In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. If there are observations with variables, then the number of distinct principal components is . This transformation is defined in such a way that the first principal component has the largest possible variance, and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.
In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix to any matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.
In geometry and group theory, a lattice in is a subgroup of the additive group which is isomorphic to the additive group , and which spans the real vector space . In other words, for any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.
In physics, the reciprocal lattice represents the Fourier transform of another lattice. In normal usage, this first lattice is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space The reciprocal of a reciprocal lattice is the original direct lattice, since the two are Fourier transforms of each other.
In linear algebra, the Gram matrix of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by .
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input produces response.
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.
The Lanczos algorithm is a direct algorithm devised by Cornelius Lanczos that is an adaptation of power methods to find the most useful eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In 1970, Ojalvo and Newman showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading. This was achieved using a method for purifying the vectors to any degree of accuracy, which when not performed, produced a series of vectors that were highly contaminated by those associated with the lowest natural frequencies. In their original work, these authors also suggested how to select a starting vector and suggested an empirically determined method for determining , the reduced number of vectors. Soon thereafter their work was followed by Paige, who also provided an error analysis. In 1988, Ojalvo produced a more detailed history of this algorithm and an efficient eigenvalue error test. Currently, the method is widely used in a variety of technical fields and has seen a number of variations.
In machine learning, kernel methods are a class of algorithms for pattern analysis, whose best known member is the support vector machine (SVM). The general task of pattern analysis is to find and study general types of relations in datasets. In its simplest form, the kernel trick means transforming data into another dimension that has a clear dividing margin between classes of data. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over pairs of data points in raw representation.
Non-negative matrix factorization, 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.
Corner detection is an approach used within computer vision systems to extract certain kinds of features and infer the contents of an image. Corner detection is frequently used in motion detection, image registration, video tracking, image mosaicing, panorama stitching, 3D modelling and object recognition. Corner detection overlaps with the topic of interest point detection.
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.
In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent. The structure tensor is often used in image processing and computer vision.
Numerical linear algebra is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to mathematical questions. It is a subfield of numerical analysis, and a type of linear algebra. Because computers use floating-point arithmetic, they cannot exactly represent irrational data, and many algorithms increase that imprecision when implemented by a computer. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize computer error while retaining efficiency and precision.
In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. Its roots lie in the classical Karhunen (1946)–Loève spectral decomposition of time series and random fields and in the Mañé (1981)–Takens (1981) embedding theorem. SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. The name "singular spectrum analysis" relates to the spectrum of eigenvalues in a singular value decomposition of a covariance matrix, and not directly to a frequency domain decomposition.
The Goldreich–Goldwasser–Halevi (GGH) lattice-based cryptosystem is an asymmetric cryptosystem based on lattices. There is also a GGH signature scheme.
A coupled map lattice (CML) is a dynamical system that models the behavior of non-linear systems. They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases.
An image texture is a set of metrics calculated in image processing designed to quantify the perceived texture of an image. Image texture gives us information about the spatial arrangement of color or intensities in an image or selected region of an image.
