on 27-Mar-2018 (Tue)

A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution

The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition.

For , a pseudoinverse of is defined as a matrix satisfying all of the following four criteria:

1. ( AA⁺ need not be the general identity matrix, but it maps all column vectors of A to themselves);
2. ( A⁺ is a weak inverse for the multiplicative semigroup);
3. ( AA⁺ is Hermitian); and
4. ( A⁺A is also Hermitian).

Moore-Penrose Pseudo-inverse

exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula.

In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as:

...

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

The eigendecomposition can be derived from the fundamental property of eigenvectors: and thus which yields .

Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.

A key fact of Gaussian processes is that they can be completely defined by their second-order statistics.

if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion.

Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'.

If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous;^[7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.

If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present.

Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)).

Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand.

If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.

A process that is concurrently stationary and isotropic is considered to be homogeneous;^[7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.

if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour.

In linear algebra, the singular-value decomposition (SVD) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar decomposition.

Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form

Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables.

Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

Variational Bayesian methods are primarily used for two purposes:

1. To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.
2. To derive a lower bound for the marginal likelihood (sometimes called the "evidence") of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved variables). This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data. (See also the Bayes factor article.)

DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design.

For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed.

SVD as change of coordinates

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : Kⁿ → K^m one can find orthonormal bases of Kⁿ and K^m such that T maps the i-th basis vector of Kⁿ to a non-negative multiple of the i-th basis vector of K^m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries.

A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.^[1][9][15] If is the singular value decomposition of A , then . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ) , where ε is the machine epsilon.

Moore-Penrose Pseudo-inverse

exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula.

In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as:

This particular pseudoinverse constitutes a left inverse, since, in this case, .

According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.^[1][9][15] If is the singular value decomposition of A , then .

In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by

1. breaking it down into a collection of simpler subproblems,
2. solving each of those subproblems just once, and
3. storing their solutions.

The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space.
(Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.)

In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option.

An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.

The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / ^[1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments.

1. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce
2. the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that
3. the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate).

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.

Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.^[1][2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The third of them, in contrast, exemplifies an entirely different rationale for logic, namely as a foundational branch of mathematics

I n the modern period, a number of philosophers came to see the nature of logic in terms of the faculties of mind.

It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.