Grammatik

Grammatik was the first grammar-checking program for home computers. Aspen Software of Albuquerque, NM, released the earliest version of this diction and style checker for personal computers. It was first released no later than 1981, and was inspired by the Writer's Workbench. Grammatik was first available for the TRS-80, and soon had versions for CP/M and the IBM PC. Reference Software International of San Francisco, California, acquired Grammatik in 1985. Development of Grammatik continued, and it became an actual grammar checker that could detect writing errors beyond simple style checking. Subsequent versions were released for MS-DOS, Windows, Macintosh, and Unix. Grammatik was ultimately acquired by WordPerfect Corporation and is integrated into the WordPerfect word processor.

Rademacher complexity

In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with respect to a probability distribution. The concept can also be extended to real valued functions. == Definitions == === Rademacher complexity of a set === Given a set A ⊆ R m {\displaystyle A\subseteq \mathbb {R} ^{m}} , the Rademacher complexity of A is defined as follows: Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]} where σ 1 , σ 2 , … , σ m {\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{m}} are independent random variables drawn from the Rademacher distribution i.e. Pr ( σ i = + 1 ) = Pr ( σ i = − 1 ) = 1 / 2 {\displaystyle \Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2} for i ∈ { 1 , 2 , … , m } {\displaystyle i\in \{1,2,\dots ,m\}} , and a = ( a 1 , … , a m ) ∈ A {\displaystyle a=(a_{1},\ldots ,a_{m})\in A} . Some authors take the absolute value of the sum before taking the supremum, but if A {\displaystyle A} is symmetric this makes no difference. === Rademacher complexity of a function class === Let S = { z 1 , z 2 , … , z m } ⊆ Z {\displaystyle S=\{z_{1},z_{2},\dots ,z_{m}\}\subseteq Z} be a sample of points and consider a function class F {\displaystyle {\mathcal {F}}} of real-valued functions over Z {\displaystyle Z} . Then, the empirical Rademacher complexity of F {\displaystyle {\mathcal {F}}} given S {\displaystyle S} is defined as: Rad S ⁡ ( F ) = 1 m E σ [ sup f ∈ F | ∑ i = 1 m σ i f ( z i ) | ] {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\left|\sum _{i=1}^{m}\sigma _{i}f(z_{i})\right|\right]} This can also be written using the previous definition: Rad S ⁡ ( F ) = Rad ⁡ ( F ∘ S ) {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})=\operatorname {Rad} ({\mathcal {F}}\circ S)} where F ∘ S {\displaystyle {\mathcal {F}}\circ S} denotes function composition, i.e.: F ∘ S := { ( f ( z 1 ) , … , f ( z m ) ) ∣ f ∈ F } {\displaystyle {\mathcal {F}}\circ S:=\{(f(z_{1}),\ldots ,f(z_{m}))\mid f\in {\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad ¯ m ( F ) = sup S = { z 1 , … , z m } Rad S ⁡ ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distribution over Z {\displaystyle Z} . The Rademacher complexity of the function class F {\displaystyle {\mathcal {F}}} with respect to P {\displaystyle P} for sample size m {\displaystyle m} is: Rad P , m ⁡ ( F ) := E S ∼ P m [ Rad S ⁡ ( F ) ] {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}}):=\mathbb {E} _{S\sim P^{m}}\left[\operatorname {Rad} _{S}({\mathcal {F}})\right]} where the above expectation is taken over an identically independently distributed (i.i.d.) sample S = ( z 1 , z 2 , … , z m ) {\displaystyle S=(z_{1},z_{2},\dots ,z_{m})} generated according to P {\displaystyle P} . == Intuition == The Rademacher complexity is typically applied on a function class of models that are used for classification, with the goal of measuring their ability to classify points drawn from a probability space under arbitrary labellings. When the function class is rich enough, it contains functions that can appropriately adapt for each arrangement of labels, simulated by the random draw of σ i {\displaystyle \sigma _{i}} under the expectation, so that this quantity in the sum is maximized. The Rademacher complexity of a set A {\displaystyle A} can be rewritten as Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] = 1 m 2 m ∑ σ ∈ { − 1 / m , + 1 / m } m [ sup a ∈ A ⟨ σ , a ⟩ ] . {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]={\frac {1}{{\sqrt {m}}2^{m}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}}\left[\sup _{a\in A}\langle \sigma ,a\rangle \right].} Each term in the summation is the farthest distance of the set A {\displaystyle A} from the origin, along a unit-length direction σ {\displaystyle \sigma } . The directions are along the vertices of a hypercube. Thus, we can also write it as Rad ⁡ ( A ) = 1 2 m 1 2 m − 1 ∑ σ ∈ { − 1 / m , + 1 / m } m / { − 1 , + 1 } [ sup a ∈ A ⟨ σ , a ⟩ − inf a ∈ A ⟨ σ , a ⟩ ] {\displaystyle \operatorname {Rad} (A)={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}}\left[\sup _{a\in A}\langle \sigma ,a\rangle -\inf _{a\in A}\langle \sigma ,a\rangle \right]} Here, the set { − 1 / m , + 1 / m } m / { − 1 , + 1 } {\displaystyle \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}} denotes half of the vertices of a hypercube, selected so that each diagonal has exactly one vertex selected. In words, this states that 2 m Rad ⁡ ( A ) {\displaystyle 2{\sqrt {m}}\operatorname {Rad} (A)} is precisely the average width of the set A {\displaystyle A} along all diagonal directions of a hypercube. == Examples == A singleton set has 0 width in any direction, so it has Rademacher complexity 0. The set A = { ( 1 , 1 ) , ( 1 , 2 ) } ⊆ R 2 {\displaystyle A=\{(1,1),(1,2)\}\subseteq \mathbb {R} ^{2}} has average width 1 / 2 {\displaystyle 1/{\sqrt {2}}} along the two diagonal directions of the square, so it has Rademacher complexity 1 / 4 {\displaystyle 1/4} . The unit cube [ 0 , 1 ] m {\displaystyle [0,1]^{m}} has constant width m {\displaystyle {\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq 1\}} has constant width 2 / m {\displaystyle 2/{\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / m {\displaystyle 1/m} . == Using the Rademacher complexity == The Rademacher complexity can be used to derive data-dependent upper-bounds on the learnability of function classes. Intuitively, a function-class with smaller Rademacher complexity is easier to learn. === Bounding the representativeness === In machine learning, it is desired to have a training set that represents the true distribution of some sample data S {\displaystyle S} . This can be quantified using the notion of representativeness. Denote by P {\displaystyle P} the probability distribution from which the samples are drawn. Denote by H {\displaystyle H} the set of hypotheses (potential classifiers) and denote by F {\displaystyle {\mathcal {F}}} the corresponding set of error functions, i.e., for every hypothesis h ∈ H {\displaystyle h\in H} , there is a function f h ∈ F {\displaystyle f_{h}\in F} , that maps each training sample (features,label) to the error of the classifier h {\displaystyle h} (note in this case hypothesis and classifier are used interchangeably). For example, in the case that h {\displaystyle h} represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function f h {\displaystyle f_{h}} returns 0 if h {\displaystyle h} correctly classifies a sample and 1 else. We omit the index and write f {\displaystyle f} instead of f h {\displaystyle f_{h}} when the underlying hypothesis is irrelevant. Define: L P ( f ) := E z ∼ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} – the expected error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the real distribution P {\displaystyle P} ; L S ( f ) := 1 m ∑ i = 1 m f ( z i ) {\displaystyle L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z_{i})} – the estimated error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the sample S {\displaystyle S} . The representativeness of the sample S {\displaystyle S} , with respect to P {\displaystyle P} and F {\displaystyle {\mathcal {F}}} , is defined as: Rep P ⁡ ( F , S ) := sup f ∈ F ( L P ( f ) − L S ( f ) ) {\displaystyle \operatorname {Rep} _{P}({\mathcal {F}},S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))} Smaller representativeness is better, since it provides a way to avoid overfitting: it means that the true error of a classifier is not much higher than its estimated error, and so selecting a classifier that has low estimated error will ensure that the true error is also low. Note however that the concept of representativeness is relative and hence can not be compared across distinct samples. The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class: If F {\displaystyle {\mathcal {F}}} is a set of functions with range within [ 0 , 1 ] {\displaystyle [0,1]} , then Rad P , m ⁡ ( F ) − ln ⁡ 2 2 m ≤ E S ∼ P m [ Rep P ⁡ ( F , S ) ] ≤ 2 Rad P , m ⁡ ( F ) {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}})-{\sqrt {\frac {\ln 2}{2m}}}\leq \mathbb {E} _{S\sim P^{m}}[\operatorname {Rep} _{P}({\

Neocognitron

The neocognitron is a hierarchical, multilayered artificial neural network proposed by Kunihiko Fukushima in 1979. It has been used for Japanese handwritten character recognition and other pattern recognition tasks, and served as the inspiration for convolutional neural networks. Previously in 1969, he published a similar architecture, but with hand-designed kernels inspired by convolutions in mammalian vision. In 1975 he improved it to the Cognitron, and in 1979 he improved it to the neocognitron, which learns all convolutional kernels by unsupervised learning (in his terminology, "self-organized by 'learning without a teacher'"). The neocognitron was inspired by the model proposed by Hubel & Wiesel in 1959. They found two types of cells in the visual primary cortex called simple cell and complex cell, and also proposed a cascading model of these two types of cells for use in pattern recognition tasks. The neocognitron is a natural extension of these cascading models. The neocognitron consists of multiple types of cells, the most important of which are called S-cells and C-cells. The local features are extracted by S-cells, and these features' deformation, such as local shifts, are tolerated by C-cells. Local features in the input are integrated gradually and classified in the higher layers. The idea of local feature integration is found in several other models, such as the Convolutional Neural Network model, the SIFT method, and the HoG method. There are various kinds of neocognitron. For example, some types of neocognitron can detect multiple patterns in the same input by using backward signals to achieve selective attention.

Multidimensional scaling

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n {\textstyle n} objects in a set into a configuration of n {\textstyle n} points mapped into an abstract Cartesian space. More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction. Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot. Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis. == Types == MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix: === Classical multidimensional scaling === It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, which is given by Strain D ( x 1 , x 2 , . . . , x n ) = ( ∑ i , j ( b i j − x i T x j ) 2 ∑ i , j b i j 2 ) 1 / 2 , {\displaystyle {\text{Strain}}_{D}(x_{1},x_{2},...,x_{n})={\Biggl (}{\frac {\sum _{i,j}{\bigl (}b_{ij}-x_{i}^{T}x_{j}{\bigr )}^{2}}{\sum _{i,j}b_{ij}^{2}}}{\Biggr )}^{1/2},} where x i {\displaystyle x_{i}} denote vectors in N-dimensional space, x i T x j {\displaystyle x_{i}^{T}x_{j}} denotes the scalar product between x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} , and b i j {\displaystyle b_{ij}} are the elements of the matrix B {\displaystyle B} defined on step 2 of the following algorithm, which are computed from the distances. Steps of a Classical MDS algorithm: Classical MDS uses the fact that the coordinate matrix X {\displaystyle X} can be derived by eigenvalue decomposition from B = X X ′ {\textstyle B=XX'} . And the matrix B {\textstyle B} can be computed from proximity matrix D {\textstyle D} by using double centering. Set up the squared proximity matrix D ( 2 ) = [ d i j 2 ] {\textstyle D^{(2)}=[d_{ij}^{2}]} Apply double centering: B = − 1 2 C D ( 2 ) C {\textstyle B=-{\frac {1}{2}}CD^{(2)}C} using the centering matrix C = I − 1 n J n {\textstyle C=I-{\frac {1}{n}}J_{n}} , where n {\textstyle n} is the number of objects, I {\textstyle I} is the n × n {\textstyle n\times n} identity matrix, and J n {\textstyle J_{n}} is an n × n {\textstyle n\times n} matrix of all ones. Determine the m {\textstyle m} largest eigenvalues λ 1 , λ 2 , . . . , λ m {\textstyle \lambda _{1},\lambda _{2},...,\lambda _{m}} and corresponding eigenvectors e 1 , e 2 , . . . , e m {\textstyle e_{1},e_{2},...,e_{m}} of B {\textstyle B} (where m {\textstyle m} is the number of dimensions desired for the output). Now, X = E m Λ m 1 / 2 {\textstyle X=E_{m}\Lambda _{m}^{1/2}} , where E m {\textstyle E_{m}} is the matrix of m {\textstyle m} eigenvectors and Λ m {\textstyle \Lambda _{m}} is the diagonal matrix of m {\textstyle m} eigenvalues of B {\textstyle B} . Classical MDS assumes metric distances. So this is not applicable for direct dissimilarity ratings. === Metric multidimensional scaling (mMDS) === It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares: Stress D ( x 1 , x 2 , . . . , x n ) = ∑ i ≠ j = 1 , . . . , n ( d i j − ‖ x i − x j ‖ ) 2 . {\displaystyle {\text{Stress}}_{D}(x_{1},x_{2},...,x_{n})={\sqrt {\sum _{i\neq j=1,...,n}{\bigl (}d_{ij}-\|x_{i}-x_{j}\|{\bigr )}^{2}}}.} Metric scaling uses a power transformation with a user-controlled exponent p {\textstyle p} : d i j p {\textstyle d_{ij}^{p}} and − d i j 2 p {\textstyle -d_{ij}^{2p}} for distance. In classical scaling p = 1. {\textstyle p=1.} Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities. === Non-metric multidimensional scaling (NMDS) === In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. Let d i j {\displaystyle d_{ij}} be the dissimilarity between points i , j {\displaystyle i,j} . Let d ^ i j = ‖ x i − x j ‖ {\displaystyle {\hat {d}}_{ij}=\|x_{i}-x_{j}\|} be the Euclidean distance between embedded points x i , x j {\displaystyle x_{i},x_{j}} . Now, for each choice of the embedded points x i {\displaystyle x_{i}} and is a monotonically increasing function f {\displaystyle f} , define the "stress" function: S ( x 1 , . . . , x n ; f ) = ∑ i < j ( f ( d i j ) − d ^ i j ) 2 ∑ i < j d ^ i j 2 . {\displaystyle S(x_{1},...,x_{n};f)={\sqrt {\frac {\sum _{i

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).

Signal transfer function

The signal transfer function (SiTF) is a measure of the signal output versus the signal input of a system such as an infrared system or sensor. There are many general applications of the SiTF. Specifically, in the field of image analysis, it gives a measure of the noise of an imaging system, and thus yields one assessment of its performance. == SiTF evaluation == In evaluating the SiTF curve, the signal input and signal output are measured differentially; meaning, the differential of the input signal and differential of the output signal are calculated and plotted against each other. An operator, using computer software, defines an arbitrary area, with a given set of data points, within the signal and background regions of the output image of the infrared sensor, i.e. of the unit under test (UUT), (see "Half Moon" image below). The average signal and background are calculated by averaging the data of each arbitrarily defined region. A second order polynomial curve is fitted to the data of each line. Then, the polynomial is subtracted from the average signal and background data to yield the new signal and background. The difference of the new signal and background data is taken to yield the net signal. Finally, the net signal is plotted versus the signal input. The signal input of the UUT is within its own spectral response. (e.g. color-correlated temperature, pixel intensity, etc.). The slope of the linear portion of this curve is then found using the method of least squares. == SiTF curve == The net signal is calculated from the average signal and background, as in signal to noise ratio (imaging)#Calculations. The SiTF curve is then given by the signal output data, (net signal data), plotted against the signal input data (see graph of SiTF to the right). All the data points in the linear region of the SiTF curve can be used in the method of least squares to find a linear approximation. Given n {\displaystyle n\,} data points ( x i , y i ) {\displaystyle (x_{i}\,,y_{i}\,)} a best fit line parameterized as y = m x + b {\displaystyle y=mx+b\,} is given by: m = ∑ x i y i n − ∑ x i n ∑ y i n ∑ x i 2 n − ( ∑ x i n ) 2 b = ∑ y i n − m ∑ x i n {\displaystyle m={\frac {{\frac {\sum x_{i}y_{i}}{n}}-{\frac {\sum x_{i}}{n}}{\frac {\sum y_{i}}{n}}}{{\frac {\sum x_{i}^{2}}{n}}-({\frac {\sum x_{i}}{n}})^{2}}}\qquad \qquad b={\frac {\sum y_{i}}{n}}-m{\frac {\sum x_{i}}{n}}}

Minimum Population Search

In evolutionary computation, Minimum Population Search (MPS) is a computational method that optimizes a problem by iteratively trying to improve a set of candidate solutions with regard to a given measure of quality. It solves a problem by evolving a small population of candidate solutions by means of relatively simple arithmetical operations. MPS is a metaheuristic as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. For problems where finding the precise global optimum is less important than finding an acceptable local optimum in a fixed amount of time, using a metaheuristic such as MPS may be preferable to alternatives such as brute-force search or gradient descent. MPS is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means MPS does not require for the optimization problem to be differentiable as is required by classic optimization methods such as gradient descent and quasi-newton methods. MPS can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. == Background == In a similar way to Differential evolution, MPS uses difference vectors between the members of the population in order to generate new solutions. It attempts to provide an efficient use of function evaluations by maintaining a small population size. If the population size is smaller than the dimensionality of the search space, then the solutions generated through difference vectors will be constrained to the n − 1 {\displaystyle n-1} dimensional hyperplane. A smaller population size will lead to a more restricted subspace. With a population size equal to the dimensionality of the problem ( n = d ) {\displaystyle (n=d)} , the “line/hyperplane points” in MPS will be generated within a d − 1 {\displaystyle d-1} dimensional hyperplane. Taking a step orthogonal to this hyperplane will allow the search process to cover all the dimensions of the search space. Population size is a fundamental parameter in the performance of population-based heuristics. Larger populations promote exploration, but they also allow fewer generations, and this can reduce the chance of convergence. Searching with a small population can increase the chances of convergence and the efficient use of function evaluations, but it can also induce the risk of premature convergence. If the risk of premature convergence can be avoided, then a population-based heuristic could benefit from the efficiency and faster convergence rate of a smaller population. To avoid premature convergence, it is important to have a diversified population. By including techniques for explicitly increasing diversity and exploration, it is possible to have smaller populations with less risk of premature convergence. === Thresheld Convergence === Thresheld Convergence (TC) is a diversification technique which attempts to separate the processes of exploration and exploitation. TC uses a “threshold” function to establish a minimum search step, and managing this step makes it possible to influence the transition from exploration to exploitation, convergence is thus “held” back until the last stages of the search process. The goal of a controlled transition is to avoid an early concentration of the population around a few search regions and avoid the loss of diversity which can cause premature convergence. Thresheld Convergence has been successfully applied to several population-based metaheuristics such as Particle Swarm Optimization, Differential evolution, Evolution strategies, Simulated annealing and Estimation of Distribution Algorithms. The ideal case for Thresheld Convergence is to have one sample solution from each attraction basin, and for each sample solution to have the same relative fitness with respect to its local optimum. Enforcing a minimum step aims to achieve this ideal case. In MPS Thresheld Convergence is specifically used to preserve diversity and avoid premature convergence by establishing a minimum search step. By disallowing new solutions which are too close to members of the current population, TC forces a strong exploration during the early stages of the search while preserving the diversity of the (small) population. == Algorithm == A basic variant of the MPS algorithm works by having a population of size equal to the dimension of the problem. New solutions are generated by exploring the hyperplane defined by the current solutions (by means of difference vectors) and performing an additional orthogonal step in order to avoid getting caught in this hyperplane. The step sizes are controlled by the Thresheld Convergence technique, which gradually reduces step sizes as the search process advances. An outline for the algorithm is given below: Generate the first initial population. Allowing these solutions to lie near the bounds of the search space generally gives good results: s k = ( r s 1 ∗ b o u n d 1 / 2 , r s 2 ∗ b o u n d 2 / 2 , . . . , r s n ∗ b o u n d n / 2 ) {\displaystyle s_{k}=(rs_{1}bound_{1}/2,rs_{2}bound_{2}/2,...,rs_{n}bound_{n}/2)} where s k {\displaystyle s_{k}} is the k {\displaystyle k} -th population member, r s i {\displaystyle rs_{i}} are random numbers which can be −1 or 1, and the b o u n d i {\displaystyle bound_{i}} are the lower and upper bounds on each dimension. While a stop condition is not reached: Update threshold convergence values ( m i n _ s t e p {\displaystyle min\_step} and m a x _ s t e p {\displaystyle max\_step} ) Calculate the centroid of the current population ( x c {\displaystyle x_{c}} ) For each member of the population ( x i {\displaystyle x_{i}} ), generate a new offspring as follows: Uniformly generate a scaling factor ( F i {\displaystyle F_{i}} ) between − m a x _ s t e p {\displaystyle -max\_step} and m a x _ s t e p {\displaystyle max\_step} Generate a vector ( x o {\displaystyle x_{o}} ) orthogonal to the difference vector between x i {\displaystyle x_{i}} and x c {\displaystyle x_{c}} Calculate a scaling factor for the orthogonal vector: m i n _ o r t h = s q r t ( m a x ( m i n _ s t e p 2 − F i 2 , 0 ) ) {\displaystyle min\_orth=sqrt(max(min\_step^{2}-F_{i}^{2},0))} m a x _ o r t h = s q r t ( m a x ( m a x _ s t e p 2 − F i 2 , 0 ) ) {\displaystyle max\_orth=sqrt(max(max\_step^{2}-F_{i}^{2},0))} o r t h _ s t e p = u n i f o r m ( m i n _ o r t h , m a x _ o r t h ) {\displaystyle orth\_step=uniform(min\_orth,max\_orth)} Generate the new solution by adding the difference and the orthogonal vectors to the original solution n e w _ s o l u t i o n = x i + F i ∗ ( x i − x c ) ∗ o r t h _ s t e p ∗ x o {\displaystyle new\_solution=x_{i}+F_{i}(x_{i}-x_{c})orth\_stepx_{o}} Pick the best members between the old population and the new one by discarding the least fit members. Return the single best solution or the best population found as the final result.