a type system with parametric polymorphism, the use of multiple dispatch as a core programming paradigm, just-in-time compilation and a parallel garbage collection implementation. Notably, Julia does not support classes with encapsulated methods but instead relies on the types of all of a function's arguments to determine which method will be called. By default, Julia is run similarly to scripting languages, using its runtime, and allows for interactions, but Julia programs can also be compiled to small binary standalone executables (or to small libraries for e.g. Python), with e.g. the JuliaC.jl compiler. Julia programs can reuse libraries from other languages, and vice versa. Julia has interoperability with C, C++, Fortran, Rust, Python, and R. Additionally, some Julia packages have bindings to be used from Python and R as libraries. Julia is supported by programmer tools like IDEs (see below) and by notebooks like Pluto.jl, Jupyter, and since 2025, Google Colab officially supports Julia natively. Julia is sometimes used in embedded systems (e.g. has been used in a satellite in space on a Raspberry Pi Compute Module 4; 64-bit Pis work best with Julia, and Julia is supported in Raspbian). == History == Work on Julia began in 2009, when Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman set out to create a free language that was both high-level and fast. On 14 February 2012, the team launched a website with a blog post explaining the language's mission. In an interview with InfoWorld in April 2012, Karpinski said about the name of the language, Julia: "There's no good reason, really. It just seemed like a pretty name." Bezanson said he chose the name on the recommendation of a friend, then years later wrote: Maybe julia stands for "Jeff's uncommon lisp is automated"? Julia's syntax is stable, since version 1.0 in 2018, and Julia has a backward compatibility guarantee for 1.x and also a stability promise for the documented (stable) API, while in the years before in the early development prior to 0.7 the syntax (and semantics) was changed in new versions. All of the (registered package) ecosystem uses the new and improved syntax, and in most cases relies on new APIs that have been added regularly, and in some cases minor additional syntax added in a forward compatible way e.g. in Julia 1.7. In the 10 years since the 2012 launch of pre-1.0 Julia, the community has grown. The Julia package ecosystem has over 11.8 million lines of code (including docs and tests). The JuliaCon academic conference for Julia users and developers has been held annually since 2014 with JuliaCon2020 welcoming over 28,900 unique viewers, and then JuliaCon2021 breaking all previous records (with more than 300 JuliaCon2021 presentations available for free on YouTube, up from 162 the year before), and 43,000 unique viewers during the conference. Three of the Julia co-creators are the recipients of the 2019 James H. Wilkinson Prize for Numerical Software (awarded every four years) "for the creation of Julia, an innovative environment for the creation of high-performance tools that enable the analysis and solution of computational science problems." Also, Alan Edelman, professor of applied mathematics at MIT, has been selected to receive the 2019 IEEE Computer Society Sidney Fernbach Award "for outstanding breakthroughs in high-performance computing, linear algebra, and computational science and for contributions to the Julia programming language." Version 0.3 was released in August 2014. Both Julia 0.7 and version 1.0 were released on 8 August 2018. Julia 1.4 added syntax for generic array indexing to handle e.g. 0-based arrays. The memory model was also changed. Julia 1.5 released in August 2020 added record and replay debugging support, for Mozilla's rr tool. The release changed the behavior in the REPL (to soft scope) to the one used in Jupyter, but keeps full compatible with non-REPL code (that retains hard scope). Julia 1.6 was the largest release since 1.0, and it was the long-term support (LTS) version for the longest time. Since Julia 1.7 development is back to time-based releases, and it was released in November 2021 with e.g. a new default random-number generator and Julia 1.7.3 fixed at least one security issue. Julia 1.8 added options for hiding source code when compiling Julia source code to executables. Julia 1.9 has added the ability to precompile packages to native machine code, done automatically; to improve precompilation of packages a new package PrecompileTools.jl was introduced, for use by package developers. Julia 1.10 was released on 25 December 2023 with new features such as parallel garbage collection. Julia 1.11 was released on 7 October 2024, and with it 1.10.5 became the next long-term support (LTS) version (i.e. those became the only two supported versions), since replaced by 1.10.10 released on 27 June, and 1.6 is no longer an LTS version. Julia 1.11 adds e.g. the new public keyword to signal safe public API (Julia users are advised to use such API, not internals, of Julia or packages, and package authors advised to use the keyword, generally indirectly, e.g. prefixed with the @compat macro, from Compat.jl, to also support older Julia versions, at least the LTS version). Julia 1.12 was released on 7 October 2025 (and 1.12.5 on 9 February 2026), and with it a JuliaC.jl package including the juliac compiler that works with it, for making rather small binary executables (much smaller than was possible before; through the use of new so-called trimming feature). Julia 1.10 LTS is an officially still-supported branch, but the 1.11 branch has also been maintained after 1.12 release, with 1.11.8 released and then 1.11.9 released on 8 February 2026. === JuliaCon === Since 2014, the Julia Community has hosted an annual Julia Conference focused on developers and users. The first JuliaCon took place in Chicago and kickstarted the annual occurrence of the conference. Since 2014, the conference has taken place across a number of locations including MIT and the University of Maryland, Baltimore. The event audience has grown from a few dozen people to over 28,900 unique attendees during JuliaCon 2020, which took place virtually. JuliaCon 2021 also took place virtually with keynote addresses from professors William Kahan, the primary architect of the IEEE 754 floating-point standard (which virtually all CPUs and languages, including Julia, use), Jan Vitek, Xiaoye Sherry Li, and Soumith Chintala, a co-creator of PyTorch. JuliaCon grew to 43,000 unique attendees and more than 300 presentations (still freely accessible, plus for older years). JuliaCon 2022 will also be virtual held between July 27 and July 29, 2022, for the first time in several languages, not just in English. === Sponsors === The Julia language became a NumFOCUS fiscally sponsored project in 2014 in an effort to ensure the project's long-term sustainability. Jeremy Kepner at MIT Lincoln Laboratory was the founding sponsor of the Julia project in its early days. In addition, funds from the Gordon and Betty Moore Foundation, the Alfred P. Sloan Foundation, Intel, and agencies such as NSF, DARPA, NIH, NASA, and FAA have been essential to the development of Julia. Mozilla, the maker of Firefox web browser, with its research grants for H1 2019, sponsored "a member of the official Julia team" for the project "Bringing Julia to the Browser", meaning to Firefox and other web browsers. The Julia language is also supported by individual donors on GitHub. === The Julia company === JuliaHub, Inc. was founded in 2015 as Julia Computing, Inc. by Viral B. Shah, Deepak Vinchhi, Alan Edelman, Jeff Bezanson, Stefan Karpinski and Keno Fischer. In June 2017, Julia Computing raised US$4.6 million in seed funding from General Catalyst and Founder Collective, the same month was "granted $910,000 by the Alfred P. Sloan Foundation to support open-source Julia development, including $160,000 to promote diversity in the Julia community", and in December 2019 the company got $1.1 million funding from the US government to "develop a neural component machine learning tool to reduce the total energy consumption of heating, ventilation, and air conditioning (HVAC) systems in buildings". In July 2021, Julia Computing announced they raised a $24 million Series A round led by Dorilton Ventures, which also owns Formula One team Williams Racing, that partnered with Julia Computing. Williams' Commercial Director said: "Investing in companies building best-in-class cloud technology is a strategic focus for Dorilton and Julia's versatile platform, with revolutionary capabilities in simulation and modelling, is hugely relevant to our business. We look forward to embedding Julia Computing in the world's most technologically advanced sport". In June 2023, JuliaHub received (again, now

    Read more →
  • DexNet

    DexNet

    Dex-net is a robotic. It uses a Grasp Quality Convolutional Neural Network to learn how to grasp unusually shaped objects. == History == Dex-net was developed by University of California, Berkeley professor Ken Goldberg and graduate student Jeff Mahler. == Design == Dex-net includes a high-resolution 3-D sensor and two arms, each controlled by a different neural network. One arm is equipped with a conventional robot gripper and another with a suction system. The robot’s software scans an object and then asks both neural networks to decide, on the fly, whether to grab or suck a particular object. It runs on an off-the-shelf industrial machine made by Swiss robotics company ABB. The software learns by attempting to pick up objects in a virtual environment. Dex-Net can generalize from an object it has seen before to a new one. The robot can "nudge" such virtual objects to examine if it is unsure how to grasp them. The trial data set was 6.7 million point clouds, grasps and analytic grasp metrics generated from thousands of 3D models. Grasps are defined as a gripper's planar position, angle and depth relative to an RGB-D sensor. == Mean picks per hour == A metric called mean picks per hour (MPPH) is calculated by multiplying the average time per pick and the average probability of success for a specific set of objects. The new metric allows labs working on picking robots to compare their results. Humans are capable of between 400 and 600 MPPH. In a contest organized by Amazon recently, the best robots were capable of between 70 and 95. Dex-net has achieved 200 to 300.

    Read more →
  • AdaBoost

    AdaBoost

    AdaBoost (short for Adaptive Boosting) is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many types of learning algorithm to improve performance. The output of multiple weak learners is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to converge to a strong learner. Although AdaBoost is typically used to combine weak base learners (such as decision stumps), it has been shown to also effectively combine strong base learners (such as deeper decision trees), producing an even more accurate model. Every learning algorithm tends to suit some problem types better than others, and typically has many different parameters and configurations to adjust before it achieves optimal performance on a dataset. AdaBoost (with decision trees as the weak learners) is often referred to as the best out-of-the-box classifier. When used with decision tree learning, information gathered at each stage of the AdaBoost algorithm about the relative 'hardness' of each training sample is fed into the tree-growing algorithm such that later trees tend to focus on harder-to-classify examples. == Training == AdaBoost refers to a particular method of training a boosted classifier. A boosted classifier is a classifier of the form F T ( x ) = ∑ t = 1 T f t ( x ) {\displaystyle F_{T}(x)=\sum _{t=1}^{T}f_{t}(x)} where each f t {\displaystyle f_{t}} is a weak learner that takes an object x {\displaystyle x} as input and returns a value indicating the class of the object. For example, in the two-class problem, the sign of the weak learner's output identifies the predicted object class and the absolute value gives the confidence in that classification. Each weak learner produces an output hypothesis h {\displaystyle h} which fixes a prediction h ( x i ) {\displaystyle h(x_{i})} for each sample in the training set. At each iteration t {\displaystyle t} , a weak learner is selected and assigned a coefficient α t {\displaystyle \alpha _{t}} such that the total training error E t {\displaystyle E_{t}} of the resulting t {\displaystyle t} -stage boosted classifier is minimized. E t = ∑ i E [ F t − 1 ( x i ) + α t h ( x i ) ] {\displaystyle E_{t}=\sum _{i}E[F_{t-1}(x_{i})+\alpha _{t}h(x_{i})]} Here F t − 1 ( x ) {\displaystyle F_{t-1}(x)} is the boosted classifier that has been built up to the previous stage of training and f t ( x ) = α t h ( x ) {\displaystyle f_{t}(x)=\alpha _{t}h(x)} is the weak learner that is being considered for addition to the final classifier. === Weighting === At each iteration of the training process, a weight w i , t {\displaystyle w_{i,t}} is assigned to each sample in the training set equal to the current error E ( F t − 1 ( x i ) ) {\displaystyle E(F_{t-1}(x_{i}))} on that sample. These weights can be used in the training of the weak learner. For instance, decision trees can be grown which favor the splitting of sets of samples with large weights. == Derivation == This derivation follows Rojas (2009): Suppose we have a data set { ( x 1 , y 1 ) , … , ( x N , y N ) } {\displaystyle \{(x_{1},y_{1}),\ldots ,(x_{N},y_{N})\}} where each item x i {\displaystyle x_{i}} has an associated class y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} , and a set of weak classifiers { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} each of which outputs a classification k j ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{j}(x_{i})\in \{-1,1\}} for each item. After the ( m − 1 ) {\displaystyle (m-1)} -th iteration our boosted classifier is a linear combination of the weak classifiers of the form: C ( m − 1 ) ( x i ) = α 1 k 1 ( x i ) + ⋯ + α m − 1 k m − 1 ( x i ) , {\displaystyle C_{(m-1)}(x_{i})=\alpha _{1}k_{1}(x_{i})+\cdots +\alpha _{m-1}k_{m-1}(x_{i}),} where the class will be the sign of C ( m − 1 ) ( x i ) {\displaystyle C_{(m-1)}(x_{i})} . At the m {\displaystyle m} -th iteration we want to extend this to a better boosted classifier by adding another weak classifier k m {\displaystyle k_{m}} , with another weight α m {\displaystyle \alpha _{m}} : C m ( x i ) = C ( m − 1 ) ( x i ) + α m k m ( x i ) {\displaystyle C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha _{m}k_{m}(x_{i})} So it remains to determine which weak classifier is the best choice for k m {\displaystyle k_{m}} , and what its weight α m {\displaystyle \alpha _{m}} should be. We define the total error E {\displaystyle E} of C m {\displaystyle C_{m}} as the sum of its exponential loss on each data point, given as follows: E = ∑ i = 1 N e − y i C m ( x i ) = ∑ i = 1 N e − y i C ( m − 1 ) ( x i ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}=\sum _{i=1}^{N}e^{-y_{i}C_{(m-1)}(x_{i})}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} Letting w i ( 1 ) = 1 {\displaystyle w_{i}^{(1)}=1} and w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} for m > 1 {\displaystyle m>1} , we have: E = ∑ i = 1 N w i ( m ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} We can split this summation between those data points that are correctly classified by k m {\displaystyle k_{m}} (so y i k m ( x i ) = 1 {\displaystyle y_{i}k_{m}(x_{i})=1} ) and those that are misclassified (so y i k m ( x i ) = − 1 {\displaystyle y_{i}k_{m}(x_{i})=-1} ): E = ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m = ∑ i = 1 N w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) ( e α m − e − α m ) {\displaystyle {\begin{aligned}E&=\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}}\\&=\sum _{i=1}^{N}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}\left(e^{\alpha _{m}}-e^{-\alpha _{m}}\right)\end{aligned}}} Since the only part of the right-hand side of this equation that depends on k m {\displaystyle k_{m}} is ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} , we see that the k m {\displaystyle k_{m}} that minimizes E {\displaystyle E} is the one in the set { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} that minimizes ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} [assuming that α m > 0 {\displaystyle \alpha _{m}>0} ], i.e. the weak classifier with the lowest weighted error (with weights w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} ). To determine the desired weight α m {\displaystyle \alpha _{m}} that minimizes E {\displaystyle E} with the k m {\displaystyle k_{m}} that we just determined, we differentiate: d E d α m = d ( ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m ) d α m {\displaystyle {\frac {dE}{d\alpha _{m}}}={\frac {d(\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}})}{d\alpha _{m}}}} The value of α m {\displaystyle \alpha _{m}} that minimizes the above expression is: α m = 1 2 ln ⁡ ( ∑ y i = k m ( x i ) w i ( m ) ∑ y i ≠ k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}}\right)} We calculate the weighted error rate of the weak classifier to be ϵ m = ∑ y i ≠ k m ( x i ) w i ( m ) ∑ i = 1 N w i ( m ) {\displaystyle \epsilon _{m}={\frac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{i=1}^{N}w_{i}^{(m)}}}} , so it follows that: α m = 1 2 ln ⁡ ( 1 − ϵ m ϵ m ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {1-\epsilon _{m}}{\epsilon _{m}}}\right)} which is the negative logit function multiplied by 0.5. Due to the convexity of E {\displaystyle E} as a function of α m {\displaystyle \alpha _{m}} , this new expression for α m {\displaystyle \alpha _{m}} gives the global minimum of the loss function. Note: This derivation only applies when k m ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{m}(x_{i})\in \{-1,1\}} , though it can be a good starting guess in other cases, such as when the weak learner is biased ( k m ( x ) ∈ { a , b } , a ≠ − b {\displaystyle k_{m}(x)\in \{a,b\},a\neq -b} ), has multiple leaves ( k m ( x ) ∈ { a , b , … , n } {\displaystyle k_{m}(x)\in \{a,b,\dots ,n\}} ) or is some other function k m ( x ) ∈ R {\displaystyle k_{m}(x)\in \mathbb {R} } . Thus we have derived the AdaBoost algorithm: At each

    Read more →
  • Multiple discriminant analysis

    Multiple discriminant analysis

    Multiple Discriminant Analysis (MDA) is a multivariate dimensionality reduction technique. It has been used to predict signals as diverse as neural memory traces and corporate failure. MDA is not directly used to perform classification. It merely supports classification by yielding a compressed signal amenable to classification. The method described in Duda et al. (2001) §3.8.3 projects the multivariate signal down to an M−1 dimensional space where M is the number of categories. MDA is useful because most classifiers are strongly affected by the curse of dimensionality. In other words, when signals are represented in very-high-dimensional spaces, the classifier's performance is catastrophically impaired by the overfitting problem. This problem is reduced by compressing the signal down to a lower-dimensional space as MDA does. MDA has been used to reveal neural codes.

    Read more →
  • Swish function

    Swish function

    The swish function is a family of mathematical function defined as follows: swish β ⁡ ( x ) = x sigmoid ⁡ ( β x ) = x 1 + e − β x . {\displaystyle \operatorname {swish} _{\beta }(x)=x\operatorname {sigmoid} (\beta x)={\frac {x}{1+e^{-\beta x}}}.} where β {\displaystyle \beta } can be constant (usually set to 1) or trainable and "sigmoid" refers to the logistic function. The swish family was designed to smoothly interpolate between a linear function and the Rectified linear unit (ReLU) function. When considering positive values, Swish is a particular case of doubly parameterized sigmoid shrinkage function defined in . Variants of the swish function include Mish. == Special values == For β = 0, the function is linear: f(x) = x/2. For β = 1, the function is the Sigmoid Linear Unit (SiLU). For β = 1.702, the function approximates GeLU. With β → ∞, the function converges to ReLU. Thus, the swish family smoothly interpolates between a linear function and the ReLU function. Since swish β ⁡ ( x ) = swish 1 ⁡ ( β x ) / β {\displaystyle \operatorname {swish} _{\beta }(x)=\operatorname {swish} _{1}(\beta x)/\beta } , all instances of swish have the same shape as the default swish 1 {\displaystyle \operatorname {swish} _{1}} , zoomed by β {\displaystyle \beta } . One usually sets β > 0 {\displaystyle \beta >0} . When β {\displaystyle \beta } is trainable, this constraint can be enforced by β = e b {\displaystyle \beta =e^{b}} , where b {\displaystyle b} is trainable. swish 1 ⁡ ( x ) = x 2 + x 2 4 − x 4 48 + x 6 480 + O ( x 8 ) {\displaystyle \operatorname {swish} _{1}(x)={\frac {x}{2}}+{\frac {x^{2}}{4}}-{\frac {x^{4}}{48}}+{\frac {x^{6}}{480}}+O\left(x^{8}\right)} swish 1 ⁡ ( x ) = x 2 tanh ⁡ ( x 2 ) + x 2 swish 1 ⁡ ( x ) + swish − 1 ⁡ ( x ) = x tanh ⁡ ( x 2 ) swish 1 ⁡ ( x ) − swish − 1 ⁡ ( x ) = x {\displaystyle {\begin{aligned}\operatorname {swish} _{1}(x)&={\frac {x}{2}}\tanh \left({\frac {x}{2}}\right)+{\frac {x}{2}}\\\operatorname {swish} _{1}(x)+\operatorname {swish} _{-1}(x)&=x\tanh \left({\frac {x}{2}}\right)\\\operatorname {swish} _{1}(x)-\operatorname {swish} _{-1}(x)&=x\end{aligned}}} == Derivatives == Because swish β ⁡ ( x ) = swish 1 ⁡ ( β x ) / β {\displaystyle \operatorname {swish} _{\beta }(x)=\operatorname {swish} _{1}(\beta x)/\beta } , it suffices to calculate its derivatives for the default case. swish 1 ′ ⁡ ( x ) = x + sinh ⁡ ( x ) 4 cosh 2 ⁡ ( x 2 ) + 1 2 {\displaystyle \operatorname {swish} _{1}'(x)={\frac {x+\sinh(x)}{4\cosh ^{2}\left({\frac {x}{2}}\right)}}+{\frac {1}{2}}} so swish 1 ′ ⁡ ( x ) − 1 2 {\displaystyle \operatorname {swish} _{1}'(x)-{\frac {1}{2}}} is odd. swish 1 ″ ⁡ ( x ) = 1 − x 2 tanh ⁡ ( x 2 ) 2 cosh 2 ⁡ ( x 2 ) {\displaystyle \operatorname {swish} _{1}''(x)={\frac {1-{\frac {x}{2}}\tanh \left({\frac {x}{2}}\right)}{2\cosh ^{2}\left({\frac {x}{2}}\right)}}} so swish 1 ″ ⁡ ( x ) {\displaystyle \operatorname {swish} _{1}''(x)} is even. == History == SiLU was first proposed alongside the GELU in 2016, then again proposed in 2017 as the Sigmoid-weighted Linear Unit (SiL) in reinforcement learning. The SiLU/SiL was then again proposed as the SWISH over a year after its initial discovery, originally proposed without the learnable parameter β, so that β implicitly equaled 1. The swish paper was then updated to propose the activation with the learnable parameter β. In 2017, after performing analysis on ImageNet data, researchers from Google indicated that using this function as an activation function in artificial neural networks improves the performance, compared to ReLU and sigmoid functions. It is believed that one reason for the improvement is that the swish function helps alleviate the vanishing gradient problem during backpropagation.

    Read more →
  • Esdat

    Esdat

    ESdat is a data management, analysis and reporting software for environmental and groundwater data, developed by EarthScience Information Systems (EScIS). It is used to manage many types of environmental data including laboratory chemistry (analytical results, QA data, lab sample planning, and electronic Chain of Custody), field chemistry (water, gas, and soil), hydrogeological data (groundwater, borehole and well construction, lithological, geotechnical and stratigraphic, and LNAPL), meteorological data (rain, wind, and temperature), emission data (dust deposition, HiVol, air quality, and noise) and logger data. Data can be compared against environmental standards or site-specific trigger levels to generate exceedence tables, time series graphs, maps, statistics, and other outputs. ESdat integrates with Power BI and ArcGIS and data can also be exported in a range of other database formats, including USEPA Regions 2,4 & 5, and NYS DEC. ESdat is used by environmental consultants, government, mining and industry for validation, interrogation, and reporting of data derived from complex environmental programs, such as contaminated sites, groundwater investigations, and regulatory compliance for landfills or mining operations.

    Read more →
  • Preference regression

    Preference regression

    Preference regression is a statistical technique used by marketers to determine consumers’ preferred core benefits. It usually supplements product positioning techniques like multi dimensional scaling or factor analysis and is used to create ideal vectors on perceptual maps. == Application == Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the survey. The dependent variable is the preference datum. Like all regression methods, the computer fits weights to best predict data. The resultant regression line is referred to as an ideal vector because the slope of the vector is the ratio of the preferences for the two dimensions. If all the data is used in the regression, the program will derive a single equation and hence a single ideal vector. This tends to be a blunt instrument so researchers refine the process with cluster analysis. This creates clusters that reflect market segments. Separate preference regressions are then done on the data within each segment. This provides an ideal vector for each segment. == Alternative methods == Self-stated importance method is an alternative method in which direct survey data is used to determine the weightings rather than statistical imputations. A third method is conjoint analysis in which an additive method is used.

    Read more →
  • Word2vec

    Word2vec

    Word2vec is a technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the word based on the surrounding words. The word2vec algorithm estimates these representations by modeling text in a large corpus. Once trained, such a model can detect synonymous words or suggest additional words for a partial sentence. Word2vec was developed by Tomáš Mikolov, Kai Chen, Greg Corrado, Ilya Sutskever and Jeff Dean at Google, and published in 2013. Word2vec represents a word as a high-dimension vector of numbers which capture relationships between words. In particular, words which appear in similar contexts are mapped to vectors which are nearby as measured by cosine similarity. This indicates the level of semantic similarity between the words, so for example the vectors for walk and ran are nearby, as are those for "but" and "however", and "Berlin" and "Germany". == Approach == Word2vec is a group of related models that are used to produce word embeddings. These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words. Word2vec takes as its input a large corpus of text and produces a mapping of the set of words to a vector space, typically of several hundred dimensions, with each unique word in the corpus being assigned a vector in the space. Word2vec can use either of two model architectures to produce these distributed representations of words: continuous bag of words (CBOW) or continuously sliding skip-gram. In both architectures, word2vec considers both individual words and a sliding context window as it iterates over the corpus. The CBOW can be viewed as a 'fill in the blank' task, where the word embedding represents the way the word influences the relative probabilities of other words in the context window. Words which are semantically similar should influence these probabilities in similar ways, because semantically similar words should be used in similar contexts. The order of context words does not influence prediction (bag of words assumption). In the continuous skip-gram architecture, the model uses the current word to predict the surrounding window of context words. The skip-gram architecture weighs nearby context words more heavily than more distant context words. According to the authors' note, CBOW is faster while skip-gram does a better job for infrequent words. After the model is trained, the learned word embeddings are positioned in the vector space such that words that share common contexts in the corpus — that is, words that are semantically and syntactically similar — are located close to one another in the space. More dissimilar words are located farther from one another in the space. == Mathematical details == This section is based on expositions. A corpus is a sequence of words. Both CBOW and skip-gram are methods to learn one vector per word appearing in the corpus. Let V {\displaystyle V} ("vocabulary") be the set of all words appearing in the corpus C {\displaystyle C} . Our goal is to learn one vector v w ∈ R d {\displaystyle v_{w}\in \mathbb {R} ^{d}} for each word w ∈ V {\displaystyle w\in V} . The idea of skip-gram is that the vector of a word should be close to the vector of each of its neighbors. The idea of CBOW is that the vector-sum of a word's neighbors should be close to the vector of the word. === Continuous bag-of-words (CBOW) === The idea of CBOW is to represent each word with a vector, such that it is possible to predict a word using the sum of the vectors of its neighbors. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i + j : j ∈ N ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i+j}\colon j\in N)} where N {\displaystyle N} is a set of (non-zero) indices representing the relative locations of nearby words considered to be in w i {\displaystyle w_{i}} 's neighborhood. For example, if we want each word in the corpus to be predicted by every other word in a small span of 4 words. The set of relative indexes of neighbor words will be: N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} , and the objective is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i − 2 , w i − 1 , w i + 1 , w i + 2 ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i-2},w_{i-1},w_{i+1},w_{i+2})} . In standard bag-of-words, a word's context is represented by a word-count (aka a word histogram) of its neighboring words. For example, the "sat" in "the cat sat on the mat" is represented as {"the": 2, "cat": 1, "on": 1}. Note that the last word "mat" is not used to represent "sat", because it is outside the neighborhood N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} . In continuous bag-of-words, the histogram is multiplied by a matrix V {\displaystyle V} to obtain a continuous representation of the word's context. The matrix V {\displaystyle V} is also called a dictionary. Its columns are the word vectors. It has D {\displaystyle D} columns, where D {\displaystyle D} is the size of the dictionary. Let d {\displaystyle d} be the length of each word vector. We have V ∈ R d × D {\displaystyle V\in \mathbb {R} ^{d\times D}} . For example, multiplying the word histogram {"the": 2, "cat": 1, "on": 1} with V {\displaystyle V} , we obtain 2 v the + v cat + v on {\displaystyle 2v_{\text{the}}+v_{\text{cat}}+v_{\text{on}}} . This is then multiplied with another matrix V ′ {\displaystyle V'} of shape R D × d {\displaystyle \mathbb {R} ^{D\times d}} . Each row of it is a word vector v ′ {\displaystyle v'} . This results in a vector of length D {\displaystyle D} , one entry per dictionary entry. Then, apply the softmax to obtain a probability distribution over the dictionary. This system can be visualized as a neural network, similar in spirit to an autoencoder, of architecture linear-linear-softmax, as depicted in the diagram. The system is trained by gradient descent to minimize the cross-entropy loss. In full formula, the cross-entropy loss is: − ∑ i ln ⁡ e v w i ′ ⋅ ( ∑ j ∈ N v w j + i ) ∑ w ′ e v w ′ ′ ⋅ ( ∑ j ∈ N v w j + i ) {\displaystyle -\sum _{i}\ln {\frac {e^{v_{w_{i}}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}{\sum _{w'}e^{v_{w'}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}}} where the outer summation ∑ i {\displaystyle \sum _{i}} is over the words in a corpus, the quantity ∑ j ∈ N v w j + i {\displaystyle \sum _{j\in N}v_{w_{j+i}}} is the sum of a word's neighbors' vectors, etc. Once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. For example, the word "sat" can be represented as either the "sat"-th column of V {\displaystyle V} or the "sat"-th row of V ′ {\displaystyle V'} . It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. === Skip-gram === The idea of skip-gram is to represent each word with a vector, such that it is possible to predict the vectors of its neighbors using the vector of a word. The architecture is still linear-linear-softmax, the same as CBOW, but the input and the output are switched. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ∑ j ∈ N ln ⁡ Pr ( w j + i ∣ w i ) {\displaystyle \sum _{i}\sum _{j\in N}\ln \Pr(w_{j+i}\mid w_{i})} . In full formula, the loss function is − ∑ i ∑ j ∈ N ln ⁡ e v w j + i ′ ⋅ v w i ∑ w ′ e v w ′ ′ ⋅ v w i {\displaystyle -\sum _{i}\sum _{j\in N}\ln {\frac {e^{v_{w_{j+i}}'\cdot v_{w_{i}}}}{\sum _{w'}e^{v_{w'}'\cdot v_{w_{i}}}}}} Same as CBOW, once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. Essentially, skip-gram and CBOW are exactly the same in architecture. They only differ in the objective function during training. == History == During the 1980s, there were some early attempts at using neural networks to represent words and concepts as vectors. In 2010, Tomáš Mikolov (then at Brno University of Technology) with co-authors applied a simple recurrent neural network with a single hidden

    Read more →
  • Stochastic block model

    Stochastic block model

    The stochastic block model is a generative model for random graphs. This model tends to produce graphs containing communities, subsets of nodes characterized by being connected with one another with particular edge densities. For example, edges may be more common within communities than between communities. Its mathematical formulation was first introduced in 1983 in the field of social network analysis by Paul W. Holland et al. The stochastic block model is important in statistics, machine learning, and network science, where it serves as a useful benchmark for the task of recovering community structure in graph data. == Definition == The stochastic block model takes the following parameters: The number n {\displaystyle n} of vertices; a partition of the vertex set { 1 , … , n } {\displaystyle \{1,\ldots ,n\}} into disjoint subsets C 1 , … , C r {\displaystyle C_{1},\ldots ,C_{r}} , called communities; a symmetric r × r {\displaystyle r\times r} matrix P {\displaystyle P} of edge probabilities. The edge set is then sampled at random as follows: any two vertices u ∈ C i {\displaystyle u\in C_{i}} and v ∈ C j {\displaystyle v\in C_{j}} are connected by an edge with probability P i j {\displaystyle P_{ij}} . An example problem is: given a graph with n {\displaystyle n} vertices, where the edges are sampled as described, recover the groups C 1 , … , C r {\displaystyle C_{1},\ldots ,C_{r}} . == Special cases == If the probability matrix is a constant, in the sense that P i j = p {\displaystyle P_{ij}=p} for all i , j {\displaystyle i,j} , then the result is the Erdős–Rényi model G ( n , p ) {\displaystyle G(n,p)} . This case is degenerate—the partition into communities becomes irrelevant—but it illustrates a close relationship to the Erdős–Rényi model. The planted partition model is the special case that the values of the probability matrix P {\displaystyle P} are a constant p {\displaystyle p} on the diagonal and another constant q {\displaystyle q} off the diagonal. Thus two vertices within the same community share an edge with probability p {\displaystyle p} , while two vertices in different communities share an edge with probability q {\displaystyle q} . Sometimes it is this restricted model that is called the stochastic block model. The case where p > q {\displaystyle p>q} is called an assortative model, while the case p < q {\displaystyle p P j k {\displaystyle P_{ii}>P_{jk}} whenever j ≠ k {\displaystyle j\neq k} : all diagonal entries dominate all off-diagonal entries. A model is called weakly assortative if P i i > P i j {\displaystyle P_{ii}>P_{ij}} whenever i ≠ j {\displaystyle i\neq j} : each diagonal entry is only required to dominate the rest of its own row and column. Disassortative forms of this terminology exist, by reversing all inequalities. For some algorithms, recovery might be easier for block models with assortative or disassortative conditions of this form. == Typical statistical tasks == Much of the literature on algorithmic community detection addresses three statistical tasks: detection, partial recovery, and exact recovery. === Detection === The goal of detection algorithms is simply to determine, given a sampled graph, whether the graph has latent community structure. More precisely, a graph might be generated, with some known prior probability, from a known stochastic block model, and otherwise from a similar Erdos-Renyi model. The algorithmic task is to correctly identify which of these two underlying models generated the graph. === Partial recovery === In partial recovery, the goal is to approximately determine the latent partition into communities, in the sense of finding a partition that is correlated with the true partition significantly better than a random guess. === Exact recovery === In exact recovery, the goal is to recover the latent partition into communities exactly. The community sizes and probability matrix may be known or unknown. == Statistical lower bounds and threshold behavior == Stochastic block models exhibit a sharp threshold effect reminiscent of percolation thresholds. Suppose that we allow the size n {\displaystyle n} of the graph to grow, keeping the community sizes in fixed proportions. If the probability matrix remains fixed, tasks such as partial and exact recovery become feasible for all non-degenerate parameter settings. However, if we scale down the probability matrix at a suitable rate as n {\displaystyle n} increases, we observe a sharp phase transition: for certain settings of the parameters, it will become possible to achieve recovery with probability tending to 1, whereas on the opposite side of the parameter threshold, the probability of recovery tends to 0 no matter what algorithm is used. For partial recovery, the appropriate scaling is to take P i j = P ~ i j / n {\displaystyle P_{ij}={\tilde {P}}_{ij}/n} for fixed P ~ {\displaystyle {\tilde {P}}} , resulting in graphs of constant average degree. In the case of two equal-sized communities, in the assortative planted partition model with probability matrix P = ( p ~ / n q ~ / n q ~ / n p ~ / n ) , {\displaystyle P=\left({\begin{array}{cc}{\tilde {p}}/n&{\tilde {q}}/n\\{\tilde {q}}/n&{\tilde {p}}/n\end{array}}\right),} partial recovery is feasible with probability 1 − o ( 1 ) {\displaystyle 1-o(1)} whenever ( p ~ − q ~ ) 2 > 2 ( p ~ + q ~ ) {\displaystyle ({\tilde {p}}-{\tilde {q}})^{2}>2({\tilde {p}}+{\tilde {q}})} , whereas any estimator fails partial recovery with probability 1 − o ( 1 ) {\displaystyle 1-o(1)} whenever ( p ~ − q ~ ) 2 < 2 ( p ~ + q ~ ) {\displaystyle ({\tilde {p}}-{\tilde {q}})^{2}<2({\tilde {p}}+{\tilde {q}})} . For exact recovery, the appropriate scaling is to take P i j = P ~ i j log ⁡ n / n {\displaystyle P_{ij}={\tilde {P}}_{ij}\log n/n} , resulting in graphs of logarithmic average degree. Here a similar threshold exists: for the assortative planted partition model with r {\displaystyle r} equal-sized communities, the threshold lies at p ~ − q ~ = r {\displaystyle {\sqrt {\tilde {p}}}-{\sqrt {\tilde {q}}}={\sqrt {r}}} . In fact, the exact recovery threshold is known for the fully general stochastic block model. == Algorithms == In principle, exact recovery can be solved in its feasible range using maximum likelihood, but this amounts to solving a constrained or regularized cut problem such as minimum bisection that is typically NP-complete. Hence, no known efficient algorithms will correctly compute the maximum-likelihood estimate in the worst case. However, a wide variety of algorithms perform well in the average case, and many high-probability performance guarantees have been proven for algorithms in both the partial and exact recovery settings. Successful algorithms include spectral clustering of the vertices, semidefinite programming, forms of belief propagation, and community detection among others. == Variants == Several variants of the model exist. One minor tweak allocates vertices to communities randomly, according to a categorical distribution, rather than in a fixed partition. More significant variants include the degree-corrected stochastic block model, the hierarchical stochastic block model, the geometric block model, censored block model and the mixed-membership block model. == Topic models == Stochastic block model have been recognised to be a topic model on bipartite networks. In a network of documents and words, Stochastic block model can identify topics: group of words with a similar meaning. == Extensions to signed graphs == Signed graphs allow for both favorable and adverse relationships and serve as a common model choice for various data analysis applications, e.g., correlation clustering. The stochastic block model can be trivially extended to signed graphs by assigning both positive and negative edge weights or equivalently using a difference of adjacency matrices of two stochastic block models. == DARPA/MIT/AWS Graph Challenge: streaming stochastic block partition == GraphChallenge encourages community approaches to developing new solutions for analyzing graphs and sparse data derived from social media, sensor feeds, and scientific data to enable relationships between events to be discovered as they unfold in the field. Streaming stochastic block partition is one of the challenges since 2017. Spectral clustering has demonstrated outstanding performance compared to the original and even improved base algorithm, matching its quality of clusters while being multiple orders of magnitude faster.

    Read more →
  • Device-independent pixel

    Device-independent pixel

    A device-independent pixel (also: density-independent pixel, dip, dp) is a unit of length. A typical use is to allow mobile device software to scale the display of information and user interaction to different screen sizes. The abstraction allows an application to work in pixels as a measurement, while the underlying graphics system converts the abstract pixel measurements of the application into real pixel measurements appropriate to the particular device. For example, on the Android operating system a device-independent pixel is equivalent to one physical pixel on a 160 dpi screen, while the Windows Presentation Foundation specifies one device-independent pixel as equivalent to 1/96th of an inch. As dp is a physical unit it has an absolute value which can be measured in traditional units, e.g. for Android devices 1 dp equals 1/160 of inch or 0.15875 mm. While traditional pixels only refer to the display of information, device-independent pixels may also be used to measure user input such as input on a touch screen device.

    Read more →
  • Quadratic classifier

    Quadratic classifier

    In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}} Read more →

  • Wolfram Mathematica

    Wolfram Mathematica

    Wolfram Mathematica (also known as Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in Mathematica. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. Mathematica's Wolfram Language is fundamentally based on Lisp; for example, the Mathematica command Most is identically equal to the Lisp command butlast. == Notebook interface == Mathematica is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The original front end, designed by Theodore Gray in 1988, consists of a notebook interface and allows the creation and editing of notebook documents that can contain code, plaintext, images, and graphics. Code development is also supported through support in a range of standard integrated development environment (IDE) including Eclipse, IntelliJ IDEA, Atom, Vim, Visual Studio Code and Git. The Mathematica Kernel also includes a command line front end. Other interfaces include JMath, based on GNU Readline and WolframScript which runs self-contained Mathematica programs (with arguments) from the UNIX command line. == High-performance computing == Capabilities for high-performance computing were extended with the introduction of packed arrays in version 4 (1999) and sparse matrices (version 5, 2003), and by adopting the GNU Multiple Precision Arithmetic Library to evaluate high-precision arithmetic. Version 5.2 (2005) added automatic multi-threading when computations are performed on multi-core computers. This release included CPU-specific optimized libraries. In addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed. In 2002, gridMathematica was introduced to allow user level parallel programming on heterogeneous clusters and multiprocessor systems and in 2008 parallel computing technology was included in all Mathematica licenses including support for grid technology such as Windows HPC Server 2008, Microsoft Compute Cluster Server and Sun Grid. Support for CUDA and OpenCL GPU hardware was added in 2010. == Extensions == As of Version 14, there are 6,602 built-in functions and symbols in the Wolfram Language. Stephen Wolfram announced the launch of the Wolfram Function Repository in June 2019 as a way for the public Wolfram community to contribute functionality to the Wolfram Language. There are currently more than 3000 functions contributed as Resource Functions. In addition to the Wolfram Function Repository, there is a Wolfram Data Repository with computable data and the Wolfram Neural Net Repository for machine learning. Wolfram Mathematica is the basis of the Combinatorica package, which adds discrete mathematics functionality in combinatorics and graph theory to the program. == Connections to other applications, programming languages, and services == Communication with other applications can be done using a protocol called Wolfram Symbolic Transfer Protocol (WSTP). It allows communication between the Wolfram Mathematica kernel and the front end and provides a general interface between the kernel and other applications. Wolfram Research freely distributes a developer kit for linking applications written in the programming language C to the Mathematica kernel through WSTP using J/Link., a Java program that can ask Mathematica to perform computations. Similar functionality is achieved with .NET /Link, but with .NET programs instead of Java programs. Other languages that connect to Mathematica include Haskell, AppleScript, Racket, Visual Basic, Python, and Clojure. Mathematica supports the generation and execution of Modelica models for systems modeling and connects with Wolfram System Modeler. Links are also available to many third-party software packages and APIs. Mathematica can also capture real-time data from a variety of sources and can read and write to public blockchains (Bitcoin, Ethereum, and ARK). It supports import and export of over 220 data, image, video, sound, computer-aided design (CAD), geographic information systems (GIS), document, and biomedical formats. In 2019, support was added for compiling Wolfram Language code to LLVM. Version 12.3 of the Wolfram Language added support for Arduino. == Computable data == Mathematica is also integrated with Wolfram Alpha, an online answer engine that provides additional data, some of which is kept updated in real time, for users who use Mathematica with an internet connection. Some of the data sets include astronomical, chemical, geopolitical, language, biomedical, airplane, and weather data, in addition to mathematical data (such as knots and polyhedra). == Reception == BYTE in 1989 listed Mathematica as among the "Distinction" winners of the BYTE Awards, stating that it "is another breakthrough Macintosh application ... it could enable you to absorb the algebra and calculus that seemed impossible to comprehend from a textbook". Mathematica has been criticized for being closed source. Wolfram Research claims keeping Mathematica closed source is central to its business model and the continuity of the software.

    Read more →