Dyme is a Dutch fintech start-up and subscription management app that allows users to cancel and renegotiate their recurring costs. In 2019, Dyme was the first independent Dutch company to receive a PSD2 licence from the Netherlands' central bank (DNB). == History == Dyme was founded in 2018 by Joran Iedema, David Knap, David Schogt and Wouter Florijn. The four had previously founded Cycleswap, a bicycle rental platform launched in 2015 and sold to the American platform Spinlister in 2016. The company gained notability in the Netherlands in 2020 when it appeared on Dutch television in Dragons Den, where Pieter Schoen made a €750,000 bid in an attempt to acquire 51.01% of the company. Dyme's Joran Iedema rejected the deal. == Recognition == Wired described Dyme as one of the "hottest start-ups in Europe" in 2021. As of 2021, the company reportedly had 350,000 registered users in the Netherlands and Great Britain.
Eat App
Eat App is a global restaurant technology company that provides a cloud-based management platform for restaurants, hotels, and other venues. The platform enables venues to accept online reservations seamlessly, manage tables, and enhance customer relationship management (CRM). It utilizes AI to improve operational efficiency, provides marketing automation, and helps build a comprehensive guestbook. The company also offers a consumer app and website for discovering and booking restaurant tables online. According to the company, the system has seated over 100 million guests, and the number continues to grow. Eat was founded by Nezar Kadhem and David Feuillard in 2015 and has raised $13M to date from Silicon Valley's 500 startups, Middle East Venture Partners (MEVP), Derayah VC, amongst other business angels. The company is currently operational across the world, with offices in Dubai and the United States. == Product overview == === For restaurants === Eat App’s reservation system allows for a digital record of all reservations, all guests that have previously visited the restaurant, as well as analytics on the performance of the restaurant. The table management feature simplifies traditional restaurant operations by providing a live snapshot of current status, seating optimization, and shift management. The CRM and analytics suite gathers and monitors data to build a segmented guestbook for personalized marketing and provides dashboards for data-driven decision-making. Additionally, the review feature makes it easy for restaurants to automatically collect reviews from their guests. Additionally, Eat App includes a chit printer function that seamlessly prints reservation details at host stands and a review management feature that allows restaurants to manage online reviews directly within the platform. == History == In February 2015, Eat App raised $300k from Bahrain-based business angel group TENMOU. In June 2018, Eat raised $1.2 million from Dubai-based Middle East Venture Partners (MEVP). In February 2020, Eat App raised $5 million in a Series B funding round led by 500 Startups, Derayah Venture Fund, and MEVP, with participation from a few angel investors and family members. In February 2021, Eat App launched its technology with The Emaar Hospitality Group, implementing it across over 50 restaurants in Emaar properties and hotels. The cloud-based system runs natively on iPads in each restaurant, providing Emaar staff access to reservations and guest information, and integrates with the U by Emaar loyalty app to personalize service. On September 28, 2022, Eat App announced the closing of an $11 million Series B funding round. The investment was led by Middle East Venture Partners (MEVP), 500 Startups, Derayah Venture Capital, Dallah Albaraka, Ali Zaid Al Quraishi & Brothers Company, and Rasameel Investment Company, with participation from existing investors.
C4.5 algorithm
C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date". It became quite popular after ranking #1 in the Top 10 Algorithms in Data Mining pre-eminent paper published by Springer LNCS in 2008. == Algorithm == C4.5 builds decision trees from a set of training data in the same way as ID3, using the concept of information entropy. The training data is a set S = s 1 , s 2 , . . . {\displaystyle S={s_{1},s_{2},...}} of already classified samples. Each sample s i {\displaystyle s_{i}} consists of a p-dimensional vector ( x 1 , i , x 2 , i , . . . , x p , i ) {\displaystyle (x_{1,i},x_{2,i},...,x_{p,i})} , where the x j {\displaystyle x_{j}} represent attribute values or features of the sample, as well as the class in which s i {\displaystyle s_{i}} falls. At each node of the tree, C4.5 chooses the attribute of the data that most effectively splits its set of samples into subsets enriched in one class or the other. The splitting criterion is the normalized information gain (difference in entropy). The attribute with the highest normalized information gain is chosen to make the decision. The C4.5 algorithm then recurses on the partitioned sublists. This algorithm has a few base cases. All the samples in the list belong to the same class. When this happens, it simply creates a leaf node for the decision tree saying to choose that class. None of the features provide any information gain. In this case, C4.5 creates a decision node higher up the tree using the expected value of the class. Instance of previously unseen class encountered. Again, C4.5 creates a decision node higher up the tree using the expected value. === Pseudocode === In pseudocode, the general algorithm for building decision trees is: Check for the above base cases. For each attribute a, find the normalized information gain ratio from splitting on a. Let a_best be the attribute with the highest normalized information gain. Create a decision node that splits on a_best. Recurse on the sublists obtained by splitting on a_best, and add those nodes as children of node. == Improvements from ID3 algorithm == C4.5 made a number of improvements to ID3. Some of these are: Handling both continuous and discrete attributes: In order to handle continuous attributes, C4.5 creates a threshold and then splits the list into those whose attribute value is above the threshold and those that are less than or equal to it. Handling training data with missing attribute values: C4.5 allows attribute values to be marked as missing. Missing attribute values are simply not used in gain and entropy calculations. Handling attributes with differing costs. Pruning trees after creation: C4.5 goes back through the tree once it's been created and attempts to remove branches that do not help by replacing them with leaf nodes. == Improvements in C5.0/See5 algorithm == Quinlan went on to create C5.0 and See5 (C5.0 for Unix/Linux, See5 for Windows) which he markets commercially. C5.0 offers a number of improvements on C4.5. Some of these are: Speed - C5.0 is significantly faster than C4.5 (several orders of magnitude) Memory usage - C5.0 is more memory efficient than C4.5 Smaller decision trees - C5.0 gets similar results to C4.5 with considerably smaller decision trees. Support for boosting - Boosting improves the trees and gives them more accuracy. Weighting - C5.0 allows you to weight different cases and misclassification types. Winnowing - a C5.0 option automatically winnows the attributes to remove those that may be unhelpful. Source for a single-threaded Linux version of C5.0 is available under the GNU General Public License (GPL).
Iris flower data set
The Iris flower data set or Fisher's Iris data set is a multivariate data set used and made famous by the British statistician and biologist Ronald Fisher in his 1936 paper The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. It is sometimes called Anderson's Iris data set because Edgar Anderson collected the data to quantify the morphologic variation of Iris flowers of three related species. Two of the three species were collected in the Gaspé Peninsula "all from the same pasture, and picked on the same day and measured at the same time by the same person with the same apparatus". The data set consists of 50 samples from each of three species of Iris (Iris setosa, Iris virginica and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimeters. Based on the combination of these four features, Fisher developed a linear discriminant model to distinguish each species. Fisher's paper was published in the Annals of Eugenics (today the Annals of Human Genetics). == Use of the data set == Originally used as an example data set on which Fisher's linear discriminant analysis was applied, it became a typical test case for many statistical classification techniques in machine learning such as support vector machines. The use of this data set in cluster analysis however is not common, since the data set only contains two clusters with rather obvious separation. One of the clusters contains Iris setosa, while the other cluster contains both Iris virginica and Iris versicolor and is not separable without the species information Fisher used. This makes the data set a good example to explain the difference between supervised and unsupervised techniques in data mining: Fisher's linear discriminant model can only be obtained when the object species are known: class labels and clusters are not necessarily the same. Nevertheless, all three species of Iris are separable in the projection on the nonlinear and branching principal component. The data set is approximated by the closest tree with some penalty for the excessive number of nodes, bending and stretching. Then the so-called "metro map" is constructed. The data points are projected into the closest node. For each node the pie diagram of the projected points is prepared. The area of the pie is proportional to the number of the projected points. It is clear from the diagram (left) that the absolute majority of the samples of the different Iris species belong to the different nodes. Only a small fraction of Iris-virginica is mixed with Iris-versicolor (the mixed blue-green nodes in the diagram). Therefore, the three species of Iris (Iris setosa, Iris virginica and Iris versicolor) are separable by the unsupervising procedures of nonlinear principal component analysis. To discriminate them, it is sufficient just to select the corresponding nodes on the principal tree. == Data set == The data set contains a set of 150 records under five attributes: sepal length, sepal width, petal length, petal width and species. The iris data set is widely used as a beginner's data set for machine learning purposes. The data set is included in R base and Python in the machine learning library scikit-learn, so that users can access it without having to find a source for it. Several versions of the data set have been published. === R code illustrating usage === The example R code shown below reproduce the scatterplot displayed at the top of this article: === Python code illustrating usage === This code gives:
Characteristic samples
Characteristic samples is a concept in the field of grammatical inference, related to passive learning. In passive learning, an inference algorithm I {\displaystyle I} is given a set of pairs of strings and labels S {\displaystyle S} , and returns a representation R {\displaystyle R} that is consistent with S {\displaystyle S} . Characteristic samples consider the scenario when the goal is not only finding a representation consistent with S {\displaystyle S} , but finding a representation that recognizes a specific target language. A characteristic sample of language L {\displaystyle L} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} where: l ( s ) = 1 {\displaystyle l(s)=1} if and only if s ∈ L {\displaystyle s\in L} l ( s ) = − 1 {\displaystyle l(s)=-1} if and only if s ∉ L {\displaystyle s\notin L} Given the characteristic sample S {\displaystyle S} , I {\displaystyle I} 's output on it is a representation R {\displaystyle R} , e.g. an automaton, that recognizes L {\displaystyle L} . == Formal Definition == === The Learning Paradigm associated with Characteristic Samples === There are three entities in the learning paradigm connected to characteristic samples, the adversary, the teacher and the inference algorithm. Given a class of languages C {\displaystyle \mathbb {C} } and a class of representations for the languages R {\displaystyle \mathbb {R} } , the paradigm goes as follows: The adversary A {\displaystyle A} selects a language L ∈ C {\displaystyle L\in \mathbb {C} } and reports it to the teacher The teacher T {\displaystyle T} then computes a set of strings and label them correctly according to L {\displaystyle L} , trying to make sure that the inference algorithm will compute L {\displaystyle L} The adversary can add correctly labeled words to the set in order to confuse the inference algorithm The inference algorithm I {\displaystyle I} gets the sample and computes a representation R ∈ R {\displaystyle R\in \mathbb {R} } consistent with the sample. The goal is that when the inference algorithm receives a characteristic sample for a language L {\displaystyle L} , or a sample that subsumes a characteristic sample for L {\displaystyle L} , it will return a representation that recognizes exactly the language L {\displaystyle L} . === Sample === Sample S {\displaystyle S} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} such that l ( s ) ∈ { − 1 , 1 } {\displaystyle l(s)\in \{-1,1\}} ==== Sample consistent with a language ==== We say that a sample S {\displaystyle S} is consistent with language L {\displaystyle L} if for every pair ( s , l ( s ) ) {\displaystyle (s,l(s))} in S {\displaystyle S} : l ( s ) = 1 if and only if s ∈ L {\displaystyle l(s)=1{\text{ if and only if }}s\in L} l ( s ) = − 1 if and only if s ∉ L {\displaystyle l(s)=-1{\text{ if and only if }}s\notin L} === Characteristic sample === Given an inference algorithm I {\displaystyle I} and a language L {\displaystyle L} , a sample S {\displaystyle S} that is consistent with L {\displaystyle L} is called a characteristic sample of L {\displaystyle L} for I {\displaystyle I} if: I {\displaystyle I} 's output on S {\displaystyle S} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . For every sample D {\displaystyle D} that is consistent with L {\displaystyle L} and also fulfils S ⊆ D {\displaystyle S\subseteq D} , I {\displaystyle I} 's output on D {\displaystyle D} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . A Class of languages C {\displaystyle \mathbb {C} } is said to have charistaristic samples if every L ∈ C {\displaystyle L\in \mathbb {C} } has a characteristic sample. == Related Theorems == === Theorem === If equivalence is undecidable for a class C {\textstyle \mathbb {C} } over Σ {\textstyle \Sigma } of cardinality bigger than 1, then C {\textstyle \mathbb {C} } doesn't have characteristic samples. ==== Proof ==== Given a class of representations C {\textstyle \mathbb {C} } such that equivalence is undecidable, for every polynomial p ( x ) {\displaystyle p(x)} and every n ∈ N {\displaystyle n\in \mathbb {N} } , there exist two representations r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} of sizes bounded by n {\displaystyle n} , that recognize different languages but are inseparable by any string of size bounded by p ( n ) {\displaystyle p(n)} . Assuming this is not the case, we can decide if r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} are equivalent by simulating their run on all strings of size smaller than p ( n ) {\displaystyle p(n)} , contradicting the assumption that equivalence is undecidable. === Theorem === If S 1 {\displaystyle S_{1}} is a characteristic sample for L 1 {\displaystyle L_{1}} and is also consistent with L 2 {\displaystyle L_{2}} , then every characteristic sample of L 2 {\displaystyle L_{2}} , is inconsistent with L 1 {\displaystyle L_{1}} . ==== Proof ==== Given a class C {\textstyle \mathbb {C} } that has characteristic samples, let R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} be representations that recognize L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} respectively. Under the assumption that there is a characteristic sample for L 1 {\displaystyle L_{1}} , S 1 {\displaystyle S_{1}} that is also consistent with L 2 {\displaystyle L_{2}} , we'll assume falsely that there exist a characteristic sample for L 2 {\displaystyle L_{2}} , S 2 {\displaystyle S_{2}} that is consistent with L 1 {\displaystyle L_{1}} . By the definition of characteristic sample, the inference algorithm I {\displaystyle I} must return a representation which recognizes the language if given a sample that subsumes the characteristic sample itself. But for the sample S 1 ∪ S 2 {\displaystyle S_{1}\cup S_{2}} , the answer of the inferring algorithm needs to recognize both L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} , in contradiction. === Theorem === If a class is polynomially learnable by example based queries, it is learnable with characteristic samples. == Polynomialy characterizable classes == === Regular languages === The proof that DFA's are learnable using characteristic samples, relies on the fact that every regular language has a finite number of equivalence classes with respect to the right congruence relation, ∼ L {\displaystyle \sim _{L}} (where x ∼ L y {\displaystyle x\sim _{L}y} for x , y ∈ Σ ∗ {\displaystyle x,y\in \Sigma ^{}} if and only if ∀ z ∈ Σ ∗ : x z ∈ L ↔ y z ∈ L {\displaystyle \forall z\in \Sigma ^{}:xz\in L\leftrightarrow yz\in L} ). Note that if x {\displaystyle x} , y {\displaystyle y} are not congruent with respect to ∼ L {\displaystyle \sim _{L}} , there exists a string z {\displaystyle z} such that x z ∈ L {\displaystyle xz\in L} but y z ∉ L {\displaystyle yz\notin L} or vice versa, this string is called a separating suffix. ==== Constructing a characteristic sample ==== The construction of a characteristic sample for a language L {\displaystyle L} by the teacher goes as follows. Firstly, by running a depth first search on a deterministic automaton A {\displaystyle A} recognizing L {\displaystyle L} , starting from its initial state, we get a suffix closed set of words, W {\displaystyle W} , ordered in shortlex order. From the fact above, we know that for every two states in the automaton, there exists a separating suffix that separates between every two strings that the run of A {\displaystyle A} on them ends in the respective states. We refer to the set of separating suffixes as S {\displaystyle S} . The labeled set (sample) of words the teacher gives the adversary is { ( w , l ( w ) ) | w ∈ W ⋅ S ∪ W ⋅ Σ ⋅ S } {\displaystyle \{(w,l(w))|w\in W\cdot S\cup W\cdot \Sigma \cdot S\}} where l ( w ) {\displaystyle l(w)} is the correct label of w {\displaystyle w} (whether it is in L {\displaystyle L} or not). We may assume that ϵ ∈ S {\displaystyle \epsilon \in S} . ==== Constructing a deterministic automata ==== Given the sample from the adversary W {\displaystyle W} , the construction of the automaton by the inference algorithm I {\displaystyle I} starts with defining P = prefix ( W ) {\displaystyle P={\text{prefix}}(W)} and S = suffix ( W ) {\displaystyle S={\text{suffix}}(W)} , which are the set of prefixes and suffixes of W {\displaystyle W} respectively. Now the algorithm constructs a matrix M {\displaystyle M} where the elements of P {\displaystyle P} function as the rows, ordered by the shortlex order, and the elements of S {\displaystyle S} function as the columns, ordered by the shortlex order. Next, the cells in the matrix are filled in the following manner for prefix p i {\displaystyle p_{i}} and suffix s j {\displaystyle s_{j}} : If p i s j ∈ W → M i j = l ( p i s j ) {\displaystyle p_{i}s_{j}\in W\rightarrow M_{ij}=l(p_{i}s_{j})} else, M i j = 0 {\displaystyle M_{ij}=0} Now, we say row i {\displaystyle i} and t {\displaystyle t} are distinguishable if there exi
YaDICs
YaDICs is a program written to perform digital image correlation on 2D and 3D tomographic images. The program was designed to be both modular, by its plugin strategy and efficient, by it multithreading strategy. It incorporates different transformations (Global, Elastic, Local), optimizing strategy (Gauss-Newton, Steepest descent), Global and/or local shape functions (Rigid-body motions, homogeneous dilatations, flexural and Brazilian test models)... == Theoretical background == === Context === In solid mechanics, digital image correlation is a tool that allows to identify the displacement field to register a reference image (called herein fixed image) to images during an experiment (mobile image). For example, it is possible to observe the face of a specimen with a painted speckle on it in order to determine its displacement fields during a tensile test. Before the appearance of such methods, researchers usually used strain gauges to measure the mechanical state of the material but strain gauges only measure the strain on a point and don't allow to understand material with an heterogeneous behavior. One can obtain a full in plane strain tensor by derivation of the displacement fields. Many methods are based upon the optical flow. In fluid mechanics a similar method is used, called Particle Image Velocimetry (PIV); the algorithms are similar to those of DIC but it is impossible to ensure that the optical flow is conserved so a vast majority of the software used the normalized cross correlation metric. In mechanics the displacement or velocity fields are the only concern, registering images is just a side effect. There is another process called image registration using the same algorithms (on monomodal images) but where the goal is to register images and thereby identifying the displacement field is just a side effect. YaDICs uses the general principle of image registration with a particular attention to the displacement fields basis. === Image registration principle === YaDICs can be explained using the classical image registration framework: === Image registration general scheme === The common idea of image registration and digital image correlation is to find the transformation between a fixed image and a moving one for a given metric using an optimization scheme. While there are many methods to achieve such a goal, Yadics focuses on registering images with the same modality. The idea behind the creation of this software is to be able to process data that comes from a μ-tomograph; i.e.: data cube over 10003 voxels. With such a size it is not possible to use naive approach usually used in a two-dimensional context. In order to get sufficient performances OpenMP parallelism is used and data are not globally stored in memory. As an extensive description of the different algorithms is given in. === Sampling === Contrary to image registration, Digital Image Correlation targets the transformation, one wants to extracted the most accurate transformation from the two images and not just match the images. Yadics uses the whole image as a sampling grid: it is thus a total sampling. === Interpolator === It is possible to choose between bilinear interpolation and bicubic interpolation for the grey level evaluation at non integer coordinates. The bi-cubic interpolation is the recommended one. === Metrics === ==== Sum of squared differences (SSD) ==== The SSD is also known as mean squared error. The equation below defines the SSD metric: S S D ( μ , I F , I M ) = 1 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) 2 , {\displaystyle SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {1}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)^{2},} where I F {\displaystyle {\mathcal {I_{F}}}} is the fixed image, I M {\displaystyle {\mathcal {I_{M}}}} the moving one, Ω F {\displaystyle \Omega _{F}} the integration area | Ω F | {\displaystyle \left|\Omega _{F}\right|} the number of pi(vo)xels (cardinal) and T μ {\displaystyle {T}_{\mu }} the transformation parametrized by μ The transformation can be written as: T μ ( x ) = x + { Φ ( x ) } t { μ } . {\displaystyle T_{\mu }(x)=x+\left\{\Phi (x)\right\}^{t}\left\{\mu \right\}.} This metric is the main one used in the YaDICs as it works well with same modality images. One has to find the minimum of this metric ==== Normalized cross-correlation ==== The normalized cross-correlation (NCC) is used when one cannot assure the optical flow conservation; it happens in case of change of lighting or if particles disappear from the scene can occur in particle images velocimetry (PIV). The NCC is defined by: N C C ( μ , I F , I M ) = ∑ x i ∈ Ω F ( I F ( x i ) − I F ¯ ) ( I M ( T μ ( x i ) ) − I M ¯ ) ∑ x i ∈ Ω F ( I F ( x i ) − I F ¯ ) 2 ∑ x i ∈ Ω F ( I M ( T μ ( x i ) ) − I M ¯ ) 2 , {\displaystyle NCC(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)}{\sqrt {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)^{2}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)^{2}}}},} where I F ¯ {\displaystyle {\overline {\mathcal {I_{F}}}}} and I M ¯ {\displaystyle {\overline {\mathcal {I_{M}}}}} are the mean values of the fixed and mobile images. This metric is only used to find local translation in Yadics. This metric with translation transform can be solved using cross-correlation methods, which are non iterative and can be accelerated using Fast Fourier Transform . === Classification of transformations === There are three categories of parametrization: elastic, global and local transformation. The elastic transformations respect the partition of unity, there are no holes created or surfaces counted several times. This is commonly used in Image Registration by the use of B-Spline functions and in solid mechanics with finite element basis. The global transformations are defined on the whole picture using rigid body or affine transformation (which is equivalent to homogeneous strain transformation). More complex transformations can be defined such as mechanically based one. These transformations have been used for stress intensity factor identification by and for rod strain by. The local transformation can be considered as the same global transformation defined on several Zone Of Interest (ZOI) of the fixed image. ==== Global ==== Several global transforms have been implemented: Rigid and homogeneous (Tx,Ty,Rz in 2D; Tx,Ty,Tz,Rx,Ry,Rz,Exx,Eyy,Ezz,Eyz,Exz,Exy in 3D) Brazilian (Only in 2D), Dynamic Flexion, ==== Elastic ==== First-order quadrangular finite elements Q4P1 are used in Yadics. ===== Local ===== Every global transform can be used on a local mesh. === Optimization === The YaDICs optimization process follows a gradient descent scheme. The first step is to compute the gradient of the metric regarding the transform parameters ∂ S S D ( μ , I F , I M ) ∂ μ = 2 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) ∂ I M ( T μ ( x i ) ∂ μ = 2 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) ( ∂ T μ ( x i ) ∂ μ ) t ∂ I M ( T μ ( x i ) ) ∂ x {\displaystyle {\begin{array}{lcl}{\dfrac {\partial SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right){\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i})}{\partial \mu }}\\&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)\left({\dfrac {\partial {T}_{\mu }(x_{i})}{\partial \mu }}\right)^{t}{\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i}))}{\partial x}}\\\end{array}}} ==== Gradient method ==== Once the metric gradient has been computed, one has to find an optimization strategy The gradient method principle is explained below: μ k + 1 = μ k + α k d k {\displaystyle \mu _{k+1}=\mu _{k}+\alpha _{k}d_{k}} The gradient step can be constant or updated at every iteration. d k = − γ k ∂ C ( μ , I F , I M ) ∂ μ {\displaystyle d_{k}=-\gamma _{k}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}} , γ k {\displaystyle \gamma _{k}} allows one to choose between the following methods : γ k {\displaystyle \gamma _{k}} ⟹ {\displaystyle \Longrightarrow } steepest descent, γ k = [ ∂ C ( μ , I F , I M ) ∂ μ ∂ C ( μ , I F , I M ) ∂ μ t ] − 1 {\displaystyle \gamma _{k}=\left[{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}^{t}\right]^{-1}} ⟹ {\displaystyle \Longrightarrow } Gauss-Newto
UIMA
UIMA ( yoo-EE-mə), short for Unstructured Information Management Architecture, is an OASIS standard for content analytics, originally developed at IBM. It provides a component software architecture for the development, discovery, composition, and deployment of multi-modal analytics for the analysis of unstructured information and integration with search technologies. == Structure == The UIMA architecture can be thought of in four dimensions: It specifies component interfaces in an analytics pipeline. It describes a set of design patterns. It suggests two data representations: an in-memory representation of annotations for high-performance analytics and an XML representation of annotations for integration with remote web services. It suggests development roles allowing tools to be used by users with diverse skills. == Implementations and uses == Apache UIMA, a reference implementation of UIMA, is maintained by the Apache Software Foundation. UIMA is used in a number of software projects: IBM Research's Watson uses UIMA for analyzing unstructured data. The Clinical Text Analysis and Knowledge Extraction System (Apache cTAKES) is a UIMA-based system for information extraction from medical records. DKPro Core is a collection of reusable UIMA components for general-purpose natural language processing.