Pixel-art scaling algorithms

Pixel art scaling algorithms are graphical filters that attempt to enhance the appearance of hand-drawn 2D pixel art graphics. These algorithms are a form of automatic image enhancement. Pixel art scaling algorithms employ methods significantly different than the common methods of image rescaling, which have the goal of preserving the appearance of images. As pixel art graphics are commonly used at very low resolutions, they employ careful coloring of individual pixels. This results in graphics that rely on a high amount of stylized visual cues to define complex shapes. Several specialized algorithms have been developed to handle re-scaling of such graphics. These specialized algorithms can improve the appearance of pixel-art graphics, but in doing so they introduce changes. Such changes may be undesirable, especially if the goal is to faithfully reproduce the original appearance. Since a typical application of this technology is improving the appearance of fourth-generation and earlier video games on arcade and console emulators, many pixel art scaling algorithms are designed to run in real-time for sufficiently small input images at 60-frames per second. This places constraints on the type of programming techniques that can be used for this sort of real-time processing. Many work only on specific scale factors. 2× is the most common scale factor, while 3×, 4×, 5×, and 6× exist but are less used. == Algorithms == === SAA5050 'Diagonal Smoothing' === The Mullard SAA5050 Teletext character generator chip (1980) used a primitive pixel scaling algorithm to generate higher-resolution characters on the screen from a lower-resolution representation from its internal ROM. Internally, each character shape was defined on a 5 × 9 pixel grid, which was then interpolated by smoothing diagonals to give a 10 × 18 pixel character, with a characteristically angular shape, surrounded to the top and the left by two pixels of blank space. The algorithm only works on monochrome source data, and assumes the source pixels will be logically true or false depending on whether they are 'on' or 'off'. Pixels 'outside the grid pattern' are assumed to be off. The algorithm works as follows: A B C --\ 1 2 D E F --/ 3 4 1 = B | (A & E & !B & !D) 2 = B | (C & E & !B & !F) 3 = E | (!A & !E & B & D) 4 = E | (!C & !E & B & F) Note that this algorithm, like the Eagle algorithm below, has a flaw: If a pattern of 4 pixels in a hollow diamond shape appears, the hollow will be obliterated by the expansion. The SAA5050's internal character ROM carefully avoids ever using this pattern. The degenerate case: becomes: === EPX/Scale2×/AdvMAME2× === Eric's Pixel Expansion (EPX) is an algorithm developed by Eric Johnston at LucasArts around 1992, when porting the SCUMM engine games from the IBM PC (which ran at 320 × 200 × 256 colors) to the early color Macintosh computers, which ran at more or less double that resolution. The algorithm works as follows, expanding P into 4 new pixels based on P's surroundings: 1=P; 2=P; 3=P; 4=P; IF C==A => 1=A IF A==B => 2=B IF D==C => 3=C IF B==D => 4=D IF of A, B, C, D, three or more are identical: 1=2=3=4=P Later implementations of this same algorithm (as AdvMAME2× and Scale2×, developed around 2001) are slightly more efficient but functionally identical: 1=P; 2=P; 3=P; 4=P; IF C==A AND C!=D AND A!=B => 1=A IF A==B AND A!=C AND B!=D => 2=B IF D==C AND D!=B AND C!=A => 3=C IF B==D AND B!=A AND D!=C => 4=D AdvMAME2× is available in DOSBox via the scaler=advmame2x dosbox.conf option. The AdvMAME4×/Scale4× algorithm is just EPX applied twice to get 4× resolution. ==== Scale3×/AdvMAME3× and ScaleFX ==== The AdvMAME3×/Scale3× algorithm (available in DOSBox via the scaler=advmame3x dosbox.conf option) can be thought of as a generalization of EPX to the 3× case. The corner pixels are calculated identically to EPX. 1=E; 2=E; 3=E; 4=E; 5=E; 6=E; 7=E; 8=E; 9=E; IF D==B AND D!=H AND B!=F => 1=D IF (D==B AND D!=H AND B!=F AND E!=C) OR (B==F AND B!=D AND F!=H AND E!=A) => 2=B IF B==F AND B!=D AND F!=H => 3=F IF (H==D AND H!=F AND D!=B AND E!=A) OR (D==B AND D!=H AND B!=F AND E!=G) => 4=D 5=E IF (B==F AND B!=D AND F!=H AND E!=I) OR (F==H AND F!=B AND H!=D AND E!=C) => 6=F IF H==D AND H!=F AND D!=B => 7=D IF (F==H AND F!=B AND H!=D AND E!=G) OR (H==D AND H!=F AND D!=B AND E!=I) => 8=H IF F==H AND F!=B AND H!=D => 9=F There is also a variant improved over Scale3× called ScaleFX, developed by Sp00kyFox, and a version combined with Reverse-AA called ScaleFX-Hybrid. === Eagle === Eagle works as follows: for every in pixel, we will generate 4 out pixels. First, set all 4 to the color of the pixel we are currently scaling (as nearest-neighbor). Next look at the three pixels above, to the left, and diagonally above left: if all three are the same color as each other, set the top left pixel of our output square to that color in preference to the nearest-neighbor color. Work similarly for all four pixels, and then move to the next one. Assume an input matrix of 3 × 3 pixels where the centermost pixel is the pixel to be scaled, and an output matrix of 2 × 2 pixels (i.e., the scaled pixel) first: |Then . . . --\ CC |S T U --\ 1 2 . C . --/ CC |V C W --/ 3 4 . . . |X Y Z | IF V==S==T => 1=S | IF T==U==W => 2=U | IF V==X==Y => 3=X | IF W==Z==Y => 4=Z Thus if we have a single black pixel on a white background it will vanish. This is a bug in the Eagle algorithm but is solved by other algorithms such as EPX, 2xSaI, and HQ2x. === 2×SaI === 2×SaI, short for 2× Scale and Interpolation engine, was inspired by Eagle. It was designed by Derek Liauw Kie Fa, also known as Kreed, primarily for use in console and computer emulators, and it has remained fairly popular in this niche. Many of the most popular emulators, including ZSNES and VisualBoyAdvance, offer this scaling algorithm as a feature. Several slightly different versions of the scaling algorithm are available, and these are often referred to as Super 2×SaI and Super Eagle. The 2xSaI family works on a 4 × 4 matrix of pixels where the pixel marked A below is scaled: I E F J G A B K --\ W X H C D L --/ Y Z M N O P For 16-bit pixels, they use pixel masks which change based on whether the 16-bit pixel format is 565 or 555. The constants colorMask, lowPixelMask, qColorMask, qLowPixelMask, redBlueMask, and greenMask are 16-bit masks. The lower 8 bits are identical in either pixel format. Two interpolation functions are described: INTERPOLATE(uint32 A, UINT32 B). -- linear midpoint of A and B if (A == B) return A; return ( ((A & colorMask) >> 1) + ((B & colorMask) >> 1) + (A & B & lowPixelMask) ); Q_INTERPOLATE(uint32 A, uint32 B, uint32 C, uint32 D) -- bilinear interpolation; A, B, C, and D's average x = ((A & qColorMask) >> 2) + ((B & qColorMask) >> 2) + ((C & qColorMask) >> 2) + ((D & qColorMask) >> 2); y = (A & qLowPixelMask) + (B & qLowPixelMask) + (C & qLowPixelMask) + (D & qLowPixelMask); y = (y >> 2) & qLowPixelMask; return x + y; The algorithm checks A, B, C, and D for a diagonal match such that A==D and B!=C, or the other way around, or if they are both diagonals or if there is no diagonal match. Within these, it checks for three or four identical pixels. Based on these conditions, the algorithm decides whether to use one of A, B, C, or D, or an interpolation among only these four, for each output pixel. The 2xSaI arbitrary scaler can enlarge any image to any resolution and uses bilinear filtering to interpolate pixels. Since Kreed released the source code under the GNU General Public License, it is freely available to anyone wishing to utilize it in a project released under that license. Developers wishing to use it in a non-GPL project would be required to rewrite the algorithm without using any of Kreed's existing code. It is available in DOSBox via scaler=2xsai option. === hqnx family === Maxim Stepin's hq2x, hq3x, and hq4x are for scale factors of 2:1, 3:1, and 4:1 respectively. Each work by comparing the color value of each pixel to those of its eight immediate neighbors, marking the neighbors as close or distant, and using a pre-generated lookup table to find the proper proportion of input pixels' values for each of the 4, 9 or 16 corresponding output pixels. The hq3x family will perfectly smooth any diagonal line whose slope is ±0.5, ±1, or ±2 and which is not anti-aliased in the input; one with any other slope will alternate between two slopes in the output. It will also smooth very tight curves. Unlike 2xSaI, it anti-aliases the output. hqnx was initially created for the Super NES emulator ZSNES. The author of bsnes has released a space-efficient implementation of hq2x to the public domain. A port to shaders, which has comparable quality to the early versions of xBR, is available. Before the port, a shader called "scalehq" has often been confused for hqx. === xBR family === There are 6 filters in this family: xBR , xBRZ, xBR-Hybrid, Super xBR, xBR+3D and Super xBR+3D. xBR ("scale by rules"), cre

Tensor (machine learning)

In machine learning, the term tensor informally refers to two different concepts: (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector space. Observations, such as images, movies, volumes, sounds, and relationships among words and concepts, stored in an M-way array ("data tensor"), may be analyzed either by artificial neural networks or tensor methods. Tensor decomposition factors data tensors into smaller tensors. Operations on data tensors can be expressed in terms of matrix multiplication and the Kronecker product. The computation of gradients, a crucial aspect of backpropagation, can be performed using software libraries such as PyTorch and TensorFlow. Computations are often performed on graphics processing units (GPUs) using CUDA, and on dedicated hardware such as Google's Tensor Processing Unit or Nvidia's Tensor core. These developments have greatly accelerated neural network architectures, and increased the size and complexity of models that can be trained. == History == A tensor is by definition a multilinear map. In mathematics, this may express a multilinear relationship between sets of algebraic objects. In physics, tensor fields, considered as tensors at each point in space, are useful in expressing mechanics such as stress or elasticity. In machine learning, the exact use of tensors depends on the statistical approach being used. In 2001, the field of signal processing and statistics were making use of tensor methods. Pierre Comon surveys the early adoption of tensor methods in the fields of telecommunications, radio surveillance, chemometrics and sensor processing. Linear tensor rank methods (such as, Parafac/CANDECOMP) analyzed M-way arrays ("data tensors") composed of higher order statistics that were employed in blind source separation problems to compute a linear model of the data. He noted several early limitations in determining the tensor rank and efficient tensor rank decomposition. In the early 2000s, multilinear tensor methods crossed over into computer vision, computer graphics and machine learning with papers by Vasilescu or in collaboration with Terzopoulos, such as Human Motion Signatures, TensorFaces TensorTextures and Multilinear Projection. Multilinear algebra, the algebra of higher-order tensors, is a suitable and transparent framework for analyzing the multifactor structure of an ensemble of observations and for addressing the difficult problem of disentangling the causal factors based on second order or higher order statistics associated with each causal factor. Tensor (multilinear) factor analysis disentangles and reduces the influence of different causal factors with multilinear subspace learning. When treating an image or a video as a 2- or 3-way array, i.e., "data matrix/tensor", tensor methods reduce spatial or time redundancies as demonstrated by Wang and Ahuja. Yoshua Bengio, Geoff Hinton and their collaborators briefly discuss the relationship between deep neural networks and tensor factor analysis beyond the use of M-way arrays ("data tensors") as inputs. One of the early uses of tensors for neural networks appeared in natural language processing. A single word can be expressed as a vector via Word2vec. Thus a relationship between two words can be encoded in a matrix. However, for more complex relationships such as subject-object-verb, it is necessary to build higher-dimensional networks. In 2009, the work of Sutskever introduced Bayesian Clustered Tensor Factorization to model relational concepts while reducing the parameter space. From 2014 to 2015, tensor methods become more common in convolutional neural networks (CNNs). Tensor methods organize neural network weights in a "data tensor", analyze and reduce the number of neural network weights. Lebedev et al. accelerated CNN networks for character classification (the recognition of letters and digits in images) by using 4D kernel tensors. == Definition == Let F {\displaystyle \mathbb {F} } be a field (such as the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } ). A tensor T ∈ F I 1 × I 2 × … × I C {\displaystyle {\mathcal {T}}\in {\mathbb {F} }^{I_{1}\times I_{2}\times \ldots \times I_{C}}} is a multilinear transformation from a set of domain vector spaces to a range vector space: T : { F I 1 × F I 2 × … F I C } ↦ F I 0 {\displaystyle {\mathcal {T}}:\{{\mathbb {F} }^{I_{1}}\times {\mathbb {F} }^{I_{2}}\times \ldots {\mathbb {F} }^{I_{C}}\}\mapsto {\mathbb {F} }^{I_{0}}} Here, C {\displaystyle C} and I 0 , I 1 , … , I C {\displaystyle I_{0},I_{1},\ldots ,I_{C}} are positive integers, and ( C + 1 ) {\displaystyle (C+1)} is the number of modes of a tensor (also known as the number of ways of a multi-way array). The dimensionality of mode c {\displaystyle c} is I c {\displaystyle I_{c}} , for 0 ≤ c ≤ C {\displaystyle 0\leq c\leq C} . In statistics and machine learning, an image is vectorized when viewed as a single observation, and a collection of vectorized images is organized as a "data tensor". For example, a set of facial images { d i p , i e , i l , i v ∈ R I X } {\displaystyle \{{\mathbb {d} }_{i_{p},i_{e},i_{l},i_{v}}\in {\mathbb {R} }^{I_{X}}\}} with I X {\displaystyle I_{X}} pixels that are the consequences of multiple causal factors, such as a facial geometry i p ( 1 ≤ i p ≤ I P ) {\displaystyle i_{p}(1\leq i_{p}\leq I_{P})} , an expression i e ( 1 ≤ i e ≤ I E ) {\displaystyle i_{e}(1\leq i_{e}\leq I_{E})} , an illumination condition i l ( 1 ≤ i l ≤ I L ) {\displaystyle i_{l}(1\leq i_{l}\leq I_{L})} , and a viewing condition i v ( 1 ≤ i v ≤ I V ) {\displaystyle i_{v}(1\leq i_{v}\leq I_{V})} may be organized into a data tensor (ie. multiway array) D ∈ R I X × I P × I E × I L × V {\displaystyle {\mathcal {D}}\in {\mathbb {R} }^{I_{X}\times I_{P}\times I_{E}\times I_{L}\times V}} where I P {\displaystyle I_{P}} are the total number of facial geometries, I E {\displaystyle I_{E}} are the total number of expressions, I L {\displaystyle I_{L}} are the total number of illumination conditions, and I V {\displaystyle I_{V}} are the total number of viewing conditions. Tensor factorizations methods such as TensorFaces and multilinear (tensor) independent component analysis factorizes the data tensor into a set of vector spaces that span the causal factor representations, where an image is the result of tensor transformation T {\displaystyle {\mathcal {T}}} that maps a set of causal factor representations to the pixel space. Another approach to using tensors in machine learning is to embed various data types directly. For example, a grayscale image, commonly represented as a discrete 2-way array D ∈ R I R X × I C X {\displaystyle {\mathbf {D} }\in {\mathbb {R} }^{I_{RX}\times I_{CX}}} with dimensionality I R X × I C X {\displaystyle I_{RX}\times I_{CX}} where I R X {\displaystyle I_{RX}} are the number of rows and I C X {\displaystyle I_{CX}} are the number of columns. When an image is treated as 2-way array or 2nd order tensor (i.e. as a collection of column/row observations), tensor factorization methods compute the image column space, the image row space and the normalized PCA coefficients or the ICA coefficients. Similarly, a color image with RGB channels, D ∈ R N × M × 3 . {\displaystyle {\mathcal {D}}\in \mathbb {R} ^{N\times M\times 3}.} may be viewed as a 3rd order data tensor or 3-way array.-------- In natural language processing, a word might be expressed as a vector v {\displaystyle v} via the Word2vec algorithm. Thus v {\displaystyle v} becomes a mode-1 tensor v ↦ A ∈ R N . {\displaystyle v\mapsto {\mathcal {A}}\in \mathbb {R} ^{N}.} The embedding of subject-object-verb semantics requires embedding relationships among three words. Because a word is itself a vector, subject-object-verb semantics could be expressed using mode-3 tensors v a × v b × v c ↦ A ∈ R N × N × N . {\displaystyle v_{a}\times v_{b}\times v_{c}\mapsto {\mathcal {A}}\in \mathbb {R} ^{N\times N\times N}.} In practice the neural network designer is primarily concerned with the specification of embeddings, the connection of tensor layers, and the operations performed on them in a network. Modern machine learning frameworks manage the optimization, tensor factorization and backpropagation automatically. === As unit values === Tensors may be used as the unit values of neural networks which extend the concept of scalar, vector and matrix values to multiple dimensions. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f} : y m = f ( ∑ n x n u m , n ) , {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right),} where y m ∈ R .

Reservoir sampling

Reservoir sampling is a family of randomized algorithms for choosing a simple random sample, without replacement, of k items from a population of unknown size n in a single pass over the items. The size of the population n is not known to the algorithm and is typically too large for all n items to fit into main memory. The population is revealed to the algorithm over time, and the algorithm cannot look back at previous items. At any point, the current state of the algorithm must permit extraction of a simple random sample without replacement of size k over the part of the population seen so far. == Motivation == Suppose we see a sequence of items, one at a time. We want to keep 10 items in memory, and we want them to be selected at random from the sequence. If we know the total number of items n and can access the items arbitrarily, then the solution is easy: select 10 distinct indices i between 1 and n with equal probability, and keep the i-th elements. The problem is that we do not always know the exact n in advance. == Simple: Algorithm R == A simple and popular but slow algorithm, Algorithm R, was created by Jeffrey Vitter. Initialize an array R {\displaystyle R} indexed from 1 {\displaystyle 1} to k {\displaystyle k} , containing the first k items of the input x 1 , . . . , x k {\displaystyle x_{1},...,x_{k}} . This is the reservoir. For each new input x i {\displaystyle x_{i}} , generate a random number j uniformly in { 1 , . . . , i } {\displaystyle \{1,...,i\}} . If j ∈ { 1 , . . . , k } {\displaystyle j\in \{1,...,k\}} , then set R [ j ] := x i . {\displaystyle R[j]:=x_{i}.} Otherwise, discard x i {\displaystyle x_{i}} . Return R {\displaystyle R} after all inputs are processed. This algorithm works by induction on i ≥ k {\displaystyle i\geq k} . While conceptually simple and easy to understand, this algorithm needs to generate a random number for each item of the input, including the items that are discarded. The algorithm's asymptotic running time is thus O ( n ) {\displaystyle O(n)} . Generating this amount of randomness and the linear run time causes the algorithm to be unnecessarily slow if the input population is large. This is Algorithm R, implemented as follows: == Optimal: Algorithm L == If we generate n {\displaystyle n} random numbers u 1 , . . . , u n ∼ U [ 0 , 1 ] {\displaystyle u_{1},...,u_{n}\sim U[0,1]} independently, then the indices of the smallest k {\displaystyle k} of them is a uniform sample of the k {\displaystyle k} -subsets of { 1 , . . . , n } {\displaystyle \{1,...,n\}} . The process can be done without knowing n {\displaystyle n} : Keep the smallest k {\displaystyle k} of u 1 , . . . , u i {\displaystyle u_{1},...,u_{i}} that has been seen so far, as well as w i {\displaystyle w_{i}} , the index of the largest among them. For each new u i + 1 {\displaystyle u_{i+1}} , compare it with u w i {\displaystyle u_{w_{i}}} . If u i + 1 < u w i {\displaystyle u_{i+1}

Information seeking

Information seeking is the process or activity of attempting to obtain information in both human and technological contexts. Information seeking is related to, but different from, information retrieval (IR). == Compared to information retrieval == Traditionally, IR tools have been designed for IR professionals to enable them to effectively and efficiently retrieve information from a source. It is assumed that the information exists in the source and that a well-formed query will retrieve it (and nothing else). It has been argued that laypersons' information seeking on the internet is very different from information retrieval as performed within the IR discourse. Yet, internet search engines are built on IR principles. Since the late 1990s a body of research on how casual users interact with internet search engines has been forming, but the topic is far from fully understood. IR can be said to be technology-oriented, focusing on algorithms and issues such as precision and recall. Information seeking may be understood as a more human-oriented and open-ended process than information retrieval. In information seeking, one does not know whether there exists an answer to one's query, so the process of seeking may provide the learning required to satisfy one's information need. == In different contexts == Much library and information science (LIS) research has focused on the information-seeking practices of practitioners within various fields of professional work. Studies have been carried out into the information-seeking behaviors of librarians, academics, medical professionals, engineers, lawyers and mini-publics(among others). Much of this research has drawn on the work done by Leckie, Pettigrew (now Fisher) and Sylvain, who in 1996 conducted an extensive review of the LIS literature (as well as the literature of other academic fields) on professionals' information seeking. The authors proposed an analytic model of professionals' information seeking behaviour, intended to be generalizable across the professions, thus providing a platform for future research in the area. The model was intended to "prompt new insights... and give rise to more refined and applicable theories of information seeking" (1996, p. 188). The model has been adapted by Wilkinson (2001) who proposes a model of the information seeking of lawyers. Recent studies in this topic address the concept of information-gathering that "provides a broader perspective that adheres better to professionals' work-related reality and desired skills." (Solomon & Bronstein, 2021). == Theories of information-seeking behavior == A variety of theories of information behavior – e.g. Zipf's Principle of Least Effort, Brenda Dervin's Sense Making, Elfreda Chatman's Life in the Round – seek to understand the processes that surround information seeking. In addition, many theories from other disciplines have been applied in investigating an aspect or whole process of information seeking behavior. A review of the literature on information seeking behavior shows that information seeking has generally been accepted as dynamic and non-linear (Foster, 2005; Kuhlthau 2006). People experience the information search process as an interplay of thoughts, feelings and actions (Kuhlthau, 2006). Donald O. Case (2007) also wrote a good book that is a review of the literature. Information seeking has been found to be linked to a variety of interpersonal communication behaviors beyond question-asking, to include strategies such as candidate answers. Robinson's (2010) research suggests that when seeking information at work, people rely on both other people and information repositories (e.g., documents and databases), and spend similar amounts of time consulting each (7.8% and 6.4% of work time, respectively; 14.2% in total). However, the distribution of time among the constituent information seeking stages differs depending on the source. When consulting other people, people spend less time locating the information source and information within that source, similar time understanding the information, and more time problem solving and decision making, than when consulting information repositories. Furthermore, the research found that people spend substantially more time receiving information passively (i.e., information that they have not requested) than actively (i.e., information that they have requested), and this pattern is also reflected when they provide others with information. == Wilson's nested model of conceptual areas == The concepts of information seeking, information retrieval, and information behaviour are objects of investigation of information science. Within this scientific discipline a variety of studies has been undertaken analyzing the interaction of an individual with information sources in case of a specific information need, task, and context. The research models developed in these studies vary in their level of scope. Wilson (1999) therefore developed a nested model of conceptual areas, which visualizes the interrelation of the here mentioned central concepts. Wilson defines models of information behavior to be "statements, often in the form of diagrams, that attempt to describe an information-seeking activity, the causes and consequences of that activity, or the relationships among stages in information-seeking behaviour" (1999: 250).

Leiden algorithm

The Leiden algorithm is a community detection algorithm developed by Traag et al at Leiden University. It was developed as a modification of the Louvain method. Like the Louvain method, the Leiden algorithm attempts to optimize modularity in extracting communities from networks; however, it addresses key issues present in the Louvain method, namely poorly connected communities and the resolution limit of modularity. == Improvement over Louvain method == Broadly, the Leiden algorithm uses the same two primary phases as the Louvain algorithm: a local node moving step (though, the method by which nodes are considered in Leiden is more efficient) and a graph aggregation step. However, to address the issues with poorly-connected communities and the merging of smaller communities into larger communities (the resolution limit of modularity), the Leiden algorithm employs an intermediate refinement phase in which communities may be split to guarantee that all communities are well-connected. Consider, for example, the following graph: Three communities are present in this graph (each color represents a community). Additionally, the center "bridge" node (represented with an extra circle) is a member of the community represented by blue nodes. Now consider the result of a node-moving step which merges the communities denoted by red and green nodes into a single community (as the two communities are highly connected): Notably, the center "bridge" node is now a member of the larger red community after node moving occurs (due to the greedy nature of the local node moving algorithm). In the Louvain method, such a merging would be followed immediately by the graph aggregation phase. However, this causes a disconnection between two different sections of the community represented by blue nodes. In the Leiden algorithm, the graph is instead refined: The Leiden algorithm's refinement step ensures that the center "bridge" node is kept in the blue community to ensure that it remains intact and connected, despite the potential improvement in modularity from adding the center "bridge" node to the red community. == Graph components == Before defining the Leiden algorithm, it will be helpful to define some of the components of a graph. === Vertices and edges === A graph is composed of vertices (nodes) and edges. Each edge is connected to two vertices, and each vertex may be connected to zero or more edges. Edges are typically represented by straight lines, while nodes are represented by circles or points. In set notation, let V {\displaystyle V} be the set of vertices, and E {\displaystyle E} be the set of edges: V := { v 1 , v 2 , … , v n } E := { e i j , e i k , … , e k l } {\displaystyle {\begin{aligned}V&:=\{v_{1},v_{2},\dots ,v_{n}\}\\E&:=\{e_{ij},e_{ik},\dots ,e_{kl}\}\end{aligned}}} where e i j {\displaystyle e_{ij}} is the directed edge from vertex v i {\displaystyle v_{i}} to vertex v j {\displaystyle v_{j}} . We can also write this as an ordered pair: e i j := ( v i , v j ) {\displaystyle {\begin{aligned}e_{ij}&:=(v_{i},v_{j})\end{aligned}}} === Community === A community is a unique set of nodes: C i ⊆ V C i ⋂ C j = ∅ ∀ i ≠ j {\displaystyle {\begin{aligned}C_{i}&\subseteq V\\C_{i}&\bigcap C_{j}=\emptyset ~\forall ~i\neq j\end{aligned}}} and the union of all communities must be the total set of vertices: V = ⋃ i = 1 C i {\displaystyle {\begin{aligned}V&=\bigcup _{i=1}C_{i}\end{aligned}}} === Partition === A partition is the set of all communities: P = { C 1 , C 2 , … , C n } {\displaystyle {\begin{aligned}{\mathcal {P}}&=\{C_{1},C_{2},\dots ,C_{n}\}\end{aligned}}} == Partition quality == How communities are partitioned is an integral part on the Leiden algorithm. How partitions are decided can depend on how their quality is measured. Additionally, many of these metrics contain parameters of their own that can change the outcome of their communities. === Modularity === Modularity is a highly used quality metric for assessing how well a set of communities partition a graph. The equation for this metric is defined for an adjacency matrix, A, as: Q = 1 2 m ∑ i j ( A i j − k i k j 2 m ) δ ( c i , c j ) {\displaystyle Q={\frac {1}{2m}}\sum _{ij}(A_{ij}-{\frac {k_{i}k_{j}}{2m}})\delta (c_{i},c_{j})} where: A i j {\displaystyle A_{ij}} represents the edge weight between nodes i {\displaystyle i} and j {\displaystyle j} ; see Adjacency matrix; k i {\displaystyle k_{i}} and k j {\displaystyle k_{j}} are the sum of the weights of the edges attached to nodes i {\displaystyle i} and j {\displaystyle j} , respectively; m {\displaystyle m} is the sum of all of the edge weights in the graph; c i {\displaystyle c_{i}} and c j {\displaystyle c_{j}} are the communities to which the nodes i {\displaystyle i} and j {\displaystyle j} belong; and δ {\displaystyle \delta } is Kronecker delta function: δ ( c i , c j ) = { 1 if c i and c j are the same community 0 otherwise {\displaystyle {\begin{aligned}\delta (c_{i},c_{j})&={\begin{cases}1&{\text{if }}c_{i}{\text{ and }}c_{j}{\text{ are the same community}}\\0&{\text{otherwise}}\end{cases}}\end{aligned}}} === Reichardt Bornholdt Potts Model (RB) === One of the most well used metrics for the Leiden algorithm is the Reichardt Bornholdt Potts Model (RB). This model is used by default in most mainstream Leiden algorithm libraries under the name RBConfigurationVertexPartition. This model introduces a resolution parameter γ {\displaystyle \gamma } and is highly similar to the equation for modularity. This model is defined by the following quality function for an adjacency matrix, A, as: Q = ∑ i j ( A i j − γ k i k j 2 m ) δ ( c i , c j ) {\displaystyle Q=\sum _{ij}(A_{ij}-\gamma {\frac {k_{i}k_{j}}{2m}})\delta (c_{i},c_{j})} where: γ {\displaystyle \gamma } represents a linear resolution parameter === Constant Potts Model (CPM) === Another metric similar to RB is the Constant Potts Model (CPM). This metric also relies on a resolution parameter γ {\displaystyle \gamma } The quality function is defined as: H = − ∑ i j ( A i j w i j − γ ) δ ( c i , c j ) {\displaystyle H=-\sum _{ij}(A_{ij}w_{ij}-\gamma )\delta (c_{i},c_{j})} === Understanding Potts Model resolution parameters/Resolution limit === Typically Potts models such as RB or CPM include a resolution parameter in their calculation. Potts models are introduced as a response to the resolution limit problem that is present in modularity maximization based community detection. The resolution limit problem is that, for some graphs, maximizing modularity may cause substructures of a graph to merge and become a single community and thus smaller structures are lost. These resolution parameters allow modularity adjacent methods to be modified to suit the requirements of the user applying the Leiden algorithm to account for small substructures at a certain granularity. The figure on the right illustrates why resolution can be a helpful parameter when using modularity based quality metrics. In the first graph, modularity only captures the large scale structures of the graph; however, in the second example, a more granular quality metric could potentially detect all substructures in a graph. == Algorithm == The Leiden algorithm starts with a graph of disorganized nodes (a) and sorts it by partitioning them to maximize modularity (the difference in quality between the generated partition and a hypothetical randomized partition of communities). The method it uses is similar to the Louvain algorithm, except that after moving each node it also considers that node's neighbors that are not already in the community it was placed in. This process results in our first partition (b), also referred to as P {\displaystyle {\mathcal {P}}} . Then the algorithm refines this partition by first placing each node into its own individual community and then moving them from one community to another to maximize modularity. It does this iteratively until each node has been visited and moved, and each community has been refined - this creates partition (c), which is the initial partition of P refined {\displaystyle {\mathcal {P}}_{\text{refined}}} . Then an aggregate network (d) is created by turning each community into a node. P refined {\displaystyle {\mathcal {P}}_{\text{refined}}} is used as the basis for the aggregate network while P {\displaystyle {\mathcal {P}}} is used to create its initial partition. Because we use the original partition P {\displaystyle {\mathcal {P}}} in this step, we must retain it so that it can be used in future iterations. These steps together form the first iteration of the algorithm. In subsequent iterations, the nodes of the aggregate network (which each represent a community) are once again placed into their own individual communities and then sorted according to modularity to form a new P refined {\displaystyle {\mathcal {P}}_{\text{refined}}} , forming (e) in the above graphic. In the case depicted by the graph, the nodes were already sorted optimally, so no change too

ROCm

ROCm is an Advanced Micro Devices (AMD) software stack for graphics processing unit (GPU) programming. ROCm spans several domains, including general-purpose computing on graphics processing units (GPGPU), high performance computing (HPC), and heterogeneous computing. It offers several programming models: HIP (GPU-kernel-based programming), OpenMP (directive-based programming), and OpenCL. ROCm is free, libre and open-source software (except the GPU firmware blobs), and it is distributed under various licenses. The name initially stood for Radeon Open Compute platform; however, due to Open Compute being a registered trademark, the name no longer functions as an acronym. == Background == The first GPGPU software stack from ATI/AMD was Close to Metal, which became Stream. ROCm was launched around 2016 with the Boltzmann Initiative. ROCm stack builds upon previous AMD GPU stacks; some tools trace back to GPUOpen and others to the Heterogeneous System Architecture (HSA). === Heterogeneous System Architecture Intermediate Language === HSAIL was aimed at producing a middle-level, hardware-agnostic intermediate representation that could be JIT-compiled to the eventual hardware (GPU, FPGA...) using the appropriate finalizer. This approach was dropped for ROCm: now it builds only GPU code, using LLVM, and its AMDGPU backend that was upstreamed, although there is still research on such enhanced modularity with LLVM MLIR. == Programming abilities == ROCm as a stack ranges from the kernel driver to the end-user applications. AMD has introductory videos about AMD GCN hardware, and ROCm programming via its learning portal. One of the best technical introductions about the stack and ROCm/HIP programming, remains, to date, to be found on Reddit. == Hardware support == ROCm is primarily targeted at discrete professional GPUs, but consumer GPUs and APUs of the same architecture as a supported professional GPU are known to work with ROCm. For example, all professional GPUs of the RDNA 2 architecture are officially supported by ROCm 5.x; users report that Consumer RDNA2 units such as the Radeon 6800M APU and the Radeon 6700XT GPU also work. === Professional-grade GPUs === === Consumer-grade GPUs === == Software ecosystem == === Machine learning === Various deep learning frameworks have a ROCm backend: PyTorch TensorFlow ONNX MXNet CuPy MIOpen Caffe Iree (which uses LLVM Multi-Level Intermediate Representation (MLIR)) llama.cpp === Supercomputing === ROCm is gaining significant traction in the top 500. ROCm is used with the Exascale supercomputers El Capitan and Frontier. Some related software is to be found at AMD Infinity hub. === Other acceleration & graphics interoperation === As of version 3.0, Blender can now use HIP compute kernels for its renderer cycles. === Other languages === ==== Julia ==== Julia has the AMDGPU.jl package, which integrates with LLVM and selects components of the ROCm stack. Instead of compiling code through HIP, AMDGPU.jl uses Julia's compiler to generate LLVM IR directly, which is later consumed by LLVM to generate native device code. AMDGPU.jl uses ROCr's HSA implementation to upload native code onto the device and execute it, similar to how HIP loads its own generated device code. AMDGPU.jl also supports integration with ROCm's rocBLAS (for BLAS), rocRAND (for random number generation), and rocFFT (for FFTs). Future integration with rocALUTION, rocSOLVER, MIOpen, and certain other ROCm libraries is planned. === Software distribution === ==== Official ==== Installation instructions are provided for Linux and Windows in the official AMD ROCm documentation. ROCm software is currently spread across several public GitHub repositories. Within the main public meta-repository, there is an XML manifest for each official release: using git-repo, a version control tool built on top of Git, is the recommended way to synchronize with the stack locally. AMD starts distributing containerized applications for ROCm, notably scientific research applications gathered under AMD Infinity Hub. AMD distributes itself packages tailored to various Linux distributions. ==== Third-party ==== There is a growing third-party ecosystem packaging ROCm. Linux distributions are officially packaging (natively) ROCm, with various degrees of advancement: Arch Linux, Gentoo, Debian, Fedora , GNU Guix, and NixOS. There are Spack packages. == Components == There is one kernel-space component, ROCk, and the rest - there is roughly a hundred components in the stack - is made of user-space modules. The unofficial typographic policy is to use: uppercase ROC lowercase following for low-level libraries, i.e. ROCt, and the contrary for user-facing libraries, i.e. rocBLAS. AMD is active developing with the LLVM community, but upstreaming is not instantaneous, and as of January 2022, is still lagging. AMD still officially packages various LLVM forks for parts that are not yet upstreamed – compiler optimizations destined to remain proprietary, debug support, OpenMP offloading, etc. === Low-level === ==== ROCk – Kernel driver ==== ==== ROCm – Device libraries ==== Support libraries implemented as LLVM bitcode. These provide various utilities and functions for math operations, atomics, queries for launch parameters, on-device kernel launch, etc. ==== ROCt – Thunk ==== The thunk is responsible for all the thinking and queuing that goes into the stack. ==== ROCr – Runtime ==== The ROC runtime is a set of APIs/libraries that allows the launch of compute kernels by host applications. It is AMD's implementation of the HSA runtime API. It is different from the ROC Common Language Runtime. ==== ROCm – CompilerSupport ==== ROCm code object manager is in charge of interacting with LLVM intermediate representation. === Mid-level === ==== ROCclr Common Language Runtime ==== The common language runtime is an indirection layer adapting calls to ROCr on Linux and PAL on windows. It used to be able to route between different compilers, like the HSAIL-compiler. It is now being absorbed by the upper indirection layers (HIP and OpenCL). ==== OpenCL ==== ROCm ships its installable client driver (ICD) loader and an OpenCL implementation bundled together. As of January 2022, ROCm 4.5.2 ships OpenCL 2.2, and is lagging behind competition. ==== HIP – Heterogeneous Interface for Portability ==== The AMD implementation for its GPUs is called HIPAMD. There is also a CPU implementation mostly for demonstration purposes. ==== HIPCC ==== HIP builds a `HIPCC` compiler that either wraps Clang and compiles with LLVM open AMDGPU backend, or redirects to the NVIDIA compiler. ==== HIPIFY ==== HIPIFY is a source-to-source compiling tool. It translates CUDA to HIP and reverse, either using a Clang-based tool, or a sed-like Perl script. ==== GPUFORT ==== Like HIPIFY, GPUFORT is a tool compiling source code into other third-generation-language sources, allowing users to migrate from CUDA Fortran to HIP Fortran. It is also in the repertoire of research projects, even more so. === High-level === ROCm high-level libraries are usually consumed directly by application software, such as machine learning frameworks. Most of the following libraries are in the General Matrix Multiply (GEMM) category, which GPU architecture excels at. The majority of these user-facing libraries comes in dual-form: hip for the indirection layer that can route to Nvidia hardware, and roc for the AMD implementation. ==== rocBLAS / hipBLAS ==== rocBLAS and hipBLAS are central in high-level libraries, it is the AMD implementation for Basic Linear Algebra Subprograms. It uses the library Tensile privately. ==== rocSOLVER / hipSOLVER ==== This pair of libraries constitutes the LAPACK implementation for ROCm and is strongly coupled to rocBLAS. === Utilities === ROCm developer tools: Debug, tracer, profiler, System Management Interface, Validation suite, Cluster management. GPUOpen tools: GPU analyzer, memory visualizer... External tools: radeontop (TUI overview) == Comparison with competitors == ROCm competes with other GPU computing stacks: Nvidia CUDA and Intel OneAPI. === Nvidia CUDA === Nvidia's CUDA is closed-source, whereas AMD ROCm is open source. There is open-source software built on top of the closed-source CUDA, for instance RAPIDS. CUDA is able to run on consumer GPUs, whereas ROCm support is mostly offered for professional hardware such as AMD Instinct and AMD Radeon Pro. Nvidia provides a C/C++-centered frontend and its Parallel Thread Execution (PTX) LLVM GPU backend as the Nvidia CUDA Compiler (NVCC). === Intel OneAPI === All the oneAPI corresponding libraries are published on its GitHub Page. ==== Unified Acceleration Foundation (UXL) ==== Unified Acceleration Foundation (UXL) is a new technology consortium that are working on the continuation of the OneAPI initiative, with the goal to create a new open standard accelerator software ecosystem, related open standards and specification projects through Working Groups and Specia

Controlled vocabulary

A controlled vocabulary provides a way to organize knowledge for subsequent retrieval. Controlled vocabularies are used in subject indexing schemes, subject headings, thesauri, taxonomies and other knowledge organization systems. Controlled vocabulary schemes mandate the use of predefined, preferred terms that have been preselected by the designers of the schemes, in contrast to natural language vocabularies, which have no such restriction. == In library and information science == In library and information science, controlled vocabulary is a carefully selected list of words and phrases, which are used to tag units of information (document or work) so that they may be more easily retrieved by a search. Controlled vocabularies solve the problems of homographs, synonyms and polysemes by a bijection between concepts and preferred terms. In short, controlled vocabularies reduce unwanted ambiguity inherent in normal human languages where the same concept can be given different names and ensure consistency. For example, in the Library of Congress Subject Headings (a subject heading system that uses a controlled vocabulary), preferred terms—subject headings in this case—have to be chosen to handle choices between variant spellings of the same word (American versus British), choice among scientific and popular terms (cockroach versus Periplaneta americana), and choices between synonyms (automobile versus car), among other difficult issues. Choices of preferred terms are based on the principles of user warrant (what terms users are likely to use), literary warrant (what terms are generally used in the literature and documents), and structural warrant (terms chosen by considering the structure, scope of the controlled vocabulary). Controlled vocabularies also typically handle the problem of homographs with qualifiers. For example, the term pool has to be qualified to refer to either swimming pool or the game pool to ensure that each preferred term or heading refers to only one concept. === Types used in libraries === There are two main kinds of controlled vocabulary tools used in libraries: subject headings and thesauri. While the differences between the two are diminishing, there are still some minor differences: Historically, subject headings were designed to describe books in library catalogs by catalogers while thesauri were used by indexers to apply index terms to documents and articles. Subject headings tend to be broader in scope describing whole books, while thesauri tend to be more specialized covering very specific disciplines. Because of the card catalog system, subject headings tend to have terms that are in indirect order (though with the rise of automated systems this is being removed), while thesaurus terms are always in direct order. Subject headings tend to use more pre-coordination of terms such that the designer of the controlled vocabulary will combine various concepts together to form one preferred subject heading. (e.g., children and terrorism) while thesauri tend to use singular direct terms. Thesauri list not only equivalent terms but also narrower, broader terms and related terms among various preferred and non-preferred (but potentially synonymous) terms, while historically most subject headings did not. For example, the Library of Congress Subject Heading itself did not have much syndetic structure until 1943, and it was not until 1985 when it began to adopt the thesauri type term "Broader term" and "Narrow term". The terms are chosen and organized by trained professionals (including librarians and information scientists) who possess expertise in the subject area. Controlled vocabulary terms can accurately describe what a given document is actually about, even if the terms themselves do not occur within the document's text. Well known subject heading systems include the Library of Congress system, Medical Subject Headings (MeSH) created by the United States National Library of Medicine, and Sears. Well known thesauri include the Art and Architecture Thesaurus and the ERIC Thesaurus. When selecting terms for a controlled vocabulary, the designer has to consider the specificity of the term chosen, whether to use direct entry, inter consistency and stability of the language. Lastly the amount of pre-coordination (in which case the degree of enumeration versus synthesis becomes an issue) and post-coordination in the system is another important issue. Controlled vocabulary elements (terms/phrases) employed as tags, to aid in the content identification process of documents, or other information system entities (e.g. DBMS, Web Services) qualifies as metadata. == Indexing languages == There are three main types of indexing languages. Controlled indexing language – only approved terms can be used by the indexer to describe the document Natural language indexing language – any term from the document in question can be used to describe the document Free indexing language – any term (not only from the document) can be used to describe the document When indexing a document, the indexer also has to choose the level of indexing exhaustivity, the level of detail in which the document is described. For example, using low indexing exhaustivity, minor aspects of the work will not be described with index terms. In general the higher the indexing exhaustivity, the more terms indexed for each document. In recent years free text search as a means of access to documents has become popular. This involves using natural language indexing with an indexing exhaustively set to maximum (every word in the text is indexed). These methods have been compared in some studies, such as the 2007 article, "A Comparative Evaluation of Full-text, Concept-based, and Context-sensitive Search". === Advantages === Controlled vocabularies are often claimed to improve the accuracy of free text searching, such as to reduce irrelevant items in the retrieval list. These irrelevant items (false positives) are often caused by the inherent ambiguity of natural language. Take the English word football for example. Football is the name given to a number of different team sports. Worldwide the most popular of these team sports is association football, which also happens to be called soccer in several countries. The word football is also applied to rugby football (rugby union and rugby league), American football, Australian rules football, Gaelic football, and Canadian football. A search for football therefore will retrieve documents that are about several completely different sports. Controlled vocabulary solves this problem by tagging the documents in such a way that the ambiguities are eliminated. Compared to free text searching, the use of a controlled vocabulary can dramatically increase the performance of an information retrieval system, if performance is measured by precision (the percentage of documents in the retrieval list that are actually relevant to the search topic). In some cases controlled vocabulary can enhance recall as well, because unlike natural language schemes, once the correct preferred term is searched, there is no need to search for other terms that might be synonyms of that term. === Disadvantages === A controlled vocabulary search may lead to unsatisfactory recall, in that it will fail to retrieve some documents that are actually relevant to the search question. This is particularly problematic when the search question involves terms that are sufficiently tangential to the subject area such that the indexer might have decided to tag it using a different term (but the searcher might consider the same). Essentially, this can be avoided only by an experienced user of controlled vocabulary whose understanding of the vocabulary coincides with that of the indexer. Another possibility is that the article is just not tagged by the indexer because indexing exhaustivity is low. For example, an article might mention football as a secondary focus, and the indexer might decide not to tag it with "football" because it is not important enough compared to the main focus. But it turns out that for the searcher that article is relevant and hence recall fails. A free text search would automatically pick up that article regardless. On the other hand, free text searches have high exhaustivity (every word is searched) so although it has much lower precision, it has potential for high recall as long as the searcher overcome the problem of synonyms by entering every combination. Controlled vocabularies may become outdated rapidly in fast developing fields of knowledge, unless the preferred terms are updated regularly. Even in an ideal scenario, a controlled vocabulary is often less specific than the words of the text itself. Indexers trying to choose the appropriate index terms might misinterpret the author, while this precise problem is not a factor in a free text, as it uses the author's own words. The use of controlled vocabularies can be costly compared to free