Marq (formerly Lucidpress) is a cloud-based software platform for brand management and templated content creation. The platform integrates with digital asset management (DAM) systems—including Aprimo and Bynder and customer relationship management (CRM) tools such as Salesforce and HubSpot. Marq also includes AI-assisted features for brand compliance and content automation. Trade publications have described the product as a brand templating and creative automation platform. == History == In October 2013, Lucid Software, Inc. announced Lucidpress as a public beta version. Following its release, Lucidpress was featured in TechCrunch, VentureBeat and PC World, with TechCrunch noting: "I had a chance to test the app before its launch and it is indeed very easy to use. If you've ever used a desktop publishing app in the past, you'll feel right at home with Marq, as it features the same kind of standard top-bar menu and layout options as most other publishing apps. In terms of features, it can also hold its own against similar desktop-based apps." In May 2021, Lucidpress announced that it had been acquired by Charles Thayne Capital ("CTC"), a growth-oriented and technology-focused private investment firm. In May 2021, following its acquisition by Charles Thayne Capital, Lucidpress became fully independent. Owen Fuller, who had served as General Manager since 2017, was appointed Chief Executive Officer. In 2022, Lucidpress was rebranded as Marq to reflect the company’s shift toward brand templating and creative automation tools, while continuing to support its publishing features. == Features == Marq integrates with customer relationship management (CRM) platforms such as Salesforce and HubSpot, enabling the creation of personalized, on-brand sales and marketing materials. The platform also connects with multiple digital asset management (DAM) systems, including Bynder, Aprimo, MediaValet, PhotoShelter, Acquia, and Canto. == Investment == Lucid Software raised $1 million in Seed in 2011, led by Google Ventures. In May 2014, the company received a $5 million investment. The round was led by Salt Lake-based Kickstart Seed Fund. In September 2016, the company received a $36 million investment from Spectrum Equity.
Turret lathe
A turret lathe is a form of metalworking lathe that is used for repetitive production of duplicate parts, which by the nature of their cutting process are usually interchangeable. It evolved from earlier lathes with the addition of the turret, which is an indexable toolholder that allows multiple cutting operations to be performed, each with a different cutting tool, in easy, rapid succession, with no need for the operator to perform set-up tasks in between (such as installing or uninstalling tools) or to control the toolpath. The latter is due to the toolpath's being controlled by the machine, either in jig-like fashion, via the mechanical limits placed on it by the turret's slide and stops, or via digitally-directed servomechanisms for computer numerical control lathes. The name derives from the way early turrets took the general form of a flattened cylindrical block mounted to the lathe's cross-slide, capable of rotating about the vertical axis and with toolholders projecting out to all sides, and thus vaguely resembled a swiveling gun turret. Capstan lathe is the usual name in the UK and Commonwealth, though the two terms are also used in contrast: see below, Capstan versus turret. == History == Turret lathes became indispensable to the production of interchangeable parts and for mass production. The first turret lathe was built by Stephen Fitch in 1845 to manufacture screws for pistol percussion parts. In the mid-nineteenth century, the need for interchangeable parts for Colt revolvers enhanced the role of turret lathes in achieving this goal as part of the "American system" of manufacturing arms. Clock-making and bicycle manufacturing had similar requirements. Christopher Spencer invented the first fully automated turret lathe in 1873, which led to designs using cam action or hydraulic mechanisms. From the late-19th through mid-20th centuries, turret lathes, both manual and automatic (i.e., screw machines and chuckers), were one of the most important classes of machine tools for mass production. They were used extensively in the mass production for the war effort in World War II. The U.S. company Warner & Swasey was one of the premier brands in heavy turret lathes between the 1910s and 1960s; it became the world's largest manufacturer of such lathes by 1928. During World War II, it employed 7,000 people and produced half of the turret lathes manufactured in the United States. == Types == There are many variants of the turret lathe. They can be most generally classified by size (small, medium, or large); method of control (manual, automated mechanically, or automated via computer (numerical control (NC) or computer numerical control (CNC)); and bed orientation (horizontal or vertical). === Archetypical: horizontal, manual === In the late 1830s a "capstan lathe" with a turret was patented in Britain. The first American turret lathe was invented by Stephen Fitch in 1845. The archetypical turret lathe, and the first in order of historical appearance, is the horizontal-bed, manual turret lathe. The term "turret lathe" without further qualification is still understood to refer to this type. The formative decades for this class of machine were the 1840s through 1860s, when the basic idea of mounting an indexable turret on a bench lathe or engine lathe was born, developed, and disseminated from the originating shops to many other factories. Some important tool-builders in this development were Stephen Fitch; Gay, Silver & Co.; Elisha K. Root of Colt; J.D. Alvord of the Sharps Armory; Frederick W. Howe, Richard S. Lawrence, and Henry D. Stone of Robbins & Lawrence; J.R. Brown of Brown & Sharpe; and Francis A. Pratt of Pratt & Whitney. Various designers at these and other firms later made further refinements. === Semi-automatic === Sometimes machines similar to those above, but with power feeds and automatic turret-indexing at the end of the return stroke, are called "semi-automatic turret lathes". This nomenclature distinction is blurry and not consistently observed. The term "turret lathe" encompasses them all. During the 1860s, when semi-automatic turret lathes were developed, they were sometimes called "automatic". What we today would call "automatics", that is, fully automatic machines, had not been developed yet. During that era both manual and semi-automatic turret lathes were sometimes called "screw machines", although we today reserve that term for fully automatic machines. === Automatic === During the 1870s through 1890s, the mechanically automated "automatic" turret lathe was developed and disseminated. These machines can execute many part-cutting cycles without human intervention. Thus the duties of the operator, which were already greatly reduced by the manual turret lathe, were even further reduced, and productivity increased. These machines use cams to automate the sliding and indexing of the turret and the opening and closing of the chuck. Thus, they execute the part-cutting cycle somewhat analogously to the way in which an elaborate cuckoo clock performs an automated theater show. Small- to medium-sized automatic turret lathes are usually called "screw machines" or "automatic screw machines", while larger ones are usually called "automatic chucking lathes", "automatic chuckers", or "chuckers". Such machine tools of the "automatic" variety, which in the pre-computer era meant mechanically automated, had already reached a highly advanced state by World War I. === Computer numerical control === When World War II ended, the digital computer was poised to develop from a colossal laboratory curiosity into a practical technology that could begin to disseminate into business and industry. The advent of computer-based automation in machine tools via numerical control (NC) and then computer numerical control (CNC) displaced to a large extent, but not at all completely, the previously existing manual and mechanically automated machines. Numerically controlled turrets allow automated selection of tools on a turret. CNC lathes may be horizontal or vertical in orientation and mount six separate tools on one or more turrets. Such machine tools can work in two axes per turret, with up to six axes being feasible for complex work. === Vertical === Vertical turret lathes have the workpiece held vertically, which allows the headstock to sit on the floor and the faceplate to become a horizontal rotating table, analogous to a huge potter's wheel. This is useful for the handling of very large, heavy, short workpieces. Vertical lathes in general are also called "vertical boring mills" or often simply "boring mills"; therefore a vertical turret lathe is a vertical boring mill equipped with a turret. == Other variations == === Capstan versus turret === The term "capstan lathe" overlaps in sense with the term "turret lathe" to a large extent. In many times and places, it has been understood to be synonymous with "turret lathe". In other times and places it has been held in technical contradistinction to "turret lathe", with the difference being in whether the turret's slide is fixed to the bed (ram-type turret) or slides on the bed's ways (saddle-type turret). The difference in terminology is mostly a matter of United Kingdom and Commonwealth usage versus United States usage. === Flat === A subtype of horizontal turret lathe is the flat-turret lathe. Its turret is flat (and analogous to a rotary table), allowing the turret to pass beneath the part. Patented by James Hartness of Jones & Lamson, and first disseminated in the 1890s, it was developed to provide more rigidity via requiring less overhang in the tool setup, especially when the part is relatively long. === Hollow-hexagon === Hollow-hexagon turret lathes competed with flat-turret lathes by taking the conventional hexagon turret and making it hollow, allowing the part to pass into it during the cut, analogously to how the part would pass over the flat turret. In both cases, the main idea is to increase rigidity by allowing a relatively long part to be turned without the tool overhang that would be needed with a conventional turret, which is not flat or hollow. === Monitor lathe === The term "monitor lathe" formerly (1860s–1940s) referred to the class of small- to medium-sized manual turret lathes used on relatively small work. The name was inspired by the monitor-class warships, which the monitor lathe's turret resembled. Today, lathes of such appearance, such as the Hardinge DSM-59 and its many clones, are still common, but the name "monitor lathe" is no longer current in the industry. === Toolpost turrets and tailstock turrets === Turrets can be added to non-turret lathes (bench lathes, engine lathes, toolroom lathes, etc.) by mounting them on the toolpost, tailstock, or both. Often these turrets are not as large as a turret lathe's, and they usually do not offer the sliding and stopping that a turret lathe's turret does; but they do offer the ability to index through successive tool
Lattice Miner
Lattice Miner is a formal concept analysis software tool for the construction, visualization and manipulation of concept lattices. It allows the generation of formal concepts and association rules as well as the transformation of formal contexts via apposition, subposition, reduction and object/attribute generalization, and the manipulation of concept lattices via approximation, projection and selection. Lattice Miner allows also the drawing of nested line diagrams. == Introduction == Formal concept analysis (FCA) is a branch of applied mathematics based on the formalization of concept and concept hierarchy and mainly used as a framework for conceptual clustering and rule mining. Over the last two decades, a collection of tools have emerged to help FCA users visualize and analyze concept lattices. They range from the earliest DOS-based implementations (e.g., ConImp and GLAD) to more recent implementations in Java like ToscanaJ, Galicia, ConExp and Coron. A main issue in the development of FCA tools is to visualize large concept lattices and provide efficient mechanisms to highlight patterns (e.g., concepts, associations) that could be relevant to the user. The initial objective of the FCA tool called Lattice Miner was to focus on visualization mechanisms for the representation of concept lattices, including nested line diagrams. Later on, many other interesting features were integrated into the tool. == Functional architecture of Lattice Miner == Lattice Miner is a Java-based platform whose functions are articulated around a core. The Lattice Miner core provides all low-level operations and structures for the representation and manipulation of contexts, lattices and association rules. Mainly, the core of Lattice Miner consists of three modules: context, concept and association rule modules. The user interface offers a context editor and concept lattice manipulator to assist the user in a set of tasks. The architecture of Lattice Miner is open and modular enough to allow the integration of new features and facilities in each one of its components. === Context module === The context module offers all the basic operations and structures to manipulate binary and valued contexts as well as context decomposition to produce nested line diagrams. Basic context operations include apposition, subposition, generalization, clarification, reduction as well as the complementary context computation. The module provides also the arrow relations (for context reduction and decomposition) [2]. The tool has an input LMB format and recognizes the binary format SLF found in Galicia and the format CEX produced by ConExp. === Concept module === The main function of the concept module is to generate the concepts of the current binary context and construct the corresponding lattice and nested structure (see Figures 2 and 3). It provides the user with basic operators such as projection, selection, and exact search as well as advanced features like pair approximation. Some known algorithms are included in this module such as Bordat’s procedure, Godin’s algorithm and NextClosure algorithm. The approximation feature implemented in Lattice Miner is based on the following idea: given a pair (X,Y) where X ⊆ G, and Y ⊆ M, is there a set of formal concepts (Ai,Bi) which are “close to” (X,Y)? To answer this question, The tool starts to identify the type of couple that the pair (X,Y) represents. It can be a formal concept, a protoconcept, a semiconcept or a preconcept. In the last case, the approximation is given by the interval [(X",X′),(Y′,Y")] and highlighted in the line diagram. === Association rule module === This module includes procedures for computing the (stem) Guigues–Duquenne base using NextClosure algorithm [3], as well as the generic and informative bases. Implications with negation can be obtained using the apposition of a context and its complementary. This module embeds also procedures for the computation of a non-redundant family C of implications and the closure of a set Y of attributes for the given implication set C. === User interface === The initial objective of Lattice Miner was to focus on lattice drawing and visualization either as a flat or nested structure by taking into account the cognitive process of human beings and known principles for lattice drawing (e.g., reducing the number of edge intersections, ensuring diagram symmetry). Some well-known visualization techniques were implemented such as focus & context and fisheye view. The basic idea behind focus & context visualization paradigm is to allow a viewer to see key (important) objects in full detail in the foreground (focus) while at the same time an overview of all the surrounding information (context) remains available in the background. Lattice Miner translates the focus & context paradigm into clear and blurred elements while the size of nodes and the intensity of their color were used to indicate their importance. Various forms of highlighting, labelling and animation are also provided. In order to better handle the display of large lattices, nested line diagrams are offered in the tool. Figure 3 shows the third level of the nested line diagram corresponding to the binary context of Figure 1 where three levels of nesting are defined. Each one of the inner nodes of this diagram represents a combination of attributes from the previous two (outer) levels. Real inner concepts (see the node on the left hand-side of the diagram) are identified by colored nodes while void elements are in grey color. Each node of levels 1 and 2 can be expanded to exhibit its internal line diagram. Both flat and nested diagrams can be saved as an image. Simple (flat) lattices can also be saved as an XML format file.
Synaptic weight
In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.
ARKA descriptors in QSAR
In computational chemistry and cheminformatics, ARKA descriptors in QSAR are a class of molecular descriptors used in quantitative structure–activity relationship (QSAR) modeling (or related approaches such as QSPR and QSTR), a computational method for predicting the biological activity or toxicity of chemical compounds based on their molecular structure. Molecular descriptors are numerical values that summarize information about a molecule's structure, topology, geometry, or physicochemical properties in a form suitable for machine learning or statistical modeling. ARKA (Arithmetic Residuals in K-Groups Analysis) descriptors differ from traditional descriptors by encoding atomic-level information through recursive autoregression techniques, which aim to capture subtle structural patterns and improve predictive accuracy. They are designed to be both interpretable and well-suited to modeling nonlinear relationships in QSAR studies. == Comparisons == While QSAR is essentially a similarity-based approach, the occurrence of activity/property cliffs may greatly reduce the predictive accuracy of the developed models. The novel Arithmetic Residuals in K-groups Analysis (ARKA) approach is a supervised dimensionality reduction technique developed by the DTC Laboratory, Jadavpur University that can easily identify activity cliffs in a data set. Activity cliffs are similar in their structures but differ considerably in their activity. The basic idea of the ARKA descriptors is to group the conventional QSAR descriptors based on a predefined criterion and then assign weightage to each descriptor in each group. ARKA descriptors have also been used to develop classification-based and regression-based QSAR models with acceptable quality statistics. The ARKA descriptors have been used for the identification of activity cliffs in QSAR studies and/or model development by multiple researchers. A tutorial presentation on the ARKA descriptors is available. Recently a multi-class ARKA framework has been proposed for improved q-RASAR model generation.
Screen space directional occlusion
Screen space directional occlusion (SSDO) is a computer graphics technique enhancing screen space ambient occlusion (SSAO) by taking direction into account to sample the ambient light (both the light coming directly at an object, as well as the light reflected off of the object directly behind it), to better approximate global illumination. SSDO was introduced by Tobias Ritschel, Thorsten Grosch, and Hans-Peter Seidel in their 2009 ACM Symposium on Interactive 3D Graphics and Games paper Approximating dynamic global illumination in image space, which describes it as extending SSAO to directional occlusion with one diffuse indirect bounce of light; later literature notes that SSDO still suffers from common screen-space artifacts such as noise and banding. == Method == The original SSDO paper describes a two-pass screen-space approach, with one pass for direct lighting and a second pass for indirect bounces. Later literature describes SSDO as assuming a general shadowing direction that allows color bleeding and a single light bounce.
LPBoost
Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → { − 1 , 1 } , {\displaystyle f:{\mathcal {X}}\to \{-1,1\},} which classifies samples from a space X {\displaystyle {\mathcal {X}}} into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains. == LPBoost overview == As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , {\displaystyle f({\boldsymbol {x}})=\sum _{j=1}^{J}\alpha _{j}h_{j}({\boldsymbol {x}}),} where α j {\displaystyle \alpha _{j}} are non-negative weightings for weak classifiers h j : X → { − 1 , 1 } {\displaystyle h_{j}:{\mathcal {X}}\to \{-1,1\}} . Each individual weak classifier h j {\displaystyle h_{j}} may be just a little bit better than random, but the resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f {\displaystyle f} by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α {\displaystyle {\boldsymbol {\alpha }}} for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower. == Linear program == More generally, let H = { h ( ⋅ ; ω ) | ω ∈ Ω } {\displaystyle {\mathcal {H}}=\{h(\cdot ;\omega )|\omega \in \Omega \}} be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α {\displaystyle {\boldsymbol {\alpha }}} , the non-negative vector ξ {\displaystyle {\boldsymbol {\xi }}} of slack variables and the margin ρ {\displaystyle \rho } is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\alpha }},{\boldsymbol {\xi }},\rho }{\min }}&-\rho +D\sum _{n=1}^{\ell }\xi _{n}\\{\textrm {sb.t.}}&\sum _{\omega \in \Omega }y_{n}\alpha _{\omega }h({\boldsymbol {x}}_{n};\omega )+\xi _{n}\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _{\omega \in \Omega }\alpha _{\omega }=1,\\&\xi _{n}\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _{\omega }\geq 0,\qquad \omega \in \Omega ,\\&\rho \in {\mathbb {R} }.\end{array}}} Note the effects of slack variables ξ ≥ 0 {\displaystyle {\boldsymbol {\xi }}\geq 0} : their one-norm is penalized in the objective function by a constant factor D {\displaystyle D} , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω {\displaystyle \Omega } , such that for a choice ω ∈ Ω {\displaystyle \omega \in \Omega } the weak classifier h ( ⋅ ; ω ) : X → { − 1 , 1 } {\displaystyle h(\cdot ;\omega ):{\mathcal {X}}\to \{-1,1\}} is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α {\displaystyle {\boldsymbol {\alpha }}} . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation. === Column generation for LPBoost === In a linear program a column corresponds to a primal variable. Column generation is a technique to solve large linear programs. It typically works in a restricted problem, dealing only with a subset of variables. By generating primal variables iteratively and on-demand, eventually the original unrestricted problem with all variables is recovered. By cleverly choosing the columns to generate the problem can be solved such that while still guaranteeing the obtained solution to be optimal for the original full problem, only a small fraction of columns has to be created. ==== LPBoost dual problem ==== Columns in the primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max λ , γ γ sb.t. ∑ n = 1 ℓ y n h ( x n ; ω ) λ n + γ ≤ 0 , ω ∈ Ω , 0 ≤ λ n ≤ D , n = 1 , … , ℓ , ∑ n = 1 ℓ λ n = 1 , γ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\lambda }},\gamma }{\max }}&\gamma \\{\textrm {sb.t.}}&\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}+\gamma \leq 0,\qquad \omega \in \Omega ,\\&0\leq \lambda _{n}\leq D,\qquad n=1,\dots ,\ell ,\\&\sum _{n=1}^{\ell }\lambda _{n}=1,\\&\gamma \in \mathbb {R} .\end{array}}} For linear programs the optimal value of the primal and dual problem are equal. For the above primal and dual problems, the optimal value is equal to the negative 'soft margin'. The soft margin is the size of the margin separating positive from negative training instances minus positive slack variables that carry penalties for margin-violating samples. Thus, the soft margin may be positive although not all samples are linearly separated by the classification function. The latter is called the 'hard margin' or 'realized margin'. ==== Convergence criterion ==== Consider a subset of the satisfied constraints in the dual problem. For any finite subset we can solve the linear program and thus satisfy all constraints. If we could prove that of all the constraints which we did not add to the dual problem no single constraint is violated, we would have proven that solving our restricted problem is equivalent to solving the original problem. More formally, let γ ∗ {\displaystyle \gamma ^{}} be the optimal objective function value for any restricted instance. Then, we can formulate a search problem for the 'most violated constraint' in the original problem space, namely finding ω ∗ ∈ Ω {\displaystyle \omega ^{}\in \Omega } as ω ∗ = argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n . {\displaystyle \omega ^{}={\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}.} That is, we search the space H {\displaystyle {\mathcal {H}}} for a single decision stump h ( ⋅ ; ω ∗ ) {\displaystyle h(\cdot ;\omega ^{})} maximizing the left hand side of the dual constraint. If the constraint cannot be violated by any choice of decision stump, none of the corresponding constraint can be active in the original problem and the restricted problem is equivalent. ==== Penalization constant ==== D {\displaystyle D} The positive value of penalization constant D {\displaystyle D} has to be found using model selection techniques. However, if we choose D = 1 ℓ ν {\displaystyle D={\frac {1}{\ell \nu }}} , where ℓ {\displaystyle \ell } is the number of training samples and 0 < ν < 1 {\displaystyle 0<\nu <1} , then the new parameter ν {\displaystyle \nu } has the following properties. ν {\displaystyle \nu } is an upper bound on the fraction of training errors; that is, if k {\displaystyle k} denotes the number of misclassified training samples, then k ℓ ≤ ν {\displaystyle {\frac {k}{\ell }}\leq \nu } . ν {\displaystyle \nu } is a lower bound on the fraction of training samples outside or on the margin. == Algorithm == Input: Training set X = { x 1 , … , x ℓ } {\displaystyle X=\{{\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{\ell }\}} , x i ∈ X {\displaystyle {\boldsymbol {x}}_{i}\in {\mathcal {X}}} Training labels Y = { y 1 , … , y ℓ } {\displaystyle Y=\{y_{1},\dots ,y_{\ell }\}} , y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} Convergence threshold θ ≥ 0 {\displaystyle \theta \geq 0} Output: Classification function f : X → { − 1 , 1 } {\displaystyle f:{\mathcal {X}}\to \{-1,1\}} Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ {\displaystyle \lambda _{n}\leftarrow {\frac {1}{\ell }},\quad n=1,\dots ,\ell } Edge γ ← 0 {\displaystyle \gamma \leftarrow 0} Hypothesis count J ← 1 {\displaystyle J\leftarrow 1} Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n {\displaystyle {\hat {h}}\leftarrow {\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}} if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ {\displaystyle \sum _{n=1}^{\ell }y_{n}{\hat {h}}({\boldsymbol {x}}_{n})\lambda _{n}+\gamma \leq \theta } then break h J ← h ^ {\displaystyle h_{J}\leftarrow {\hat {h}}} J