Digital image processing

Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allows a much wider range of algorithms to be applied to the input data and can avoid problems such as the build-up of noise and distortion during processing. Since images are defined over two dimensions (perhaps more), digital image processing may be modeled in the form of multidimensional systems. The generation and development of digital image processing are mainly affected by three factors: first, the development of computers; second, the development of mathematics (especially the creation and improvement of discrete mathematics theory); and third, the demand for a wide range of applications in environment, agriculture, military, industry and medical science has increased. == History == Many of the techniques of digital image processing, or digital picture processing as it often was called, were developed in the 1960s, at Bell Laboratories, the Jet Propulsion Laboratory, Massachusetts Institute of Technology, University of Maryland, and a few other research facilities, with application to satellite imagery, wire-photo standards conversion, medical imaging, videophone, character recognition, and photograph enhancement. The purpose of early image processing was to improve the quality of the image. In image processing, the input is a low-quality image, and the output is an image with improved quality. Common image processing includes image enhancement, restoration, encoding, and compression. The first successful application was the American Jet Propulsion Laboratory (JPL). They used image processing techniques such as geometric correction, gradation transformation, noise removal, etc. on the thousands of lunar photos sent back by the Space Detector Ranger 7 in 1964, taking into account the position of the Sun and the environment of the Moon. The impact of the successful mapping of the Moon's surface map by the computer has been a success. Later, more complex image processing was performed on the nearly 100,000 photos sent back by the spacecraft, so that the topographic map, color map and panoramic mosaic of the Moon were obtained, which achieved extraordinary results and laid a solid foundation for human landing on the Moon. The cost of processing was fairly high, however, with the computing equipment of that era. That changed in the 1970s, when digital image processing proliferated as cheaper computers and dedicated hardware became available. This led to images being processed in real-time, for some dedicated problems such as television standards conversion. As general-purpose computers became faster, they started to take over the role of dedicated hardware for all but the most specialized and computer-intensive operations. With the fast computers and signal processors available in the 2000s, digital image processing has become the most common form of image processing, and is generally used because it is not only the most versatile method, but also the cheapest. === Image sensors === The basis for modern image sensors is metal–oxide–semiconductor (MOS) technology, invented at Bell Labs between 1955 and 1960, This led to the development of digital semiconductor image sensors, including the charge-coupled device (CCD) and later the CMOS sensor. The charge-coupled device was invented by Willard S. Boyle and George E. Smith at Bell Labs in 1969. While researching MOS technology, they realized that an electric charge was the analogy of the magnetic bubble and that it could be stored on a tiny MOS capacitor. As it was fairly straightforward to fabricate a series of MOS capacitors in a row, they connected a suitable voltage to them so that the charge could be stepped along from one to the next. The CCD is a semiconductor circuit that was later used in the first digital video cameras for television broadcasting. The NMOS active-pixel sensor (APS) was invented by Olympus in Japan during the mid-1980s. This was enabled by advances in MOS semiconductor device fabrication, with MOSFET scaling reaching smaller micron and then sub-micron levels. The NMOS APS was fabricated by Tsutomu Nakamura's team at Olympus in 1985. The CMOS active-pixel sensor (CMOS sensor) was later developed by Eric Fossum's team at the NASA Jet Propulsion Laboratory in 1993. By 2007, sales of CMOS sensors had surpassed CCD sensors. MOS image sensors are widely used in optical mouse technology. The first optical mouse, invented by Richard F. Lyon at Xerox in 1980, used a 5 μm NMOS integrated circuit sensor chip. Since the first commercial optical mouse, the IntelliMouse introduced in 1999, most optical mouse devices use CMOS sensors. === Image compression === An important development in digital image compression technology was the discrete cosine transform (DCT), a lossy compression technique first proposed by Nasir Ahmed in 1972. DCT compression became the basis for JPEG, which was introduced by the Joint Photographic Experts Group in 1992. JPEG compresses images down to much smaller file sizes, and has become the most widely used image file format on the Internet. Its highly efficient DCT compression algorithm was largely responsible for the wide proliferation of digital images and digital photos, with several billion JPEG images produced every day as of 2015. Medical imaging techniques produce very large amounts of data, especially from CT, MRI and PET modalities. As a result, storage and communications of electronic image data are prohibitive without the use of compression. JPEG 2000 image compression is used by the DICOM standard for storage and transmission of medical images. The cost and feasibility of accessing large image data sets over low or various bandwidths are further addressed by use of another DICOM standard, called JPIP, to enable efficient streaming of the JPEG 2000 compressed image data. === Digital signal processor (DSP) === Electronic signal processing was revolutionized by the wide adoption of MOS technology in the 1970s. MOS integrated circuit technology was the basis for the first single-chip microprocessors and microcontrollers in the early 1970s, and then the first single-chip digital signal processor (DSP) chips in the late 1970s. DSP chips have since been widely used in digital image processing. The discrete cosine transform (DCT) image compression algorithm has been widely implemented in DSP chips, with many companies developing DSP chips based on DCT technology. DCTs are widely used for encoding, decoding, video coding, audio coding, multiplexing, control signals, signaling, analog-to-digital conversion, formatting luminance and color differences, and color formats such as YUV444 and YUV411. DCTs are also used for encoding operations such as motion estimation, motion compensation, inter-frame prediction, quantization, perceptual weighting, entropy encoding, variable encoding, and motion vectors, and decoding operations such as the inverse operation between different color formats (YIQ, YUV and RGB) for display purposes. DCTs are also commonly used for high-definition television (HDTV) encoder/decoder chips. == Tasks == Digital image processing allows the use of much more complex algorithms, and hence, can offer both more sophisticated performance at simple tasks, and the implementation of methods which would be impossible by analogue means. In particular, digital image processing is a concrete application of, and a practical technology based on: Classification Feature extraction Multi-scale signal analysis Pattern recognition Projection Some techniques that are used in digital image processing include: Anisotropic diffusion Hidden Markov models Image editing Image restoration Independent component analysis Linear filtering Neural networks Partial differential equations Pixelation Point feature matching Principal components analysis Self-organizing maps Wavelets == Digital image transformations == === Filtering === Digital filters are used to blur and sharpen digital images. Filtering can be performed by: convolution with specifically designed kernels (filter array) in the spatial domain masking specific frequency regions in the frequency (Fourier) domain The following examples show both methods: ==== Image padding in Fourier domain filtering ==== Images are typically padded before being transformed to the Fourier space, the highpass filtered images below illustrate the consequences of different padding techniques: Notice that the highpass filter shows extra edges when zero padded compared to the repeated edge padding. ==== Filtering code examples ==== MATLAB example for spatial domain highpass filtering. === Affine transformations === Affine transformations enable basic image transformations including scale, rotate, translate, mirror and shear as is shown in the following examples: To apply the affine

Kolmogorov–Arnold Networks

Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.

Out-of-bag error

Out-of-bag (OOB) error, also called out-of-bag estimate, is a method of measuring the prediction error of random forests, boosted decision trees, and other machine learning models utilizing bootstrap aggregating (bagging). Bagging uses subsampling with replacement to create training samples for the model to learn from. OOB error is the mean prediction error on each training sample xi, using only the trees that did not have xi in their bootstrap sample. Bootstrap aggregating allows one to define an out-of-bag estimate of the prediction performance improvement by evaluating predictions on those observations that were not used in the building of the next base learner. == Out-of-bag dataset == When bootstrap aggregating is performed, two independent sets are created. One set, the bootstrap sample, is the data chosen to be "in-the-bag" by sampling with replacement. The out-of-bag set is all data not chosen in the sampling process. When this process is repeated, such as when building a random forest, many bootstrap samples and OOB sets are created. The OOB sets can be aggregated into one dataset, but each sample is only considered out-of-bag for the trees that do not include it in their bootstrap sample. The picture below shows that for each bag sampled, the data is separated into two groups. This example shows how bagging could be used in the context of diagnosing disease. A set of patients are the original dataset, but each model is trained only by the patients in its bag. The patients in each out-of-bag set can be used to test their respective models. The test would consider whether the model can accurately determine if the patient has the disease. == Calculating out-of-bag error == Since each out-of-bag set is not used to train the model, it is a good test for the performance of the model. The specific calculation of OOB error depends on the implementation of the model, but a general calculation is as follows. Find all models (or trees, in the case of a random forest) that are not trained by the OOB instance. Take the majority vote of these models' result for the OOB instance, compared to the true value of the OOB instance. Compile the OOB error for all instances in the OOB dataset. The bagging process can be customized to fit the needs of a model. To ensure an accurate model, the bootstrap training sample size should be close to that of the original set. Also, the number of iterations (trees) of the model (forest) should be considered to find the true OOB error. The OOB error will stabilize over many iterations so starting with a high number of iterations is a good idea. Shown in the example to the right, the OOB error can be found using the method above once the forest is set up. == Comparison to cross-validation == Out-of-bag error and cross-validation (CV) are different methods of measuring the error estimate of a machine learning model. Over many iterations, the two methods should produce a very similar error estimate. That is, once the OOB error stabilizes, it will converge to the cross-validation (specifically leave-one-out cross-validation) error. The advantage of the OOB method is that it requires less computation and allows one to test the model as it is being trained. == Accuracy and Consistency == Out-of-bag error is used frequently for error estimation within random forests but with the conclusion of a study done by Silke Janitza and Roman Hornung, out-of-bag error has shown to overestimate in settings that include an equal number of observations from all response classes (balanced samples), small sample sizes, a large number of predictor variables, small correlation between predictors, and weak effects.

Relief (feature selection)

Relief is an algorithm developed by Kenji Kira and Larry Rendell in 1992 that takes a filter-method approach to feature selection that is notably sensitive to feature interactions. It was originally designed for application to binary classification problems with discrete or numerical features. Relief calculates a feature score for each feature which can then be applied to rank and select top scoring features for feature selection. Alternatively, these scores may be applied as feature weights to guide downstream modeling. Relief feature scoring is based on the identification of feature value differences between nearest neighbor instance pairs. If a feature value difference is observed in a neighboring instance pair with the same class (a 'hit'), the feature score decreases. Alternatively, if a feature value difference is observed in a neighboring instance pair with different class values (a 'miss'), the feature score increases. The original Relief algorithm has since inspired a family of Relief-based feature selection algorithms (RBAs), including the ReliefF algorithm. Beyond the original Relief algorithm, RBAs have been adapted to (1) perform more reliably in noisy problems, (2) generalize to multi-class problems (3) generalize to numerical outcome (i.e. regression) problems, and (4) to make them robust to incomplete (i.e. missing) data. To date, the development of RBA variants and extensions has focused on four areas; (1) improving performance of the 'core' Relief algorithm, i.e. examining strategies for neighbor selection and instance weighting, (2) improving scalability of the 'core' Relief algorithm to larger feature spaces through iterative approaches, (3) methods for flexibly adapting Relief to different data types, and (4) improving Relief run efficiency. Their strengths are that they are not dependent on heuristics, they run in low-order polynomial time, and they are noise-tolerant and robust to feature interactions, as well as being applicable for binary or continuous data; however, it does not discriminate between redundant features, and low numbers of training instances fool the algorithm. == Relief Algorithm == Take a data set with n instances of p features, belonging to two known classes. Within the data set, each feature should be scaled to the interval [0 1] (binary data should remain as 0 and 1). The algorithm will be repeated m times. Start with a p-long weight vector (W) of zeros. At each iteration, take the feature vector (X) belonging to one random instance, and the feature vectors of the instance closest to X (by Euclidean distance) from each class. The closest same-class instance is called 'near-hit', and the closest different-class instance is called 'near-miss'. Update the weight vector such that W i = W i − ( x i − n e a r H i t i ) 2 + ( x i − n e a r M i s s i ) 2 , {\displaystyle W_{i}=W_{i}-(x_{i}-\mathrm {nearHit} _{i})^{2}+(x_{i}-\mathrm {nearMiss} _{i})^{2},} where i {\displaystyle i} indexes the components and runs from 1 to p. Thus the weight of any given feature decreases if it differs from that feature in nearby instances of the same class more than nearby instances of the other class, and increases in the reverse case. After m iterations, divide each element of the weight vector by m. This becomes the relevance vector. Features are selected if their relevance is greater than a threshold τ. Kira and Rendell's experiments showed a clear contrast between relevant and irrelevant features, allowing τ to be determined by inspection. However, it can also be determined by Chebyshev's inequality for a given confidence level (α) that a τ of 1/sqrt(αm) is good enough to make the probability of a Type I error less than α, although it is stated that τ can be much smaller than that. Relief was also described as generalizable to multinomial classification by decomposition into a number of binary problems. == ReliefF Algorithm == Kononenko et al. propose a number of updates to Relief. Firstly, they find the near-hit and near-miss instances using the Manhattan (L1) norm rather than the Euclidean (L2) norm, although the rationale is not specified. Furthermore, they found taking the absolute differences between xi and near-hiti, and xi and near-missi to be sufficient when updating the weight vector (rather than the square of those differences). === Reliable probability estimation === Rather than repeating the algorithm m times, implement it exhaustively (i.e. n times, once for each instance) for relatively small n (up to one thousand). Furthermore, rather than finding the single nearest hit and single nearest miss, which may cause redundant and noisy attributes to affect the selection of the nearest neighbors, ReliefF searches for k nearest hits and misses and averages their contribution to the weights of each feature. k can be tuned for any individual problem. === Incomplete data === In ReliefF, the contribution of missing values to the feature weight is determined using the conditional probability that two values should be the same or different, approximated with relative frequencies from the data set. This can be calculated if one or both features are missing. === Multi-class problems === Rather than use Kira and Rendell's proposed decomposition of a multinomial classification into a number of binomial problems, ReliefF searches for k near misses from each different class and averages their contributions for updating W, weighted with the prior probability of each class. == Other Relief-based Algorithm Extensions/Derivatives == The following RBAs are arranged chronologically from oldest to most recent. They include methods for improving (1) the core Relief algorithm concept, (2) iterative approaches for scalability, (3) adaptations to different data types, (4) strategies for computational efficiency, or (5) some combination of these goals. For more on RBAs see these book chapters or this most recent review paper. === RRELIEFF === Robnik-Šikonja and Kononenko propose further updates to ReliefF, making it appropriate for regression. === Relieved-F === Introduced deterministic neighbor selection approach and a new approach for incomplete data handling. === Iterative Relief === Implemented method to address bias against non-monotonic features. Introduced the first iterative Relief approach. For the first time, neighbors were uniquely determined by a radius threshold and instances were weighted by their distance from the target instance. === I-RELIEF === Introduced sigmoidal weighting based on distance from target instance. All instance pairs (not just a defined subset of neighbors) contributed to score updates. Proposed an on-line learning variant of Relief. Extended the iterative Relief concept. Introduced local-learning updates between iterations for improved convergence. === TuRF (a.k.a. Tuned ReliefF) === Specifically sought to address noise in large feature spaces through the recursive elimination of features and the iterative application of ReliefF. === Evaporative Cooling ReliefF === Similarly seeking to address noise in large feature spaces. Utilized an iterative `evaporative' removal of lowest quality features using ReliefF scores in association with mutual information. === EReliefF (a.k.a. Extended ReliefF) === Addressing issues related to incomplete and multi-class data. === VLSReliefF (a.k.a. Very Large Scale ReliefF) === Dramatically improves the efficiency of detecting 2-way feature interactions in very large feature spaces by scoring random feature subsets rather than the entire feature space. === ReliefMSS === Introduced calculation of feature weights relative to average feature 'diff' between instance pairs. === SURF === SURF identifies nearest neighbors (both hits and misses) based on a distance threshold from the target instance defined by the average distance between all pairs of instances in the training data. Results suggest improved power to detect 2-way epistatic interactions over ReliefF. === SURF (a.k.a. SURFStar) === SURF extends the SURF algorithm to not only utilized 'near' neighbors in scoring updates, but 'far' instances as well, but employing inverted scoring updates for 'far instance pairs. Results suggest improved power to detect 2-way epistatic interactions over SURF, but an inability to detect simple main effects (i.e. univariate associations). === SWRF === SWRF extends the SURF algorithm adopting sigmoid weighting to take distance from the threshold into account. Also introduced a modular framework for further developing RBAs called MoRF. === MultiSURF (a.k.a. MultiSURFStar) === MultiSURF extends the SURF algorithm adapting the near/far neighborhood boundaries based on the average and standard deviation of distances from the target instance to all others. MultiSURF uses the standard deviation to define a dead-band zone where 'middle-distance' instances do not contribute to scoring. Evidence suggests MultiSURF performs best in detecting pure 2-way feature interactions. === Reli

Vapnik–Chervonenkis theory

Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view. == Introduction == VC theory covers at least four parts (as explained in The Nature of Statistical Learning Theory): Theory of consistency of learning processes What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? Nonasymptotic theory of the rate of convergence of learning processes How fast is the rate of convergence of the learning process? Theory of controlling the generalization ability of learning processes How can one control the rate of convergence (the generalization ability) of the learning process? Theory of constructing learning machines How can one construct algorithms that can control the generalization ability? VC Theory is a major subbranch of statistical learning theory. One of its main applications in statistical learning theory is to provide generalization conditions for learning algorithms. From this point of view, VC theory is related to stability, which is an alternative approach for characterizing generalization. In addition, VC theory and VC dimension are instrumental in the theory of empirical processes, in the case of processes indexed by VC classes. Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book Weak Convergence and Empirical Processes: With Applications to Statistics. == Overview of VC theory in empirical processes == === Background on empirical processes === Let ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} be a measurable space. For any measure Q {\displaystyle Q} on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} , and any measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } , define Q f = ∫ f d Q {\displaystyle Qf=\int fdQ} Measurability issues will be ignored here, for more technical detail see. Let F {\displaystyle {\mathcal {F}}} be a class of measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } and define: ‖ Q ‖ F = sup { | Q f | : f ∈ F } . {\displaystyle \|Q\|_{\mathcal {F}}=\sup\{\vert Qf\vert \ :\ f\in {\mathcal {F}}\}.} Let X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} be independent, identically distributed random elements of ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Then define the empirical measure P n = n − 1 ∑ i = 1 n δ X i , {\displaystyle \mathbb {P} _{n}=n^{-1}\sum _{i=1}^{n}\delta _{X_{i}},} where δ here stands for the Dirac measure. The empirical measure induces a map F → R {\displaystyle {\mathcal {F}}\to \mathbf {R} } given by: f ↦ P n f = 1 n ( f ( X 1 ) + . . . + f ( X n ) ) {\displaystyle f\mapsto \mathbb {P} _{n}f={\frac {1}{n}}(f(X_{1})+...+f(X_{n}))} Now suppose P is the underlying true distribution of the data, which is unknown. Empirical Processes theory aims at identifying classes F {\displaystyle {\mathcal {F}}} for which statements such as the following hold: uniform law of large numbers: ‖ P n − P ‖ F → n 0 , {\displaystyle \|\mathbb {P} _{n}-P\|_{\mathcal {F}}{\underset {n}{\to }}0,} That is, as n → ∞ {\displaystyle n\to \infty } , | 1 n ( f ( X 1 ) + . . . + f ( X n ) ) − ∫ f d P | → 0 {\displaystyle \left|{\frac {1}{n}}(f(X_{1})+...+f(X_{n}))-\int fdP\right|\to 0} uniformly for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . uniform central limit theorem: G n = n ( P n − P ) ⇝ G , in ℓ ∞ ( F ) {\displaystyle \mathbb {G} _{n}={\sqrt {n}}(\mathbb {P} _{n}-P)\rightsquigarrow \mathbb {G} ,\quad {\text{in }}\ell ^{\infty }({\mathcal {F}})} In the former case F {\displaystyle {\mathcal {F}}} is called Glivenko–Cantelli class, and in the latter case (under the assumption ∀ x , sup f ∈ F | f ( x ) − P f | < ∞ {\displaystyle \forall x,\sup \nolimits _{f\in {\mathcal {F}}}\vert f(x)-Pf\vert <\infty } ) the class F {\displaystyle {\mathcal {F}}} is called Donsker or P-Donsker. A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem. These statements are true for a single f {\displaystyle f} , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . Intuitively then, the set F {\displaystyle {\mathcal {F}}} cannot be too large, and as it turns out that the geometry of F {\displaystyle {\mathcal {F}}} plays a very important role. One way of measuring how big the function set F {\displaystyle {\mathcal {F}}} is to use the so-called covering numbers. The covering number N ( ε , F , ‖ ⋅ ‖ ) {\displaystyle N(\varepsilon ,{\mathcal {F}},\|\cdot \|)} is the minimal number of balls { g : ‖ g − f ‖ < ε } {\displaystyle \{g:\|g-f\|<\varepsilon \}} needed to cover the set F {\displaystyle {\mathcal {F}}} (here it is obviously assumed that there is an underlying norm on F {\displaystyle {\mathcal {F}}} ). The entropy is the logarithm of the covering number. Two sufficient conditions are provided below, under which it can be proved that the set F {\displaystyle {\mathcal {F}}} is Glivenko–Cantelli or Donsker. A class F {\displaystyle {\mathcal {F}}} is P-Glivenko–Cantelli if it is P-measurable with envelope F such that P ∗ F < ∞ {\displaystyle P^{\ast }F<\infty } and satisfies: ∀ ε > 0 sup Q N ( ε ‖ F ‖ Q , F , L 1 ( Q ) ) < ∞ . {\displaystyle \forall \varepsilon >0\quad \sup \nolimits _{Q}N(\varepsilon \|F\|_{Q},{\mathcal {F}},L_{1}(Q))<\infty .} The next condition is a version of Dudley's theorem. If F {\displaystyle {\mathcal {F}}} is a class of functions such that ∫ 0 ∞ sup Q log ⁡ N ( ε ‖ F ‖ Q , 2 , F , L 2 ( Q ) ) d ε < ∞ {\displaystyle \int _{0}^{\infty }\sup \nolimits _{Q}{\sqrt {\log N\left(\varepsilon \|F\|_{Q,2},{\mathcal {F}},L_{2}(Q)\right)}}d\varepsilon <\infty } then F {\displaystyle {\mathcal {F}}} is P-Donsker for every probability measure P such that P ∗ F 2 < ∞ {\displaystyle P^{\ast }F^{2}<\infty } . In the last integral, the notation means ‖ f ‖ Q , 2 = ( ∫ | f | 2 d Q ) 1 2 {\displaystyle \|f\|_{Q,2}=\left(\int |f|^{2}dQ\right)^{\frac {1}{2}}} . === Symmetrization === The majority of the arguments about how to bound the empirical process rely on symmetrization, maximal and concentration inequalities, and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality which is discussed in the next section). It is presented here: Consider the empirical process: f ↦ ( P n − P ) f = 1 n ∑ i = 1 n ( f ( X i ) − P f ) {\displaystyle f\mapsto (\mathbb {P} _{n}-P)f={\dfrac {1}{n}}\sum _{i=1}^{n}(f(X_{i})-Pf)} Turns out that there is a connection between the empirical and the following symmetrized process: f ↦ P n 0 f = 1 n ∑ i = 1 n ε i f ( X i ) {\displaystyle f\mapsto \mathbb {P} _{n}^{0}f={\dfrac {1}{n}}\sum _{i=1}^{n}\varepsilon _{i}f(X_{i})} The symmetrized process is a Rademacher process, conditionally on the data X i {\displaystyle X_{i}} . Therefore, it is a sub-Gaussian process by Hoeffding's inequality. Lemma (Symmetrization). For every nondecreasing, convex Φ: R → R and class of measurable functions F {\displaystyle {\mathcal {F}}} , E Φ ( ‖ P n − P ‖ F ) ≤ E Φ ( 2 ‖ P n 0 ‖ F ) {\displaystyle \mathbb {E} \Phi (\|\mathbb {P} _{n}-P\|_{\mathcal {F}})\leq \mathbb {E} \Phi \left(2\left\|\mathbb {P} _{n}^{0}\right\|_{\mathcal {F}}\right)} The proof of the Symmetrization lemma relies on introducing independent copies of the original variables X i {\displaystyle X_{i}} (sometimes referred to as a ghost sample) and replacing the inner expectation of the LHS by these copies. After an application of Jensen's inequality different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature. The same proof method can be used to prove the Glivenko–Cantelli theorem. A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to P n 0 {\displaystyle \mathbb {P} _{n}^{0}} and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties. === VC Connection === It turns out that there is a fascinating connection between certain combinatorial properties of the set F {\displaystyle {\mathcal {F}}} and the entropy numbers. Uniform covering numbers can be controlled by the notion of Vapnik–Chervonenkis classes of sets – or shortly VC sets. Consider a collection C {\displaystyle {\mathcal {C}}} of subsets of the sample space X {\displaystyle

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