AltStore is an alternative app store for the iOS and iPadOS[1] mobile operating systems, which allows users to download applications that are not available on the App Store, most commonly tweaked apps, jailbreak apps, and apps including paid apps on the app store. It was publicly announced on September 25, 2019, and launched on September 28. == History == Riley Testut is an American developer who began to work on AltStore after Apple declined to allow his Nintendo emulator Delta on the App Store. Since Xcode allowed him to temporarily install his Delta app to his iOS device for 7 days of testing, he created AltStore in 2019 to replicate this functionality, which could be extended to other .ipa files. As of 2022, AltStore had been downloaded 1.5 million times. In the following years, AltStore expanded beyond its initial sideloading functionality. The platform was founded by Testut, with Shane Gill later joining as co-founder. AltStore was initially supported through Patreon contributions from its user community, and later saw increased adoption following regulatory developments in the European Union that enabled broader third-party app distribution. The project has also been involved in notable industry collaborations, including a partnership with Epic Games. == Features == AltStore exploits a loophole in the Xcode developer platform, which allows developers to sideload their own apps which they are working on without needing to jailbreak. Sideloaded apps are signed like a developer project for testing and will expire after 7 days with a free account or one year with a paid developer account, by which they will need to be refreshed or reinstalled.
Path tracing
Path tracing is a rendering algorithm in computer graphics that simulates how light interacts with objects and participating media to generate realistic (physically plausible) images. It is based on earlier, more limited, ray tracing algorithms. Path tracing is used to create photorealistic images for artistic purposes, and for applications such as architectural rendering and product design. It is also used to render frames for animated films, and visual effects for film and television. Because it can be very accurate and unbiased, it is commonly used to generate reference images when testing the quality of other rendering algorithms. The technique uses the Monte Carlo method to compute estimates of global illumination and simulate the ways different materials reflect (or scatter), transmit, absorb, and emit light. It can incorporate simple modeling of the effects of aperture and lens (depth of field, and bokeh) and shutter speed (motion blur), or more realistic simulation of the optical components in a camera. The algorithm works by describing illumination in a scene using the rendering equation, or light transport equation, and finding an approximate solution using Monte Carlo integration. An inefficient (but accurate) version of the algorithm can be very simple, and involves tracing a ray from the camera, allowing this ray to bounce in random directions as it hits different objects in the scene, and computing the amount of light transmitted along the path to the camera whenever the path encounters a light source. This process is repeated many times for each pixel (each repetition, with generated path and transmitted light, is called a sample), and the results are averaged. One main difference between this algorithm and standard ray tracing is that a single unbranching path is traced each time, while "Whitted-style" or "Cook-style" ray tracing recursively samples branching paths (e.g. when light is both reflected and refracted by a glass object). More practical versions incorporate improvements such as quasi-Monte Carlo methods (techniques that distribute samples more evenly), importance sampling (take more samples of paths that are likely to transport more light), and next event estimation (allow a very limited form of branching, and sample additional paths that connect to the lights more directly). Because path tracing uses random samples there is noise in the final image, which decreases as more samples are taken. Images commonly require many thousands of samples per pixel (spp) to reduce noise to an acceptable level, and denoising techniques (e.g. based on neural networks) are often used. Denoising is usually necessary when path tracing is used for real-time rendering in video games, because relatively few samples can be taken. Many alternative algorithms for path tracing have been developed, although they do not always outperform more straightforward implementations. These include bidirectional path tracing (which traces paths forwards from the light source as well as backwards from the camera), Metropolis light transport, and ways of combining path tracing with photon mapping. Video games often use biased versions of path tracing to improve performance (e.g. limiting the number of bounces in each path). A family of techniques called ReSTIR has been developed that can help real-time path tracing by sharing data between nearby pixels and consecutive frames. == History == Like all ray tracing methods, path tracing is based on ray casting, which Arthur Appel used for computer graphics rendering in the late 1960s. In 1980, John Turner Whitted published a recursive ray tracing algorithm that allows rendering images of scenes containing mirrored surfaces and refractive transparent objects. In 1984, Cook et al. described a form of ray tracing called distributed ray tracing, which uses Monte Carlo integration to render effects such as depth of field, motion blur, reflection from rough surfaces, and area lights. The same year, the radiosity method (not a ray tracing method) was published, which was the first physically based method for rendering diffuse global illumination. In 1986, Jim Kajiya published a paper exploring how to use distributed ray tracing to render physically-based global illumination, and this paper also introduced and named the method called "path tracing". Path tracing and other distributed ray tracing techniques were further refined in the late 1980s and early 1990s by researchers such as James Arvo and Peter Shirley, and by Greg Ward in the open source Radiance software. Despite being theoretically able to render any lighting, the original form of path tracing can sometimes be very inefficient (or noisy) for rendering light that is reflected or refracted before illuminating a visible surface, including diffuse global illumination where light enters an area through narrow gaps, because it traces paths only from the camera. To address this, variations of path tracing that trace paths from both the camera and from light sources, called bidirectional path tracing, were published in 1993 by Eric Lafortune and Yves Willems, and in 1997 by Eric Veach and Leonidas Guibas. In 1997 Veach and Guibas also published an alternative method called Metropolis light transport, which combines bidirectional path tracing with the Metropolis method. Veach's lengthy Ph.D. dissertation described both techniques, along with the theoretical background of path tracing; later, the book Physically Based Rendering (which won an Academy Award for Technical Achievement in 2014) helped to make information about path tracing more widely available. Path tracing requires tracing a large number of paths of light in order to produce an image with a visually acceptable amount of noise. This made path tracing very slow on computers available in the 1980s and 1990s, and noise remained a problem when trying to reproduce the style of earlier computer graphics animated films. Most animated films produced until around 2010, by studios such as Pixar, used rasterization-based rendering, with ray tracing used selectively for reflections (and later for precomputed or cached global illumination). However the speed of computers rapidly increased during the 1990s. Blue Sky Studios pioneered using Monte Carlo ray tracing for global illumination in animation, including in the 1998 short film "Bunny", but they did not disclose the precise techniques used. Path tracing gradually become more practical for film production in the early 2000s. The Arnold renderer, developed by Marcos Fajardo, was used by Sony Pictures Imageworks to produce the feature-length film Monster House, released in 2006. Pixar rewrote their RenderMan software to use path tracing, and released their first feature-length path-traced film Finding Dory in 2016. Although path tracing still had a large computational cost, animation studios discovered that less human labor was required when using it, for example because global illumination no longer needed to be faked by manually placing lights. The amount of noise present in path traced images still caused difficulties, particularly when rendering motion blur (which was used extensively by earlier animated films) but denoising techniques were developed to address this. New techniques were also needed for rendering hair and fur, and to handle the extremely large scenes sometimes required by films. Renderers such as Arnold, and Disney's Hyperion, originally only used CPUs for rendering, but as GPUs became more capable (and APIs such as CUDA, OpenCL, and OptiX were released) researchers and developers began adapting algorithms and implementations to use GPUs. GPUs can dramatically reduce rendering time: for example using a high-end GPU to accelerate portions of the rendering code can make it over 30 times faster than using only a high-end CPU. == Description == Kajiya's 1986 paper defined a recursive integral equation called the rendering equation, which describes a simplified form of light transport. Using Monte Carlo integration for the integral on the right side of the equation leads fairly directly to the path tracing algorithm: I ( x , x ′ ) = g ( x , x ′ ) [ ϵ ( x , x ′ ) + ∫ S ρ ( x , x ′ , x ″ ) I ( x ′ , x ″ ) d x ″ ] {\displaystyle I(x,x')=g(x,x')\left[\epsilon (x,x')+\int _{S}\rho (x,x',x'')I(x',x'')dx''\right]} This expresses I(x,x'), the light arriving at point x from point x', as the product of a geometry term, g(x,x'), which is 0 if there is something blocking the light between the two points and 1 otherwise, and the amount of light leaving point x' and traveling towards x. The light leaving point x' is the sum of the light emitted by the surface at x', and the integral of the light arriving at x' from all other points in the scene (the integration domain S) and being reflected towards x. The factor ρ(x,x',x''), which calculates how much light is reflected, must take into account the angles at which the light is arriving and leaving, and
Variable kernel density estimation
In statistics, adaptive or "variable-bandwidth" kernel density estimation is a form of kernel density estimation in which the size of the kernels used in the estimate are varied depending upon either the location of the samples or the location of the test point. It is a particularly effective technique when the sample space is multi-dimensional. == Rationale == Given a set of samples, { x → i } {\displaystyle \lbrace {\vec {x}}_{i}\rbrace } , we wish to estimate the density, P ( x → ) {\displaystyle P({\vec {x}})} , at a test point, x → {\displaystyle {\vec {x}}} : P ( x → ) ≈ W n h D {\displaystyle P({\vec {x}})\approx {\frac {W}{nh^{D}}}} W = ∑ i = 1 n w i {\displaystyle W=\sum _{i=1}^{n}w_{i}} w i = K ( x → − x → i h ) {\displaystyle w_{i}=K\left({\frac {{\vec {x}}-{\vec {x}}_{i}}{h}}\right)} where n is the number of samples, K is the "kernel", h is its width and D is the number of dimensions in x → {\displaystyle {\vec {x}}} . The kernel can be thought of as a simple, linear filter. Using a fixed filter width may mean that in regions of low density, all samples will fall in the tails of the filter with very low weighting, while regions of high density will find an excessive number of samples in the central region with weighting close to unity. To fix this problem, we vary the width of the kernel in different regions of the sample space. There are two methods of doing this: balloon and pointwise estimation. In a balloon estimator, the kernel width is varied depending on the location of the test point. In a pointwise estimator, the kernel width is varied depending on the location of the sample. For multivariate estimators, the parameter, h, can be generalized to vary not just the size, but also the shape of the kernel. This more complicated approach will not be covered here. == Balloon estimators == A common method of varying the kernel width is to make it inversely proportional to the density at the test point: h = k [ n P ( x → ) ] 1 / D {\displaystyle h={\frac {k}{\left[nP({\vec {x}})\right]^{1/D}}}} where k is a constant. If we back-substitute the estimated PDF, and assuming a Gaussian kernel function, we can show that W is a constant: W = k D ( 2 π ) D / 2 {\displaystyle W=k^{D}(2\pi )^{D/2}} A similar derivation holds for any kernel whose normalising function is of the order hD, although with a different constant factor in place of the (2 π)D/2 term. This produces a generalization of the k-nearest neighbour algorithm. That is, a uniform kernel function will return the KNN technique. There are two components to the error: a variance term and a bias term. The variance term is given as: e 1 = P ∫ K 2 n h D {\displaystyle e_{1}={\frac {P\int K^{2}}{nh^{D}}}} . The bias term is found by evaluating the approximated function in the limit as the kernel width becomes much larger than the sample spacing. By using a Taylor expansion for the real function, the bias term drops out: e 2 = h 2 n ∇ 2 P {\displaystyle e_{2}={\frac {h^{2}}{n}}\nabla ^{2}P} An optimal kernel width that minimizes the error of each estimate can thus be derived. == Use for statistical classification == The method is particularly effective when applied to statistical classification. There are two ways we can proceed: the first is to compute the PDFs of each class separately, using different bandwidth parameters, and then compare them as in Taylor. Alternatively, we can divide up the sum based on the class of each sample: P ( j , x → ) ≈ 1 n ∑ i = 1 , c i = j n w i {\displaystyle P(j,{\vec {x}})\approx {\frac {1}{n}}\sum _{i=1,c_{i}=j}^{n}w_{i}} where ci is the class of the ith sample. The class of the test point may be estimated through maximum likelihood.
Representer theorem
For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle f^{}} of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data. == Formal statement == The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } on a non-empty set X {\displaystyle {\mathcal {X}}} with a corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} . Let there be given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\dotsc ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } , a strictly increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R} } , and an arbitrary error function E : ( X × R 2 ) n → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{n}\to \mathbb {R} \cup \lbrace \infty \rbrace } , which together define the following regularized empirical risk functional on H k {\displaystyle H_{k}} : f ↦ E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) . {\displaystyle f\mapsto E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right).} Then, any minimizer of the empirical risk f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) } , ( ∗ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right)\right\rbrace ,\quad ()} admits a representation of the form: f ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } for all 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . Proof: Define a mapping φ : X → H k φ ( x ) = k ( ⋅ , x ) {\displaystyle {\begin{aligned}\varphi \colon {\mathcal {X}}&\to H_{k}\\\varphi (x)&=k(\cdot ,x)\end{aligned}}} (so that φ ( x ) = k ( ⋅ , x ) {\displaystyle \varphi (x)=k(\cdot ,x)} is itself a map X → R {\displaystyle {\mathcal {X}}\to \mathbb {R} } ). Since k {\displaystyle k} is a reproducing kernel, then φ ( x ) ( x ′ ) = k ( x ′ , x ) = ⟨ φ ( x ′ ) , φ ( x ) ⟩ , {\displaystyle \varphi (x)(x')=k(x',x)=\langle \varphi (x'),\varphi (x)\rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product on H k {\displaystyle H_{k}} . Given any x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , one can use orthogonal projection to decompose any f ∈ H k {\displaystyle f\in H_{k}} into a sum of two functions, one lying in span { φ ( x 1 ) , … , φ ( x n ) } {\displaystyle \operatorname {span} \left\lbrace \varphi (x_{1}),\ldots ,\varphi (x_{n})\right\rbrace } , and the other lying in the orthogonal complement: f = ∑ i = 1 n α i φ ( x i ) + v , {\displaystyle f=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,} where ⟨ v , φ ( x i ) ⟩ = 0 {\displaystyle \langle v,\varphi (x_{i})\rangle =0} for all i {\displaystyle i} . The above orthogonal decomposition and the reproducing property together show that applying f {\displaystyle f} to any training point x j {\displaystyle x_{j}} produces f ( x j ) = ⟨ ∑ i = 1 n α i φ ( x i ) + v , φ ( x j ) ⟩ = ∑ i = 1 n α i ⟨ φ ( x i ) , φ ( x j ) ⟩ , {\displaystyle f(x_{j})=\left\langle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,\varphi (x_{j})\right\rangle =\sum _{i=1}^{n}\alpha _{i}\langle \varphi (x_{i}),\varphi (x_{j})\rangle ,} which we observe is independent of v {\displaystyle v} . Consequently, the value of the error function E {\displaystyle E} in () is likewise independent of v {\displaystyle v} . For the second term (the regularization term), since v {\displaystyle v} is orthogonal to ∑ i = 1 n α i φ ( x i ) {\displaystyle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})} and g {\displaystyle g} is strictly monotonic, we have g ( ‖ f ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) + v ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ 2 + ‖ v ‖ 2 ) ≥ g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ ) . {\displaystyle {\begin{aligned}g\left(\lVert f\rVert \right)&=g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v\rVert \right)\\&=g\left({\sqrt {\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert ^{2}+\lVert v\rVert ^{2}}}\right)\\&\geq g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert \right).\end{aligned}}} Therefore, setting v = 0 {\displaystyle v=0} does not affect the first term of (), while it strictly decreases the second term. Consequently, any minimizer f ∗ {\displaystyle f^{}} in () must have v = 0 {\displaystyle v=0} , i.e., it must be of the form f ∗ ( ⋅ ) = ∑ i = 1 n α i φ ( x i ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} which is the desired result. == Generalizations == The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such. The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) = 1 n ∑ i = 1 n ( f ( x i ) − y i ) 2 , g ( ‖ f ‖ ) = λ ‖ f ‖ 2 {\displaystyle {\begin{aligned}E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)&={\frac {1}{n}}\sum _{i=1}^{n}(f(x_{i})-y_{i})^{2},\\g(\lVert f\rVert )&=\lambda \lVert f\rVert ^{2}\end{aligned}}} for λ > 0 {\displaystyle \lambda >0} . Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function g ( ⋅ ) {\displaystyle g(\cdot )} of the Hilbert space norm. It is possible to generalize further by augmenting the regularized empirical risk functional through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization f ~ ∗ = argmin { E ( ( x 1 , y 1 , f ~ ( x 1 ) ) , … , ( x n , y n , f ~ ( x n ) ) ) + g ( ‖ f ‖ ) ∣ f ~ = f + h ∈ H k ⊕ span { ψ p ∣ 1 ≤ p ≤ M } } , ( † ) {\displaystyle {\tilde {f}}^{}=\operatorname {argmin} \left\lbrace E\left((x_{1},y_{1},{\tilde {f}}(x_{1})),\ldots ,(x_{n},y_{n},{\tilde {f}}(x_{n}))\right)+g\left(\lVert f\rVert \right)\mid {\tilde {f}}=f+h\in H_{k}\oplus \operatorname {span} \lbrace \psi _{p}\mid 1\leq p\leq M\rbrace \right\rbrace ,\quad (\dagger )} i.e., we consider functions of the form f ~ = f + h {\displaystyle {\tilde {f}}=f+h} , where f ∈ H k {\displaystyle f\in H_{k}} and h {\displaystyle h} is an unpenalized function lying in the span of a finite set of real-valued functions { ψ p : X → R ∣ 1 ≤ p ≤ M } {\displaystyle \lbrace \psi _{p}\colon {\mathcal {X}}\to \mathbb {R} \mid 1\leq p\leq M\rbrace } . Under the assumption that the n × M {\displaystyle n\times M} matrix ( ψ p ( x i ) ) i p {\displaystyle \left(\psi _{p}(x_{i})\right)_{ip}} has rank M {\displaystyle M} , they show that the minimizer f ~ ∗ {\displaystyle {\tilde {f}}^{}} in ( † ) {\displaystyle (\dagger )} admits a representation of the form f ~ ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) + ∑ p = 1 M β p ψ p ( ⋅ ) {\displaystyle {\tilde {f}}^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i})+\sum _{p=1}^{M}\beta _{p}\psi _{p}(\cdot )} where α i , β p ∈ R {\displaystyle \alpha _{i},\beta _{p}\in \mathbb {R} } and the β p {\displaystyle \beta _{p}} are all uniquely determined. The conditions under which a representer theorem exists were investigated by Argyriou, Micchelli, and Pontil, who proved the following: Theorem: Let X {\displaystyle {\mathcal {X}}} be a nonempty set, k {\displaystyle k} a positive-definite real-valued kernel on X × X {\displaystyle {\mathcal {X}}\times {\mathcal {X}}} with corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} , and let R : H k → R {\displaystyle R\colon H_{k}\to \mathbb {R} } be a differentiable regularization function. Then given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } and an arbitrary error function E : ( X × R 2 ) m → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{m}\to \mathbb {R} \cup \lbrace \infty \rbrace } , a minimizer f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + R ( f ) } ( ‡ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+R(f)\right\rbrace \quad (\ddagger )} of the regularized empirical risk admits a repr
Non-negative matrix factorization
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically. NMF finds applications in such fields as astronomy, computer vision, document clustering, missing data imputation, chemometrics, audio signal processing, recommender systems, and bioinformatics. == History == In chemometrics non-negative matrix factorization has a long history under the name "self modeling curve resolution". In this framework the vectors in the right matrix are continuous curves rather than discrete vectors. Also early work on non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more widely known as non-negative matrix factorization after Lee and Seung investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations. == Background == Let matrix V be the product of the matrices W and H, V = W H . {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} Matrix multiplication can be implemented as computing the column vectors of V as linear combinations of the column vectors in W using coefficients supplied by columns of H. That is, each column of V can be computed as follows: v i = W h i , {\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} where vi is the i-th column vector of the product matrix V and hi is the i-th column vector of the matrix H. When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it is this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. For example, if V is an m × n matrix, W is an m × p matrix, and H is a p × n matrix then p can be significantly less than both m and n. Here is an example based on a text-mining application: Let the input matrix (the matrix to be factored) be V with 10000 rows and 500 columns where words are in rows and documents are in columns. That is, we have 500 documents indexed by 10000 words. It follows that a column vector v in V represents a document. Assume we ask the algorithm to find 10 features in order to generate a features matrix W with 10000 rows and 10 columns and a coefficients matrix H with 10 rows and 500 columns. The product of W and H is a matrix with 10000 rows and 500 columns, the same shape as the input matrix V and, if the factorization worked, it is a reasonable approximation to the input matrix V. From the treatment of matrix multiplication above it follows that each column in the product matrix WH is a linear combination of the 10 column vectors in the features matrix W with coefficients supplied by the coefficients matrix H. This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix W as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix H represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in W) where each feature is weighted by the feature's cell value from the document's column in H. == Clustering property == NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data V = ( v 1 , … , v n ) {\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} . More specifically, the approximation of V {\displaystyle \mathbf {V} } by V ≃ W H {\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } is achieved by finding W {\displaystyle W} and H {\displaystyle H} that minimize the error function (using the Frobenius norm) ‖ V − W H ‖ F , {\displaystyle \left\|V-WH\right\|_{F},} subject to W ≥ 0 , H ≥ 0. {\displaystyle W\geq 0,H\geq 0.} , If we furthermore impose an orthogonality constraint on H {\displaystyle \mathbf {H} } , i.e. H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed H {\displaystyle H} gives the cluster membership, i.e., if H k j > H i j {\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} for all i ≠ k, this suggests that the input data v j {\displaystyle v_{j}} belongs to k {\displaystyle k} -th cluster. The computed W {\displaystyle W} gives the cluster centroids, i.e., the k {\displaystyle k} -th column gives the cluster centroid of k {\displaystyle k} -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document clustering method. == Types == === Approximate non-negative matrix factorization === Usually the number of columns of W and the number of rows of H in NMF are selected so the product WH will become an approximation to V. The full decomposition of V then amounts to the two non-negative matrices W and H as well as a residual U, such that: V = WH + U. The elements of the residual matrix can either be negative or positive. When W and H are smaller than V they become easier to store and manipulate. Another reason for factorizing V into smaller matrices W and H, is that if one's goal is to approximately represent the elements of V by significantly less data, then one has to infer some latent structure in the data. === Convex non-negative matrix factorization === In standard NMF, matrix factor W ∈ R+m × k， i.e., W can be anything in that space. Convex NMF restricts the columns of W to convex combinations of the input data vectors ( v 1 , … , v n ) {\displaystyle (v_{1},\dots ,v_{n})} . This greatly improves the quality of data representation of W. Furthermore, the resulting matrix factor H becomes more sparse and orthogonal. === Nonnegative rank factorization === In case the nonnegative rank of V is equal to its actual rank, V = WH is called a nonnegative rank factorization (NRF). The problem of finding the NRF of V, if it exists, is known to be NP-hard. === Different cost functions and regularizations === There are different types of non-negative matrix factorizations. The different types arise from using different cost functions for measuring the divergence between V and WH and possibly by regularization of the W and/or H matrices. Two simple divergence functions studied by Lee and Seung are the squared error (or Frobenius norm) and an extension of the Kullback–Leibler divergence to positive matrices (the original Kullback–Leibler divergence is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules. The factorization problem in the squared error version of NMF may be stated as: Given a matrix V {\displaystyle \mathbf {V} } find nonnegative matrices W and H that minimize the function F ( W , H ) = ‖ V − W H ‖ F 2 {\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} Another type of NMF for images is based on the total variation norm. When L1 regularization (akin to Lasso) is added to NMF with the mean squared error cost function, the resulting problem may be called non-negative sparse coding due to the similarity to the sparse coding problem, although it may also still be referred to as NMF. === Online NMF === Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in streaming fashion. One such use is for collaborative filtering in recommendation systems, where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for o
Systems development life cycle
The systems development life cycle (SDLC) describes the typical phases and progression between phases during the development of a computer-based system. These phases progress from inception to retirement. At base, there is just one life cycle, but the taxonomy used to describe it may vary; the cycle may be classified into different numbers of phases and various names may be used for those phases. The SDLC is analogous to the life cycle of a living organism from its birth to its death. In particular, the SDLC varies by system in much the same way that each living organism has a unique path through its life. The SDLC does not prescribe how engineers should go about their work to move the system through its life cycle. Prescriptive techniques are referred to using various terms such as methodology, model, framework, and formal process. Other terms are used for the same concept as SDLC, including software development life cycle (also SDLC), application development life cycle (ADLC), and system design life cycle (also SDLC). These other terms focus on a different scope of development and are associated with different prescriptive techniques, but are about the same essential life cycle. The term "life cycle" is often written without a space, as "lifecycle", with the former more popular in the past and in non-engineering contexts. The acronym SDLC was coined when the longer form was more popular and has remained associated with the expansion, even though the shorter form is popular in engineering. Also, SDLC is relatively unique as opposed to the TLA SDL, which is highly overloaded. == Phases == Depending on the source, the SDLC is described as having different phases and using different terms. Even so, there are common aspects. The following attempts to describe notable phases using notable terminology. The phases are somewhat ordered by the natural sequence of development, although they can be overlapping and iterative. === Conceptualization === During conceptualization (a.k.a. conceptual design, system investigation, feasibility), options and priorities are considered. A feasibility study can determine whether the development effort is worthwhile via activities such as understanding user needs, cost estimation, benefit analysis, and resource analysis. A study should address operational, financial, technical, human factors, and legal/political concerns. === Requirements analysis === Requirements analysis (a.k.a. preliminary design) involves understanding the problem and determining what is needed. Often this involves engaging users to define the requirements and recording them in a document known as a requirements specification. === Design === During the design phase (a.k.a. detail design), a solution is planned. The plan can include relatively high-level information such as describing the major components of the system. The plan can include relatively low-level information such as describing functions, screen layout, business rules, and process flow. The design phase is informed by the requirements of the system. The design must satisfy each requirement. The design may be recorded in textual documents as well as functional hierarchy diagrams, example screen images, business rules, process diagrams, pseudo-code, and data models. === Construction === During construction (a.k.a. implementation, production), the system is realized. Based on the design, hardware and software components are created and integrated. This phase includes testing sub-components, components and the integration of some components, but typically does not include testing at the complete system level. This phase may include the development of training materials, including user manuals and help files. === Acceptance === The acceptance phase (a.k.a. system testing) is about testing the complete system to ensure that it meets customer expectations (requirements). === Deployment === The deployment phase (a.k.a. implementation) involves the logistics of delivery to the customer. Some systems are deployed as a single instance (i.e. in the cloud), and deployment may be ad hoc and manual. Some systems are built in quantity and are associated with manufacturing process and commissioning. This phase may include training users to use the system. It may include transitioning future development to support staff. === Maintenance === During the maintenance phase (a.k.a. operation, utilization, support) development is largely inactive, although this phase does include customer support for resolving user issues and recording suggestions for improvement. Fixes and enhancements are handled by returning to the first phase, conceptualization. For minor changes, the cycle may be significantly abbreviated compared to initial development. === Decommission === Decommission (a.k.a. disposition, retirement, phase-out) is when the system is removed from use, i.e., when it reaches end-of-life. == Practices == === Management and control === SDLC phase objectives are described in this section with key deliverables, a description of recommended tasks, and a summary of related control objectives for effective management. It is critical for the project manager to establish and monitor control objectives while executing projects. Control objectives are clear statements of the desired result or purpose and should be defined and monitored throughout a project. Control objectives can be grouped into major categories (domains), and relate to the SDLC phases as shown in the figure. To manage and control a substantial SDLC initiative, a work breakdown structure (WBS) captures and schedules the work. The WBS and all programmatic material should be kept in the "project description" section of the project notebook. The project manager chooses a WBS format that best describes the project. The diagram shows that coverage spans numerous phases of the SDLC, but the associated MCD (Management Control Domains) shows mappings to SDLC phases. For example, Analysis and Design is primarily performed as part of the Acquisition and Implementation Domain, and System Build and Prototype is primarily performed as part of delivery and support. === Work breakdown structured organization === The upper section of the WBS provides an overview of the project scope and timeline. It should also summarize the major phases and milestones. The middle section is based on the SDLC phases. WBS elements consist of milestones and tasks to be completed rather than activities to be undertaken, and have a deadline. Each task has a measurable output (e.g., an analysis document). A WBS task may rely on one or more activities (e.g., coding). Parts of the project needing support from contractors should have a statement of work (SOW). The development of an SOW does not occur during a specific phase of SDLC but is developed to include the work from the SDLC process that may be conducted by contractors. === Baselines === Baselines are established after four of the five phases of the SDLC, and are critical to the iterative nature of the model. Baselines become milestones. functional baseline: established after the conceptual design phase. allocated baseline: established after the preliminary design phase. product baseline: established after the detailed design and development phase. updated product baseline: established after the production construction phase. In the following diagram, these stages are divided into ten steps, from definition to creation and modification of IT work products:
Spiking neural network
Spiking neural networks (SNNs) are artificial neural networks (ANN) that mimic natural neural networks. These models leverage timing of discrete spikes as the main information carrier. In addition to neuronal and synaptic state, SNNs incorporate the concept of time into their operating model. The idea is that neurons in the SNN do not transmit information at each propagation cycle (as it happens with typical multi-layer perceptron networks), but rather transmit information only when a membrane potential—an intrinsic quality of the neuron related to its membrane electrical charge—reaches a specific value, called the threshold. When the membrane potential reaches the threshold, the neuron fires, and generates a signal that travels to other neurons which, in turn, increase or decrease their potentials in response to this signal. A neuron model that fires at the moment of threshold crossing is also called a spiking neuron model. While spike rates can be considered the analogue of the variable output of a traditional ANN, neurobiology research indicated that high speed processing cannot be performed solely through a rate-based scheme. For example humans can perform an image recognition task requiring no more than 10ms of processing time per neuron through the successive layers (going from the retina to the temporal lobe). This time window is too short for rate-based encoding. The precise spike timings in a small set of spiking neurons also has a higher information coding capacity compared with a rate-based approach. The most prominent spiking neuron model is the leaky integrate-and-fire model. In that model, the momentary activation level (modeled as a differential equation) is normally considered to be the neuron's state, with incoming spikes pushing this value higher or lower, until the state eventually either decays or—if the firing threshold is reached—the neuron fires. After firing, the state variable is reset to a lower value. Various decoding methods exist for interpreting the outgoing spike train as a real-value number, relying on either the frequency of spikes (rate-code), the time-to-first-spike after stimulation, or the interval between spikes. == History == Many multi-layer artificial neural networks are fully connected, receiving input from every neuron in the previous layer and signalling every neuron in the subsequent layer. Although these networks have achieved breakthroughs, they do not match biological networks and do not mimic neurons. The biology-inspired Hodgkin–Huxley model of a spiking neuron was proposed in 1952. This model described how action potentials are initiated and propagated. Communication between neurons, which requires the exchange of chemical neurotransmitters in the synaptic gap, is described in models such as the integrate-and-fire model, FitzHugh–Nagumo model (1961–1962), and Hindmarsh–Rose model (1984). The leaky integrate-and-fire model (or a derivative) is commonly used as it is easier to compute than Hodgkin–Huxley. While the notion of an artificial spiking neural network became popular only in the twenty-first century, studies between 1980 and 1995 supported the concept. The first models of this type of ANN appeared to simulate non-algorithmic intelligent information processing systems. However, the notion of the spiking neural network as a mathematical model was first worked on in the early 1970s. As of 2019 SNNs lagged behind ANNs in accuracy, but the gap is decreasing, and has vanished on some tasks. == Underpinnings == Information in the brain is represented as action potentials (neuron spikes), which may group into spike trains or coordinated waves. A fundamental question of neuroscience is to determine whether neurons communicate by a rate or temporal code. Temporal coding implies that a single spiking neuron can replace hundreds of hidden units on a conventional neural net. SNNs define a neuron's current state as its potential (possibly modeled as a differential equation). An input pulse causes the potential to rise and then gradually decline. Encoding schemes can interpret these pulse sequences as a number, considering pulse frequency and pulse interval. Using the precise time of pulse occurrence, a neural network can consider more information and offer better computing properties. SNNs compute in the continuous domain. Such neurons test for activation only when their potentials reach a certain value. When a neuron is activated, it produces a signal that is passed to connected neurons, accordingly raising or lowering their potentials. The SNN approach produces a continuous output instead of the binary output of traditional ANNs. Pulse trains are not easily interpretable, hence the need for encoding schemes. However, a pulse train representation may be more suited for processing spatiotemporal data (or real-world sensory data classification). SNNs connect neurons only to nearby neurons so that they process input blocks separately (similar to CNN using filters). They consider time by encoding information as pulse trains so as not to lose information. This avoids the complexity of a recurrent neural network (RNN). Impulse neurons are more powerful computational units than traditional artificial neurons. SNNs are theoretically more powerful than so called "second-generation networks" defined as ANNs "based on computational units that apply activation function with a continuous set of possible output values to a weighted sum (or polynomial) of the inputs"; however, SNN training issues and hardware requirements limit their use. Although unsupervised biologically inspired learning methods are available such as Hebbian learning and STDP, no effective supervised training method is suitable for SNNs that can provide better performance than second-generation networks. Spike-based activation of SNNs is not differentiable, thus gradient descent-based backpropagation (BP) is not available. SNNs have much larger computational costs for simulating realistic neural models than traditional ANNs. Pulse-coupled neural networks (PCNN) are often confused with SNNs. A PCNN can be seen as a kind of SNN. Researchers are actively working on various topics. The first concerns differentiability. The expressions for both the forward- and backward-learning methods contain the derivative of the neural activation function which is not differentiable because a neuron's output is either 1 when it spikes, and 0 otherwise. This all-or-nothing behavior disrupts gradients and makes these neurons unsuitable for gradient-based optimization. Approaches to resolving it include: resorting to entirely biologically inspired local learning rules for the hidden units translating conventionally trained "rate-based" NNs to SNNs smoothing the network model to be continuously differentiable defining an SG (Surrogate Gradient) as a continuous relaxation of the real gradients The second concerns the optimization algorithm. Standard BP can be expensive in terms of computation, memory, and communication and may be poorly suited to the hardware that implements it (e.g., a computer, brain, or neuromorphic device). Incorporating additional neuron dynamics such as Spike Frequency Adaptation (SFA) is a notable advance, enhancing efficiency and computational power. These neurons sit between biological complexity and computational complexity. Originating from biological insights, SFA offers significant computational benefits by reducing power usage, especially in cases of repetitive or intense stimuli. This adaptation improves signal/noise clarity and introduces an elementary short-term memory at the neuron level, which in turn, improves accuracy and efficiency. This was mostly achieved using compartmental neuron models. The simpler versions are of neuron models with adaptive thresholds, are an indirect way of achieving SFA. It equips SNNs with improved learning capabilities, even with constrained synaptic plasticity, and elevates computational efficiency. This feature lessens the demand on network layers by decreasing the need for spike processing, thus lowering computational load and memory access time—essential aspects of neural computation. Moreover, SNNs utilizing neurons capable of SFA achieve levels of accuracy that rival those of conventional ANNs, while also requiring fewer neurons for comparable tasks. This efficiency streamlines the computational workflow and conserves space and energy, while maintaining technical integrity. High-performance deep spiking neural networks can operate with 0.3 spikes per neuron. == Applications == SNNs can in principle be applied to the same applications as traditional ANNs. In addition, SNNs can model the central nervous system of biological organisms, such as an insect seeking food without prior knowledge of the environment. Due to their relative realism, they can be used to study biological neural circuits. Starting with a hypothesis about the topology of a biological neuronal circuit and its functi