In image processing, a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between the kernel and an image. Or more simply, when each pixel in the output image is a function of the nearby pixels (including itself) in the input image, the kernel is that function. == Details == The general expression of a convolution is g x , y = ω ∗ f x , y = ∑ i = − a a ∑ j = − b b ω i , j f x − i , y − j , {\displaystyle g_{x,y}=\omega f_{x,y}=\sum _{i=-a}^{a}{\sum _{j=-b}^{b}{\omega _{i,j}f_{x-i,y-j}}},} where g ( x , y ) {\displaystyle g(x,y)} is the filtered image, f ( x , y ) {\displaystyle f(x,y)} is the original image, ω {\displaystyle \omega } is the filter kernel. Every element of the filter kernel is considered by − a ≤ i ≤ a {\displaystyle -a\leq i\leq a} and − b ≤ j ≤ b {\displaystyle -b\leq j\leq b} . Depending on the element values, a kernel can cause a wide range of effects: The above are just a few examples of effects achievable by convolving kernels and images. === Origin === The origin is the position of the kernel which is above (conceptually) the current output pixel. This could be outside of the actual kernel, though usually it corresponds to one of the kernel elements. For a symmetric kernel, the origin is usually the center element. == Convolution == Convolution is the process of adding each element of the image to its local neighbors, weighted by the kernel. This is related to a form of mathematical convolution. The matrix operation being performed—convolution—is not traditional matrix multiplication, despite being similarly denoted by . For example, if we have two three-by-three matrices, the first a kernel, and the second an image piece, convolution is the process of flipping both the rows and columns of the kernel and multiplying locally similar entries and summing. The element at coordinates [2, 2] (that is, the central element) of the resulting image would be a weighted combination of all the entries of the image matrix, with weights given by the kernel: ( [ a b c d e f g h i ] ∗ [ 1 2 3 4 5 6 7 8 9 ] ) [ 2 , 2 ] = {\displaystyle \left({\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}{\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}\right)[2,2]=} ( i ⋅ 1 ) + ( h ⋅ 2 ) + ( g ⋅ 3 ) + ( f ⋅ 4 ) + ( e ⋅ 5 ) + ( d ⋅ 6 ) + ( c ⋅ 7 ) + ( b ⋅ 8 ) + ( a ⋅ 9 ) . {\displaystyle (i\cdot 1)+(h\cdot 2)+(g\cdot 3)+(f\cdot 4)+(e\cdot 5)+(d\cdot 6)+(c\cdot 7)+(b\cdot 8)+(a\cdot 9).} The other entries would be similarly weighted, where we position the center of the kernel on each of the boundary points of the image, and compute a weighted sum. The values of a given pixel in the output image are calculated by multiplying each kernel value by the corresponding input image pixel values. This can be described algorithmically with the following pseudo-code: for each image row in input image: for each pixel in image row: set accumulator to zero for each kernel row in kernel: for each element in kernel row: if element position corresponding to pixel position then multiply element value corresponding to pixel value add result to accumulator endif set output image pixel to accumulator corresponding input image pixels are found relative to the kernel's origin. If the kernel is symmetric then place the center (origin) of the kernel on the current pixel. The kernel will overlap the neighboring pixels around the origin. Each kernel element should be multiplied with the pixel value it overlaps with and all of the obtained values should be summed. This resultant sum will be the new value for the current pixel currently overlapped with the center of the kernel. If the kernel is not symmetric, it has to be flipped both around its horizontal and vertical axis before calculating the convolution as above. The general form for matrix convolution is [ x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋮ ⋮ ⋱ ⋮ x m 1 x m 2 ⋯ x m n ] ∗ [ y 11 y 12 ⋯ y 1 n y 21 y 22 ⋯ y 2 n ⋮ ⋮ ⋱ ⋮ y m 1 y m 2 ⋯ y m n ] = ∑ i = 0 m − 1 ∑ j = 0 n − 1 x ( m − i ) ( n − j ) y ( 1 + i ) ( 1 + j ) {\displaystyle {\begin{bmatrix}x_{11}&x_{12}&\cdots &x_{1n}\\x_{21}&x_{22}&\cdots &x_{2n}\\\vdots &\vdots &\ddots &\vdots \\x_{m1}&x_{m2}&\cdots &x_{mn}\\\end{bmatrix}}{\begin{bmatrix}y_{11}&y_{12}&\cdots &y_{1n}\\y_{21}&y_{22}&\cdots &y_{2n}\\\vdots &\vdots &\ddots &\vdots \\y_{m1}&y_{m2}&\cdots &y_{mn}\\\end{bmatrix}}=\sum _{i=0}^{m-1}\sum _{j=0}^{n-1}x_{(m-i)(n-j)}y_{(1+i)(1+j)}} === Edge handling === Kernel convolution usually requires values from pixels outside of the image boundaries. There are a variety of methods for handling image edges. Extend The nearest border pixels are conceptually extended as far as necessary to provide values for the convolution. Corner pixels are extended in 90° wedges. Other edge pixels are extended in lines. Wrap The image is conceptually wrapped (or tiled) and values are taken from the opposite edge or corner. Mirror The image is conceptually mirrored at the edges. For example, attempting to read a pixel 3 units outside an edge reads one 3 units inside the edge instead. Crop / Avoid overlap Any pixel in the output image which would require values from beyond the edge is skipped. This method can result in the output image being slightly smaller, with the edges having been cropped. Move kernel so that values from outside of image is never required. Machine learning mainly uses this approach. Example: Kernel size 10x10, image size 32x32, result image is 23x23. Kernel Crop Any pixel in the kernel that extends past the input image isn't used and the normalizing is adjusted to compensate. Constant Use constant value for pixels outside of image. Usually black or sometimes gray is used. Generally this depends on application. === Normalization === Normalization is defined as the division of each element in the kernel by the sum of all kernel elements, so that the sum of the elements of a normalized kernel is unity. This will ensure the average pixel in the modified image is as bright as the average pixel in the original image. === Optimization === Fast convolution algorithms include: separable convolution ==== Separable convolution ==== 2D convolution with an M × N kernel requires M × N multiplications for each sample (pixel). If the kernel is separable, then the computation can be reduced to M + N multiplications. Using separable convolutions can significantly decrease the computation by doing 1D convolution twice instead of one 2D convolution. === Implementation === Here a concrete convolution implementation done with the GLSL shading language :
Intelligent database
Until the 1980s, databases were viewed as computer systems that stored record-oriented and business data such as manufacturing inventories, bank records, and sales transactions. A database system was not expected to merge numeric data with text, images, or multimedia information, nor was it expected to automatically notice patterns in the data it stored. In the late 1980s the concept of an intelligent database was put forward as a system that manages information (rather than data) in a way that appears natural to users and which goes beyond simple record keeping. The term was introduced in 1989 by the book Intelligent Databases by Kamran Parsaye, Mark Chignell, Setrag Khoshafian and Harry Wong. The concept postulated three levels of intelligence for such systems: high level tools, the user interface and the database engine. The high level tools manage data quality and automatically discover relevant patterns in the data with a process called data mining. This layer often relies on the use of artificial intelligence techniques. The user interface uses hypermedia in a form that uniformly manages text, images and numeric data. The intelligent database engine supports the other two layers, often merging relational database techniques with object orientation. In the twenty-first century, intelligent databases have now become widespread, e.g. hospital databases can now call up patient histories consisting of charts, text and x-ray images just with a few mouse clicks, and many corporate databases include decision support tools based on sales pattern analysis.
Jeremy Renner Official
Jeremy Renner Official (or Jeremy Renner on the Google Play Store) was a mobile app created by American actor Jeremy Renner. He created the app in March 2017 to hear the input and comments of his fans. The app was shut down in September 2019 in part due to the frequent bullying and trolling that the platform had experienced. The app featured optional microtransactions, with some ranging up to roughly US$400 despite the app itself being free. Upon shutting down the app, Renner issued a mass-refund for the collectible "stars" in the app for purchases made within the last ninety days, from the day the announcement was posted. He then posted an apology to the app itself, and the app was deleted from both the Google Play Store and the App Store shortly after. == Usage == Upon downloading the app, the user was faced with a video of Renner speaking about his fans and superfans, regular giveaways, and real-life updates. While the app was active, Renner posted regular questions and comments for fans. Renner occasionally livestreamed about his work and day-to-day life. The community developed to include memes, selfies, and a "Happy Rennsday" event on Wednesdays. == History == === 2017–2019 === The app launched in March 2017 with a promotional contest. Renner's fans were encouraged to download the app and create comments about being Renner's biggest fan; Renner would then choose a winner and transport the winner and a guest to have lunch with him at the Calgary Expo. In the first few months Renner teased behind-the-scenes of projects he was working on, which he now sporadically does on Instagram. The app was similarly designed to Instagram as well, with a near identically styled layout. Around midway through 2019, a hoax account of Renner was made to mock the celebrity, joking about masturbating to porn and defending another hoax account of Casey Anthony. FastCompany wrote extensively about Renner's app in April 2019, calling it "a surprising new kind of social media". The Ringer stated "Jeremy Renner's Jeremy Renner app is the Jeremy Renner of apps." === After deletion (2019–2020) === After the shutdown of the app, a comedy-based pseudo-app with modular endings was released, called "The Jeremy Renner App Experience", in which the player plays as Jeremy Renner on the day of the Jeremy Renner Official app's shutdown. The app details several different choices on how Renner handles the situation. A six-part podcast was also created to mock the app's deletion, called The Renner Files, featuring Carolyn Goldfarb and Sarah Ramos. == Controversies == === Marketing === One of the main controversies of Renner's app was its marketing. The app's developers, Escapex, specialized in and grew famous for making similar monetized apps for celebrities. The marketing campaign was based on direct contact with Renner, whose chances were increased with regular payments for "stars", although very few encounters seemed to happen with Renner himself. The multiple problems with the app led the CEO of Escapex, Sephi Shapira, to call the app a "freak situation", and added "Am I concerned about this? Not more than I'm concerned about 50 other things I'm dealing with as a startup company." Along with the marketing failures, the app was seen as misrepresenting itself as seemingly erotic with some advertisements featuring Renner suggestively staring at the camera, despite the actual app being initially considered safe for children. === Harassment === After its release in 2017, the app was met with waves of harassment and bullying by many users on the app, most frequently by using impersonation — referenced in Renner's apology/deletion notice. Some death threats were made across the app by fraud accounts pretending to be several controversial celebrities, including O. J. Simpson and Casey Anthony. As early as October 2017, there were claims of censorship, bullying, and "contest-rigging". In September 2019, comedian Stefan Heck publicized his discovery of the fact that replies through the app appeared as if they were sent by Renner himself in push notifications. After several users abused this feature, Renner asked Escapex to shut down the app.
List of security-focused operating systems
This is a list of operating systems specifically focused on security. Similar concepts include security-evaluated operating systems that have achieved certification from an auditing organization, and trusted operating systems that provide sufficient support for multilevel security and evidence of correctness to meet a particular set of requirements. == Linux == === Android-based === GrapheneOS is a security-focused, Android-based mobile OS that uses a hardened kernel, C library, custom memory allocator (hardened_malloc), and a hardened Chromium-based browser named Vanadium. It also offers privacy/security features, such as Duress PIN/Password or disabling the USB-C port at a driver/hardware level to avoid exploitation. It deploys exploit mitigations such as hardware-based memory tagging, secure app spawning, restricted dynamic code loading, and more. === Debian-based === Linux Kodachi is a security-focused operating system. Tails is aimed at preserving privacy and anonymity. KickSecure is a security-focused Linux distribution that aims to be "hardened by default". It uses network hardening, kernel hardening, Strong Linux User Account Isolation, better randomness, root access restrictions, and app-specific hardening. Whonix is an anonymity focused operating system based on KickSecure. It consists of two virtual machines, And all communications are routed through Tor. === Other Linux distributions === Alpine Linux is designed to be small, simple, and secure. It uses musl, BusyBox, and OpenRC instead of the more commonly used glibc, GNU Core Utilities, and systemd. Owl - Openwall GNU/Linux, a security-enhanced Linux distribution for servers. Secureblue, a Fedora Silverblue based distro that uses a hardened kernel, custom memory allocator (hardened_malloc), Trivalent, a security-focused, Chromium-based browser inspired by Vanadium, and many other exploit mitigations. == BSD == OpenBSD is a Unix-like operating system that emphasizes portability, standardization, correctness, proactive security, and integrated cryptography. == Xen == Qubes OS aims to provide security through isolation. Isolation is provided through the use of virtualization technology. This allows the segmentation of applications into secure virtual machines.
Global serializability
In concurrency control of databases, transaction processing (transaction management), and other transactional distributed applications, global serializability (or modular serializability) is a property of a global schedule of transactions. A global schedule is the unified schedule of all the individual database (and other transactional object) schedules in a multidatabase environment (e.g., federated database). Complying with global serializability means that the global schedule is serializable, has the serializability property, while each component database (module) has a serializable schedule as well. In other words, a collection of serializable components provides overall system serializability, which is usually incorrect. A need in correctness across databases in multidatabase systems makes global serializability a major goal for global concurrency control (or modular concurrency control). With the proliferation of the Internet, Cloud computing, Grid computing, and small, portable, powerful computing devices (e.g., smartphones), as well as increase in systems management sophistication, the need for atomic distributed transactions and thus effective global serializability techniques, to ensure correctness in and among distributed transactional applications, seems to increase. In a federated database system or any other more loosely defined multidatabase system, which are typically distributed in a communication network, transactions span multiple (and possibly distributed) databases. Enforcing global serializability in such system, where different databases may use different types of concurrency control, is problematic. Even if every local schedule of a single database is serializable, the global schedule of a whole system is not necessarily serializable. The massive communication exchanges of conflict information needed between databases to reach conflict serializability globally would lead to unacceptable performance, primarily due to computer and communication latency. Achieving global serializability effectively over different types of concurrency control has been open for several years. == The global serializability problem == === Problem statement === The difficulties described above translate into the following problem: Find an efficient (high-performance and fault tolerant) method to enforce Global serializability (global conflict serializability) in a heterogeneous distributed environment of multiple autonomous database systems. The database systems may employ different concurrency control methods. No limitation should be imposed on the operations of either local transactions (confined to a single database system) or global transactions (span two or more database systems). === Quotations === Lack of an appropriate solution for the global serializability problem has driven researchers to look for alternatives to serializability as a correctness criterion in a multidatabase environment (e.g., see Relaxing global serializability below), and the problem has been characterized as difficult and open. The following two quotations demonstrate the mindset about it by the end of the year 1991, with similar quotations in numerous other articles: "Without knowledge about local as well as global transactions, it is highly unlikely that efficient global concurrency control can be provided... Additional complications occur when different component DBMSs [Database Management Systems] and the FDBMSs [Federated Database Management Systems] support different concurrency mechanisms... It is unlikely that a theoretically elegant solution that provides conflict serializability without sacrificing performance (i.e., concurrency and/or response time) and availability exists." === Proposed solutions === Several solutions, some partial, have been proposed for the global serializability problem. Among them: Global conflict graph (serializability graph, precedence graph) checking Distributed Two-phase locking (Distributed 2PL) Distributed Timestamp ordering Tickets (local logical timestamps which define local total orders, and are propagated to determine global partial order of transactions) == Relaxing global serializability == Some techniques have been developed for relaxed global serializability (i.e., they do not guarantee global serializability; see also Relaxing serializability). Among them (with several publications each): Quasi serializability Two-level serializability Another common reason nowadays for Global serializability relaxation is the requirement of availability of internet products and services. This requirement is typically answered by large scale data replication. The straightforward solution for synchronizing replicas' updates of a same database object is including all these updates in a single atomic distributed transaction. However, with many replicas such a transaction is very large, and may span several computers and networks that some of them are likely to be unavailable. Thus such a transaction is likely to end with abort and miss its purpose. Consequently, Optimistic replication (Lazy replication) is often utilized (e.g., in many products and services by Google, Amazon, Yahoo, and alike), while global serializability is relaxed and compromised for eventual consistency. In this case relaxation is done only for applications that are not expected to be harmed by it. Classes of schedules defined by relaxed global serializability properties either contain the global serializability class, or are incomparable with it. What differentiates techniques for relaxed global conflict serializability (RGCSR) properties from those of relaxed conflict serializability (RCSR) properties that are not RGCSR is typically the different way global cycles (span two or more databases) in the global conflict graph are handled. No distinction between global and local cycles exists for RCSR properties that are not RGCSR. RCSR contains RGCSR. Typically RGCSR techniques eliminate local cycles, i.e., provide local serializability (which can be achieved effectively by regular, known concurrency control methods); however, obviously they do not eliminate all global cycles (which would achieve global serializability).
Statistical learning theory
Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. == Introduction == The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input–output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to predict the output from future input. Depending on the type of output, supervised learning problems are either problems of regression or problems of classification. If the output takes a continuous range of values, it is a regression problem. Using Ohm's law as an example, a regression could be performed with voltage as input and current as an output. The regression would find the functional relationship between voltage and current to be R {\displaystyle R} , such that V = I R {\displaystyle V=IR} Classification problems are those for which the output will be an element from a discrete set of labels. Classification is very common for machine learning applications. In facial recognition, for instance, a picture of a person's face would be the input, and the output label would be that person's name. The input would be represented by a large multidimensional vector whose elements represent pixels in the picture. After learning a function based on the training set data, that function is validated on a test set of data, data that did not appear in the training set. == Formal description == Take X {\displaystyle X} to be the vector space of all possible inputs, and Y {\displaystyle Y} to be the vector space of all possible outputs. Statistical learning theory takes the perspective that there is some unknown probability distribution over the product space Z = X × Y {\displaystyle Z=X\times Y} , i.e. there exists some unknown p ( z ) = p ( x , y ) {\displaystyle p(z)=p(\mathbf {x} ,y)} . The training set is made up of n {\displaystyle n} samples from this probability distribution, and is notated S = { ( x 1 , y 1 ) , … , ( x n , y n ) } = { z 1 , … , z n } {\displaystyle S=\{(\mathbf {x} _{1},y_{1}),\dots ,(\mathbf {x} _{n},y_{n})\}=\{\mathbf {z} _{1},\dots ,\mathbf {z} _{n}\}} Every x i {\displaystyle \mathbf {x} _{i}} is an input vector from the training data, and y i {\displaystyle y_{i}} is the output that corresponds to it. In this formalism, the inference problem consists of finding a function f : X → Y {\displaystyle f:X\to Y} such that f ( x ) ∼ y {\displaystyle f(\mathbf {x} )\sim y} . Let H {\displaystyle {\mathcal {H}}} be a space of functions f : X → Y {\displaystyle f:X\to Y} called the hypothesis space. The hypothesis space is the space of functions the algorithm will search through. Let V ( f ( x ) , y ) {\displaystyle V(f(\mathbf {x} ),y)} be the loss function, a metric for the difference between the predicted value f ( x ) {\displaystyle f(\mathbf {x} )} and the actual value y {\displaystyle y} . The expected risk is defined to be I [ f ] = ∫ X × Y V ( f ( x ) , y ) p ( x , y ) d x d y {\displaystyle I[f]=\int _{X\times Y}V(f(\mathbf {x} ),y)\,p(\mathbf {x} ,y)\,d\mathbf {x} \,dy} The target function, the best possible function f {\displaystyle f} that can be chosen, is given by the f {\displaystyle f} that satisfies f = argmin h ∈ H I [ h ] {\displaystyle f=\mathop {\operatorname {argmin} } _{h\in {\mathcal {H}}}I[h]} Because the probability distribution p ( x , y ) {\displaystyle p(\mathbf {x} ,y)} is unknown, a proxy measure for the expected risk must be used. This measure is based on the training set, a sample from this unknown probability distribution. It is called the empirical risk I S [ f ] = 1 n ∑ i = 1 n V ( f ( x i ) , y i ) {\displaystyle I_{S}[f]={\frac {1}{n}}\sum _{i=1}^{n}V(f(\mathbf {x} _{i}),y_{i})} A learning algorithm that chooses the function f S {\displaystyle f_{S}} that minimizes the empirical risk is called empirical risk minimization. == Loss functions == The choice of loss function is a determining factor on the function f S {\displaystyle f_{S}} that will be chosen by the learning algorithm. The loss function also affects the convergence rate for an algorithm. It is important for the loss function to be convex. Different loss functions are used depending on whether the problem is one of regression or one of classification. === Regression === The most common loss function for regression is the square loss function (also known as the L2-norm). This familiar loss function is used in Ordinary Least Squares regression. The form is: V ( f ( x ) , y ) = ( y − f ( x ) ) 2 {\displaystyle V(f(\mathbf {x} ),y)=(y-f(\mathbf {x} ))^{2}} The absolute value loss (also known as the L1-norm) is also sometimes used: V ( f ( x ) , y ) = | y − f ( x ) | {\displaystyle V(f(\mathbf {x} ),y)=|y-f(\mathbf {x} )|} === Classification === In some sense the 0-1 indicator function is the most natural loss function for classification. It takes the value 0 if the predicted output is the same as the actual output, and it takes the value 1 if the predicted output is different from the actual output. For binary classification with Y = { − 1 , 1 } {\displaystyle Y=\{-1,1\}} , this is: V ( f ( x ) , y ) = θ ( − y f ( x ) ) {\displaystyle V(f(\mathbf {x} ),y)=\theta (-yf(\mathbf {x} ))} where θ {\displaystyle \theta } is the Heaviside step function. == Regularization == In machine learning problems, a major problem that arises is that of overfitting. Because learning is a prediction problem, the goal is not to find a function that most closely fits the (previously observed) data, but to find one that will most accurately predict output from future input. Empirical risk minimization runs this risk of overfitting: finding a function that matches the data exactly but does not predict future output well. Overfitting is symptomatic of unstable solutions; a small perturbation in the training set data would cause a large variation in the learned function. It can be shown that if the stability for the solution can be guaranteed, generalization and consistency are guaranteed as well. Regularization can solve the overfitting problem and give the problem stability. Regularization can be accomplished by restricting the hypothesis space H {\displaystyle {\mathcal {H}}} . A common example would be restricting H {\displaystyle {\mathcal {H}}} to linear functions: this can be seen as a reduction to the standard problem of linear regression. H {\displaystyle {\mathcal {H}}} could also be restricted to polynomial of degree p {\displaystyle p} , exponentials, or bounded functions on L1. Restriction of the hypothesis space avoids overfitting because the form of the potential functions are limited, and so does not allow for the choice of a function that gives empirical risk arbitrarily close to zero. One example of regularization is Tikhonov regularization. This consists of minimizing 1 n ∑ i = 1 n V ( f ( x i ) , y i ) + γ ‖ f ‖ H 2 {\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}V(f(\mathbf {x} _{i}),y_{i})+\gamma \left\|f\right\|_{\mathcal {H}}^{2}} where γ {\displaystyle \gamma } is a fixed and positive parameter, the regularization parameter. Tikhonov regularization ensures existence, uniqueness, and stability of the solution. == Bounding empirical risk == Consider a binary classifier f : X → { 0 , 1 } {\displaystyle f:{\mathcal {X}}\to \{0,1\}} . We can apply Hoeffding's inequality to bound the probability that the empirical risk deviates from the true risk to be a Sub-Gaussian distribution. P ( | R ^ ( f ) − R ( f ) | ≥ ϵ ) ≤ 2 e − 2 n ϵ 2 {\displaystyle \mathbb {P} (|{\hat {R}}(f)-R(f)|\geq \epsilon )\leq 2e^{-2n\epsilon ^{2}}} But generally, when we do empirical risk minimization, we are not given a classifier; we must choose it. Therefore, a more useful result is to bound the probability of the supremum of the difference over the whole class. P ( sup f ∈ F | R ^ ( f ) − R ( f ) | ≥ ϵ ) ≤ 2 S ( F , n ) e − n ϵ 2 / 8 ≈ n d e − n ϵ 2 / 8 {\displaystyle \mathbb {P} {\bigg (}\sup _{f\in {\mathcal {F}}}|{\hat {R}}(f)-R(f)|\geq \epsilon {\bigg )}\leq 2S({\mathcal {F}},n)e^{-n\epsilon ^{2}/8}\approx n^{d}e^{-n\epsilon ^{2}/8}} where S ( F , n ) {\displaystyle S({\mathcal {F}},n)} is the shattering number and n {\displaystyle n} is the number of samples in your dataset. The exponential term comes from Hoeffding but there is an extra cost of taking the supremum over the whole cla
Simulation noise
Simulation noise is a function that creates a divergence-free vector field. This signal can be used in artistic simulations for the purpose of increasing the perception of extra detail. The function can be calculated in three dimensions by dividing the space into a regular lattice grid. With each edge is associated a random value, indicating a rotational component of material revolving around the edge. By following rotating material into and out of faces, one can quickly sum the flux passing through each face of the lattice. Flux values at lattice faces are then interpolated to create a field value for all positions. Perlin noise is the earliest form of lattice noise, which has become very popular in computer graphics. Perlin Noise is not suited for simulation because it is not divergence-free. Noises based on lattices, such as simulation noise and Perlin noise, are often calculated at different frequencies and summed together to form band-limited fractal signals. Other approaches developed later that use vector calculus identities to produce divergence free fields, such as "Curl-Noise" as suggested by Rook Bridson, and "Divergence-Free Noise" due to Ivan DeWolf. These often require calculation of lattice noise gradients, which sometimes are not readily available. A naive implementation would call a lattice noise function several times to calculate its gradient, resulting in more computation than is strictly necessary. Unlike these noises, simulation noise has a geometric rationale in addition to its mathematical properties. It simulates vortices scattered in space, to produce its pleasing aesthetic. == Curl noise == The vector field is created as follows, for every point (x,y,z) in the space a vector field G is created, every component x, y and z of the vector field (Gx, Gy, Gz) is defined by a 3D perlin or simplex noise function with x, y and z as parameters. The partial derivative of Gx, Gy, and Gz respect to x, y and z is obtained with the gradient of the perlin or simplex noise by finite differences of implicit calculation inside the simplex noise. The partial derivatives are used to calculate F as the curl of G given by F = ( ∂ G z ∂ y − ∂ G y ∂ z , ∂ G x ∂ z − ∂ G z ∂ x , ∂ G y ∂ x − ∂ G x ∂ y ) {\displaystyle F=({\frac {\partial Gz}{\partial y}}-{\frac {\partial Gy}{\partial z}},{\frac {\partial Gx}{\partial z}}-{\frac {\partial Gz}{\partial x}},{\frac {\partial Gy}{\partial x}}-{\frac {\partial Gx}{\partial y}})} == Bitangent noise == This method is based in the fact that the curl of the gradient of scalar field is zero and the identity that expand the divergence of a cross product of two vectors A and B as the difference of the dot products of each vector with the curl of the other: ∇ × ( ∇ φ ) = 0 . {\displaystyle \nabla \times (\nabla \varphi )=\mathbf {0} .} ∇ ⋅ ( A × B ) = ( ∇ × A ) ⋅ B − A ⋅ ( ∇ × B ) {\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=\ (\nabla {\times }\mathbf {A} )\cdot \mathbf {B} \,-\,\mathbf {A} \cdot (\nabla {\times }\mathbf {B} )} which means that if the curl of both vector fields is zero then the divergence of the product of two vectors that are the gradients of scalar fields is zero too. This result in a divergence free vector field by construction only calling two noise functions to create the scalar fields. The vector field es created as follows, two scalar fields are calculated ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } using 3D perlin or simplex noise functions, then the gradients A and B of each of this fields is calculated, the cross product of A and B gives a divergence free vector field. == Signed distance noise == The vector field is created based on a closed and differentiable implicit surface S = F(x,y,z) = 0. For every point in the space, frequently outside or near the surface, we get a vector g that is normal to the surface, this is the gradient of S or the partial derivatives respect to x, y and z, this vector is not unitary, but we can get a unitary normal n by dividing each component of the point by the magnitude of the gradient g. Outside of the surface all these normals point away from the surface. g = ∇ F ( x , y , z ) = ( ∂ F ∂ x , ∂ F ∂ y , ∂ F ∂ z ) {\displaystyle g=\nabla F(x,y,z)=\left({\frac {\partial F}{\partial x}},{\frac {\partial F}{\partial y}},{\frac {\partial F}{\partial z}}\right)} n = g ( x , y , z ) ‖ ∇ F ( x , y , z ) ‖ {\displaystyle \mathbf {n} ={\frac {g(x,y,z)}{\|\nabla F(x,y,z)\|}}} ‖ ∇ F ( x , y , z ) ‖ = ( ∂ F ∂ x ) 2 + ( ∂ F ∂ y ) 2 + ( ∂ F ∂ z ) 2 {\displaystyle \|\nabla F(x,y,z)\|={\sqrt {\left({\frac {\partial F}{\partial x}}\right)^{2}+\left({\frac {\partial F}{\partial y}}\right)^{2}+\left({\frac {\partial F}{\partial z}}\right)^{2}}}} Afterwards we calculate a scalar value p for that point in the space using a 3D perlin or simplex noise function. Now we create a vector field V = pn pointing outside of the surface. The curl of this vector field gives the direction in every point in the space where the particles should move. S D N = ( ∂ V z ∂ y − ∂ V y ∂ z , ∂ V x ∂ z − ∂ V z ∂ x , ∂ V y ∂ x − ∂ V x ∂ y ) {\displaystyle SDN=({\frac {\partial Vz}{\partial y}}-{\frac {\partial Vy}{\partial z}},{\frac {\partial Vx}{\partial z}}-{\frac {\partial Vz}{\partial x}},{\frac {\partial Vy}{\partial x}}-{\frac {\partial Vx}{\partial y}})} By construction this vector SDN will point in a tangent direction to an isosurface at the level of the signed distance to the original surface and can be used to confine the movements of the particles to stay in that surface.