In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, the singular value decomposition (SVD) is a
factorization of a
real or
complex matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. It generalizes the
eigendecomposition of a square
normal matrix with an orthonormal eigenbasis to any
matrix. It is related to the
polar decomposition.
Specifically, the singular value decomposition of an
complex matrix is a factorization of the form
where is an
complex
unitary matrix,
is an
rectangular diagonal matrix with non-negative real numbers on the diagonal, is an
complex unitary matrix, and
is the
conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real
orthogonal matrices; in such contexts, the SVD is often denoted
The diagonal entries
of
are uniquely determined by and are known as the
singular values of . The number of non-zero singular values is equal to the
rank of . The columns of and the columns of are called left-singular vectors and right-singular vectors of , respectively. They form two sets of
orthonormal bases
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
and , and if they are sorted so that the singular values
with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as
where
is the rank of .
The SVD is not unique. It is always possible to choose the decomposition so that the singular values
are in descending order. In this case,
(but not and ) is uniquely determined by .
The term sometimes refers to the compact SVD, a similar decomposition
in which
is square diagonal of size
, where
is the rank of , and has only the non-zero singular values. In this variant, is an
semi-unitary matrix and
is an
semi-unitary matrix, such that
Mathematical applications of the SVD include computing the
pseudoinverse, matrix approximation, and determining the rank,
range, and
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
of a matrix. The SVD is also extremely useful in all areas of science,
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, such as
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
,
least squares fitting of data, and
process control.
Intuitive interpretations
Rotation, coordinate scaling, and reflection
In the special case when is an real
square matrix, the matrices and can be chosen to be real matrices too. In that case, "unitary" is the same as "
orthogonal". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as , as a
linear transformation of the space , the matrices and represent
rotations or
reflection of the space, while
represents the
scaling of each coordinate by the factor . Thus the SVD decomposition breaks down any linear transformation of into a
composition of three geometrical
transformations: a rotation or reflection (), followed by a coordinate-by-coordinate
scaling (
), followed by another rotation or reflection ().
In particular, if has a positive determinant, then and can be chosen to be both rotations with reflections, or both rotations without reflections. If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type.
If the matrix is real but not square, namely with , it can be interpreted as a linear transformation from to . Then and can be chosen to be rotations/reflections of and , respectively; and
, besides scaling the first
coordinates, also extends the vector with zeros, i.e. removes trailing coordinates, so as to turn into .
Singular values as semiaxes of an ellipse or ellipsoid
As shown in the figure, the
singular values In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self ...
can be interpreted as the magnitude of the semiaxes of an
ellipse in 2D. This concept can be generalized to -dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, with the singular values of any
square matrix being viewed as the magnitude of the semiaxis of an -dimensional
ellipsoid. Similarly, the singular values of any matrix can be viewed as the magnitude of the semiaxis of an -dimensional
ellipsoid in -dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction. See
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
for further details.
The columns of and are orthonormal bases
Since and are unitary, the columns of each of them form a set of
orthonormal vectors, which can be regarded as
basis vectors
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
. The matrix maps the basis vector to the stretched unit vector . By the definition of a unitary matrix, the same is true for their conjugate transposes and , except the geometric interpretation of the singular values as stretches is lost. In short, the columns of , and are
orthonormal bases
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
. When the
is a
positive-semidefinite Hermitian matrix, and are both equal to the unitary matrix used to diagonalize
. However, when
is not positive-semidefinite and Hermitian but still
diagonalizable, its
eigendecomposition and singular value decomposition are distinct.
Geometric meaning
Because and are unitary, we know that the columns of yield an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of and the columns of yield an orthonormal basis of (with respect to the standard
scalar products on these spaces).
The
linear transformation
:
has a particularly simple description with respect to these orthonormal bases: we have
:
where is the -th diagonal entry of
, and for .
The geometric content of the SVD theorem can thus be summarized as follows: for every linear map one can find orthonormal bases of and such that maps the -th basis vector of to a non-negative multiple of the -th basis vector of , and sends the left-over basis vectors to zero. With respect to these bases, the map is therefore represented by a diagonal matrix with non-negative real diagonal entries.
To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere of radius one in . The linear map maps this sphere onto an
ellipsoid in . Non-zero singular values are simply the lengths of the
semi-axes of this ellipsoid. Especially when , and all the singular values are distinct and non-zero, the SVD of the linear map can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid and specifically its axes; then consider the directions in sent by onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry sending these directions to the coordinate axes of . On a second move, apply an
endomorphism diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of as stretching coefficients. The composition then sends the unit-sphere onto an ellipsoid isometric to . To define the third and last move, apply an isometry to this ellipsoid to obtain . As can be easily checked, the composition coincides with .
Example
Consider the matrix
:
A singular value decomposition of this matrix is given by
:
The scaling matrix
is zero outside of the diagonal (grey italics) and one diagonal element is zero (red bold). Furthermore, because the matrices and are
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
, multiplying by their respective conjugate transposes yields
identity matrices, as shown below. In this case, because and are real valued, each is an
orthogonal matrix.
:
This particular singular value decomposition is not unique. Choosing
such that
:
is also a valid singular value decomposition.
SVD and spectral decomposition
Singular values, singular vectors, and their relation to the SVD
A non-negative real number is a
singular value for if and only if there exist unit-length vectors
in ''K
m'' and
in ''K
n'' such that
:
The vectors
and
are called left-singular and right-singular vectors for , respectively.
In any singular value decomposition
:
the diagonal entries of
are equal to the singular values of . The first columns of and are, respectively, left- and right-singular vectors for the corresponding singular values. Consequently, the above theorem implies that:
* An matrix has at most distinct singular values.
* It is always possible to find a
unitary basis for with a subset of basis vectors spanning the left-singular vectors of each singular value of .
* It is always possible to find a unitary basis for with a subset of basis vectors spanning the right-singular vectors of each singular value of .
A singular value for which we can find two left (or right) singular vectors that are linearly independent is called ''degenerate''. If
and
are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ. The similar statement is true for right-singular vectors. The number of independent left and right-singular vectors coincides, and these singular vectors appear in the same columns of and corresponding to diagonal elements of
all with the same value ''σ''.
As an exception, the left and right-singular vectors of singular value 0 comprise all unit vectors in the
cokernel and
kernel, respectively, of , which by the
rank–nullity theorem cannot be the same dimension if . Even if all singular values are nonzero, if then the cokernel is nontrivial, in which case is padded with orthogonal vectors from the cokernel. Conversely, if , then is padded by orthogonal vectors from the kernel. However, if the singular value of 0 exists, the extra columns of or already appear as left or right-singular vectors.
Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor ''e''
i''φ'' (for the real case up to a sign). Consequently, if all singular values of a square matrix are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of by a unit-phase factor and simultaneous multiplication of the corresponding column of by the same unit-phase factor.
In general, the SVD is unique up to arbitrary unitary transformations applied uniformly to the column vectors of both and spanning the subspaces of each singular value, and up to arbitrary unitary transformations on vectors of and spanning the kernel and cokernel, respectively, of .
Relation to eigenvalue decomposition
The singular value decomposition is very general in the sense that it can be applied to any matrix, whereas
eigenvalue decomposition can only be applied to
diagonalizable matrices. Nevertheless, the two decompositions are related.
Given an SVD of , as described above, the following two relations hold:
:
The right-hand sides of these relations describe the eigenvalue decompositions of the left-hand sides. Consequently:
* The columns of (right-singular vectors) are
eigenvectors of .
* The columns of (left-singular vectors) are eigenvectors of .
* The non-zero elements of
(non-zero singular values) are the square roots of the non-zero
eigenvalues of or .
In the special case that is a
normal matrix, which by definition must be square, the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
says that it can be
unitarily diagonalized using a basis of
eigenvectors, so that it can be written for a unitary matrix and a diagonal matrix with complex elements along the diagonal. When is
positive semi-definite, will be non-negative real numbers so that the decomposition is also a singular value decomposition. Otherwise, it can be recast as an SVD by moving the phase of each to either its corresponding or . The natural connection of the SVD to non-normal matrices is through the
polar decomposition theorem: , where is positive semidefinite and normal, and is unitary.
Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of , while related, differ: the eigenvalue decomposition is , where is not necessarily unitary and is not necessarily positive semi-definite, while the SVD is , where
is diagonal and positive semi-definite, and and are unitary matrices that are not necessarily related except through the matrix . While only
non-defective square matrices have an eigenvalue decomposition, any
matrix has a SVD.
Applications of the SVD
Pseudoinverse
The singular value decomposition can be used for computing the
pseudoinverse of a matrix. (Various authors use different notation for the pseudoinverse; here we use .) Indeed, the pseudoinverse of the matrix with singular value decomposition is
:
where is the pseudoinverse of , which is formed by replacing every non-zero diagonal entry by its
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
and transposing the resulting matrix. The pseudoinverse is one way to solve
linear least squares problems.
Solving homogeneous linear equations
A set of
homogeneous linear equations can be written as for a matrix and vector . A typical situation is that is known and a non-zero is to be determined which satisfies the equation. Such an belongs to 's
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
and is sometimes called a (right) null vector of . The vector can be characterized as a right-singular vector corresponding to a singular value of that is zero. This observation means that if is a
square matrix and has no vanishing singular value, the equation has no non-zero as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero satisfying , with denoting the conjugate transpose of , is called a left null vector of .
Total least squares minimization
A
total least squares
In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalizati ...
problem seeks the vector that minimizes the
2-norm of a vector under the constraint . The solution turns out to be the right-singular vector of corresponding to the smallest singular value.
Range, null space and rank
Another application of the SVD is that it provides an explicit representation of the
range and
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
of a matrix . The right-singular vectors corresponding to vanishing singular values of span the null space of and the left-singular vectors corresponding to the non-zero singular values of span the range of . For example, in the above
example the null space is spanned by the last two rows of and the range is spanned by the first three columns of .
As a consequence, the
rank of equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in
. In numerical linear algebra the singular values can be used to determine the ''effective rank'' of a matrix, as
rounding error may lead to small but non-zero singular values in a rank deficient matrix. Singular values beyond a significant gap are assumed to be numerically equivalent to zero.
Low-rank matrix approximation
Some practical applications need to solve the problem of
approximating a matrix with another matrix
, said to be
truncated, which has a specific rank . In the case that the approximation is based on minimizing the
Frobenius norm of the difference between and
under the constraint that
, it turns out that the solution is given by the SVD of , namely
:
where
is the same matrix as
except that it contains only the largest singular values (the other singular values are replaced by zero). This is known as the
Eckart–Young theorem, as it was proved by those two authors in 1936 (although it was later found to have been known to earlier authors; see ).
Separable models
The SVD can be thought of as decomposing a matrix into a weighted, ordered sum of separable matrices. By separable, we mean that a matrix can be written as an
outer product of two vectors , or, in coordinates,
. Specifically, the matrix can be decomposed as
:
Here and are the -th columns of the corresponding SVD matrices, are the ordered singular values, and each is separable. The SVD can be used to find the decomposition of an image processing filter into separable horizontal and vertical filters. Note that the number of non-zero is exactly the rank of the matrix.
Separable models often arise in biological systems, and the SVD factorization is useful to analyze such systems. For example, some visual area V1 simple cells' receptive fields can be well described by a
Gabor filter in the space domain multiplied by a modulation function in the time domain. Thus, given a linear filter evaluated through, for example,
reverse correlation, one can rearrange the two spatial dimensions into one dimension, thus yielding a two-dimensional filter (space, time) which can be decomposed through SVD. The first column of in the SVD factorization is then a Gabor while the first column of represents the time modulation (or vice versa). One may then define an index of separability
:
which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.
Nearest orthogonal matrix
It is possible to use the SVD of a square matrix to determine the
orthogonal matrix closest to . The closeness of fit is measured by the
Frobenius norm of . The solution is the product . This intuitively makes sense because an orthogonal matrix would have the decomposition where is the identity matrix, so that if then the product amounts to replacing the singular values with ones. Equivalently, the solution is the unitary matrix of the Polar Decomposition in either order of stretch and rotation, as described above.
A similar problem, with interesting applications in
shape analysis, is the
orthogonal Procrustes problem, which consists of finding an orthogonal matrix which most closely maps to . Specifically,
:
where
denotes the Frobenius norm.
This problem is equivalent to finding the nearest orthogonal matrix to a given matrix .
The Kabsch algorithm
The
Kabsch algorithm The Kabsch algorithm, named after Wolfgang Kabsch, is a method for calculating the optimal rotation matrix that minimizes the RMSD (root mean squared deviation) between two paired sets of points. It is useful in graphics, cheminformatics to compa ...
(called
Wahba's problem in other fields) uses SVD to compute the optimal rotation (with respect to least-squares minimization) that will align a set of points with a corresponding set of points. It is used, among other applications, to compare the structures of molecules.
Signal processing
The SVD and pseudoinverse have been successfully applied to
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
,
image processing and
big data (e.g., in genomic signal processing).
Other examples
The SVD is also applied extensively to the study of linear
inverse problems and is useful in the analysis of regularization methods such as that of
Tikhonov. It is widely used in statistics, where it is related to
principal component analysis and to
correspondence analysis, and in
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and
pattern recognition. It is also used in output-only
modal analysis, where the non-scaled
mode shapes can be determined from the singular vectors. Yet another usage is
latent semantic indexing in natural-language text processing.
In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number"
. It often controls the error rate or convergence rate of a given computational scheme on such systems.
The SVD also plays a crucial role in the field of
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both ...
, in a form often referred to as the
Schmidt decomposition. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be
entangled: if the rank of the
matrix is larger than one.
One application of SVD to rather large matrices is in
numerical weather prediction, where
Lanczos methods are used to estimate the most linearly quickly growing few perturbations to the central numerical weather prediction over a given initial forward time period; i.e., the singular vectors corresponding to the largest singular values of the linearized propagator for the global weather over that time interval. The output singular vectors in this case are entire weather systems. These perturbations are then run through the full nonlinear model to generate an
ensemble forecast, giving a handle on some of the uncertainty that should be allowed for around the current central prediction.
SVD has also been applied to reduced order modelling. The aim of reduced order modelling is to reduce the number of degrees of freedom in a complex system which is to be modeled. SVD was coupled with
radial basis functions to interpolate solutions to three-dimensional unsteady flow problems.
Interestingly, SVD has been used to improve gravitational waveform modeling by the ground-based gravitational-wave interferometer aLIGO. SVD can help to increase the accuracy and speed of waveform generation to support gravitational-waves searches and update two different waveform models.
Singular value decomposition is used in
recommender systems to predict people's item ratings. Distributed algorithms have been developed for the purpose of calculating the SVD on clusters of commodity machines.
Low-rank SVD has been applied for hotspot detection from spatiotemporal data with application to disease
outbreak detection. A combination of SVD and
higher-order SVD also has been applied for real time event detection from complex data streams (multivariate data with space and time dimensions) in
disease surveillance
Disease surveillance is an epidemiological practice by which the spread of disease is monitored in order to establish patterns of progression. The main role of disease surveillance is to predict, observe, and minimize the harm caused by outbreak, ...
.
Existence proofs
An eigenvalue of a matrix is characterized by the algebraic relation . When is
Hermitian, a variational characterization is also available. Let be a real
symmetric matrix. Define
:
By the
extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> s ...
, this continuous function attains a maximum at some u when restricted to the unit sphere {, , x, , = 1}. By the
Lagrange multipliers theorem, u necessarily satisfies
:
for some real number . The nabla symbol, , is the
del operator (differentiation with respect to x). Using the symmetry of we obtain
:
Therefore , so u is a unit length eigenvector of . For every unit length eigenvector v of its eigenvalue is ''f''(v), so is the largest eigenvalue of . The same calculation performed on the orthogonal complement of ''u'' gives the next largest eigenvalue and so on. The complex Hermitian case is similar; there is a real-valued function of real variables.
Singular values are similar in that they can be described algebraically or from variational principles. Although, unlike the eigenvalue case, Hermiticity, or symmetry, of is no longer required.
This section gives these two arguments for existence of singular value decomposition.
Based on the spectral theorem
Let
be an complex matrix. Since
is positive semi-definite and Hermitian, by the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
, there exists an unitary matrix
such that
:
where
is diagonal and positive definite, of dimension
, with
the number of non-zero eigenvalues of
(which can be shown to verify
). Note that
is here by definition a matrix whose
-th column is the
-th eigenvector of
, corresponding to the eigenvalue
. Moreover, the
-th column of
, for
, is an eigenvector of
with eigenvalue
. This can be expressed by writing
as
, where the columns of
and
therefore contain the eigenvectors of
corresponding to non-zero and zero eigenvalues, respectively. Using this rewriting of
, the equation becomes:
:
This implies that
:
Moreover, the second equation implies
. Finally, the unitary-ness of
translates, in terms of
and
, into the following conditions:
:
where the subscripts on the identity matrices are used to remark that they are of different dimensions.
Let us now define
:
Then,
:
since
This can be also seen as immediate consequence of the fact that
. This is equivalent to the observation that if
is the set of eigenvectors of
corresponding to non-vanishing eigenvalues
, then
is a set of orthogonal vectors, and
is a (generally not complete) set of ''orthonormal'' vectors. This matches with the matrix formalism used above denoting with
the matrix whose columns are
, with
the matrix whose columns are the eigenvectors of
with vanishing eigenvalue, and
the matrix whose columns are the vectors
.
We see that this is almost the desired result, except that
and
are in general not unitary, since they might not be square. However, we do know that the number of rows of
is no smaller than the number of columns, since the dimensions of
is no greater than
and
. Also, since
:
the columns in
are orthonormal and can be extended to an orthonormal basis. This means that we can choose
such that
is unitary.
For we already have to make it unitary. Now, define
:
where extra zero rows are added or removed to make the number of zero rows equal the number of columns of , and hence the overall dimensions of
equal to
. Then
:
which is the desired result:
:
Notice the argument could begin with diagonalizing rather than (This shows directly that and have the same non-zero eigenvalues).
Based on variational characterization
The singular values can also be characterized as the maxima of , considered as a function of and , over particular subspaces. The singular vectors are the values of and where these maxima are attained.
Let denote an matrix with real entries. Let be the unit
-sphere in
, and define
Consider the function restricted to . Since both and are
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
sets, their
product is also compact. Furthermore, since is continuous, it attains a largest value for at least one pair of vectors and . This largest value is denoted and the corresponding vectors are denoted and . Since is the largest value of it must be non-negative. If it were negative, changing the sign of either or would make it positive and therefore larger.
Statement. are left and right-singular vectors of with corresponding singular value ''σ''
1.
Proof. Similar to the eigenvalues case, by assumption the two vectors satisfy the Lagrange multiplier equation:
:
After some algebra, this becomes
:
Multiplying the first equation from left by
and the second equation from left by
and taking into account gives
:
Plugging this into the pair of equations above, we have
:
This proves the statement.
More singular vectors and singular values can be found by maximizing over normalized which are orthogonal to and , respectively.
The passage from real to complex is similar to the eigenvalue case.
Calculating the SVD
The singular value decomposition can be computed using the following observations:
* The left-singular vectors of are a set of
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
eigenvectors of .
* The right-singular vectors of are a set of orthonormal eigenvectors of .
* The non-zero singular values of (found on the diagonal entries of
) are the square roots of the non-zero
eigenvalues of both and .
Numerical approach
The SVD of a matrix is typically computed by a two-step procedure. In the first step, the matrix is reduced to a
bidiagonal matrix In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and ''either'' the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix.
When the diagonal abo ...
. This takes
O(''mn''
2) floating-point operations (flop), assuming that ''m'' ≥ ''n''. The second step is to compute the SVD of the bidiagonal matrix. This step can only be done with an
iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
(as with
eigenvalue algorithms). However, in practice it suffices to compute the SVD up to a certain precision, like the
machine epsilon. If this precision is considered constant, then the second step takes O(''n'') iterations, each costing O(''n'') flops. Thus, the first step is more expensive, and the overall cost is O(''mn''
2) flops .
The first step can be done using
Householder reflections for a cost of 4''mn''
2 − 4''n''
3/3 flops, assuming that only the singular values are needed and not the singular vectors. If ''m'' is much larger than ''n'' then it is advantageous to first reduce the matrix to a triangular matrix with the
QR decomposition and then use Householder reflections to further reduce the matrix to bidiagonal form; the combined cost is 2''mn''
2 + 2''n''
3 flops .
The second step can be done by a variant of the
QR algorithm for the computation of eigenvalues, which was first described by . The
LAPACK subroutine DBDSQR implements this iterative method, with some modifications to cover the case where the singular values are very small . Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition.
The same algorithm is implemented in the
GNU Scientific Library
The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C; wrappers are available for other programming languages. The GSL is part of the GNU Project and is ...
(GSL). The GSL also offers an alternative method that uses a one-sided
Jacobi orthogonalization Jacobi may refer to:
* People with the surname Jacobi
Mathematics:
* Jacobi sum, a type of character sum
* Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
* Jacobi eigenvalue algorithm, a ...
in step 2 . This method computes the SVD of the bidiagonal matrix by solving a sequence of 2 × 2 SVD problems, similar to how the
Jacobi eigenvalue algorithm solves a sequence of 2 × 2 eigenvalue methods . Yet another method for step 2 uses the idea of
divide-and-conquer eigenvalue algorithms .
There is an alternative way that does not explicitly use the eigenvalue decomposition. Usually the singular value problem of a matrix is converted into an equivalent symmetric eigenvalue problem such as , , or
:
The approaches that use eigenvalue decompositions are based on the
QR algorithm, which is well-developed to be stable and fast.
Note that the singular values are real and right- and left- singular vectors are not required to form similarity transformations. One can iteratively alternate between the
QR decomposition and the
LQ decomposition to find the real diagonal
Hermitian matrices. The
QR decomposition gives and the
LQ decomposition of gives . Thus, at every iteration, we have , update and repeat the orthogonalizations. Eventually, this iteration between
QR decomposition and
LQ decomposition produces left- and right- unitary singular matrices. This approach cannot readily be accelerated, as the QR algorithm can with spectral shifts or deflation. This is because the shift method is not easily defined without using similarity transformations. However, this iterative approach is very simple to implement, so is a good choice when speed does not matter. This method also provides insight into how purely orthogonal/unitary transformations can obtain the SVD.
Analytic result of 2 × 2 SVD
The singular values of a 2 × 2 matrix can be found analytically. Let the matrix be
where
are complex numbers that parameterize the matrix, is the identity matrix, and
denote the
Pauli matrices. Then its two singular values are given by
:
Reduced SVDs
In applications it is quite unusual for the full SVD, including a full unitary decomposition of the null-space of the matrix, to be required. Instead, it is often sufficient (as well as faster, and more economical for storage) to compute a reduced version of the SVD. The following can be distinguished for an ''m''×''n'' matrix ''M'' of rank ''r'':
Thin SVD
The thin, or economy-sized, SVD of a matrix ''M'' is given by
:
where
:
,
the matrices ''U''
''k'' and ''V''
''k'' contain only the first ''k'' columns of ''U'' and ''V'', and Σ
''k'' contains only the first ''k'' singular values from Σ. The matrix ''U''
''k'' is thus ''m''×''k'', Σ
''k'' is ''k''×''k'' diagonal, and ''V''
''k''* is ''k''×''n''.
The thin SVD uses significantly less space and computation time if ''k'' ≪ max(''m'', ''n''). The first stage in its calculation will usually be a
QR decomposition of ''M'', which can make for a significantly quicker calculation in this case.
Compact SVD
:
Only the ''r'' column vectors of ''U'' and ''r'' row vectors of ''V*'' corresponding to the non-zero singular values Σ
''r'' are calculated. The remaining vectors of ''U'' and ''V*'' are not calculated. This is quicker and more economical than the thin SVD if ''r'' ≪ min(''m'', ''n''). The matrix ''U''
''r'' is thus ''m''×''r'', Σ
''r'' is ''r''×''r'' diagonal, and ''V''
''r''* is ''r''×''n''.
Truncated SVD
In many applications the number ''r'' of the non-zero singular values is large making even the Compact SVD impractical to compute. In such cases, the smallest singular values may need to be truncated to compute only ''t'' ≪ ''r'' non-zero singular values. The truncated SVD is no longer an exact decomposition of the original matrix ''M'', but rather provides the optimal
low-rank matrix approximation by any matrix of a fixed rank ''t''
:
,
where matrix ''U''
''t'' is ''m''×''t'', Σ
''t'' is ''t''×''t'' diagonal, and ''V''
''t''* is ''t''×''n''.
Only the ''t'' column vectors of ''U'' and ''t'' row vectors of ''V*'' corresponding to the ''t'' largest singular values Σ
''t'' are calculated. This can be much quicker and more economical than the compact SVD if ''t''≪''r'', but requires a completely different toolset of numerical solvers.
In applications that require an approximation to the
Moore–Penrose inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Ro ...
of the matrix ''M'', the smallest singular values of ''M'' are of interest, which are more challenging to compute compared to the largest ones.
Truncated SVD is employed in
latent semantic indexing.
Norms
Ky Fan norms
The sum of the ''k'' largest singular values of ''M'' is a
matrix norm, the
Ky Fan
Ky Fan (樊𰋀, , September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician. He was a professor of mathematics at the University of California, Santa Barbara.
Biography
Fan was born in Hangzhou, the capital of Zhejiang ...
''k''-norm of ''M''.
The first of the Ky Fan norms, the Ky Fan 1-norm, is the same as the
operator norm of ''M'' as a linear operator with respect to the Euclidean norms of ''K''
''m'' and ''K''
''n''. In other words, the Ky Fan 1-norm is the operator norm induced by the standard ''ℓ''
2 Euclidean inner product. For this reason, it is also called the operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator ''M'' on (possibly infinite-dimensional) Hilbert spaces
:
But, in the matrix case, (''M* M'')
1/2 is a
normal matrix, so , , ''M* M'', ,
1/2 is the largest eigenvalue of (''M* M'')
1/2, i.e. the largest singular value of ''M''.
The last of the Ky Fan norms, the sum of all singular values, is the
trace norm (also known as the 'nuclear norm'), defined by , , ''M'', , = Tr
1/2">''M* M'')1/2(the eigenvalues of ''M* M'' are the squares of the singular values).
Hilbert–Schmidt norm
The singular values are related to another norm on the space of operators. Consider the
Hilbert–Schmidt inner product on the matrices, defined by
:
So the induced norm is
:
Since the trace is invariant under unitary equivalence, this shows
:
where are the singular values of . This is called the
Frobenius norm, Schatten 2-norm, or Hilbert–Schmidt norm of . Direct calculation shows that the Frobenius norm of coincides with:
:
In addition, the Frobenius norm and the trace norm (the nuclear norm) are special cases of the
Schatten norm.
Variations and generalizations
Mode-''k'' representation
can be represented using
mode-''k'' multiplication of matrix
applying
then
on the result; that is
.
Tensor SVD
Two types of tensor decompositions exist, which generalise the SVD to multi-way arrays. One of them decomposes a tensor into a sum of rank-1 tensors, which is called a
tensor rank decomposition In multilinear algebra, the tensor rank decomposition or the rank-R decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum R rank-1 tensors. This is an open problem.
Canonical polyadic decomposition (CPD) is a var ...
. The second type of decomposition computes the orthonormal subspaces associated with the different factors appearing in the tensor product of vector spaces in which the tensor lives. This decomposition is referred to in the literature as the
higher-order SVD (HOSVD) or
Tucker3/TuckerM. In addition,
multilinear principal component analysis in
multilinear subspace learning involves the same mathematical operations as Tucker decomposition, being used in a different context of
dimensionality reduction.
Scale-invariant SVD
The singular values of a matrix ''A'' are uniquely defined and are invariant with respect to left and/or right unitary transformations of ''A''. In other words, the singular values of ''UAV'', for unitary ''U'' and ''V'', are equal to the singular values of ''A''. This is an important property for applications in which it is necessary to preserve Euclidean distances and invariance with respect to rotations.
The Scale-Invariant SVD, or SI-SVD, is analogous to the conventional SVD except that its uniquely-determined singular values are invariant with respect to diagonal transformations of ''A''. In other words, the singular values of ''DAE'', for invertible diagonal matrices ''D'' and ''E'', are equal to the singular values of ''A''. This is an important property for applications for which invariance to the choice of units on variables (e.g., metric versus imperial units) is needed.
Higher-order SVD of functions (HOSVD)
Tensor product (TP) model transformation numerically reconstruct the
HOSVD
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. It has applications in c ...
of functions. For further details please visit:
*
Tensor product model transformation In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. It transforms a function (which can be given via closed formulas or neural net ...
*
HOSVD-based canonical form of TP functions and qLPV models
*
TP model transformation in control theory Baranyi and Yam proposed the TP model transformation as a new concept in quasi-LPV (qLPV) based control, which plays a central role in the highly desirable bridging between identification and polytopic systems theories. It is also used as a TS (Taka ...
Bounded operators on Hilbert spaces
The factorization can be extended to a
bounded operator ''M'' on a separable Hilbert space ''H''. Namely, for any bounded operator ''M'', there exist a
partial isometry In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.
The orthogonal complement of its kernel is called the initial subspace and its range is call ...
''U'', a unitary ''V'', a measure space (''X'', ''μ''), and a non-negative measurable ''f'' such that
:
where
is the
multiplication by ''f'' on ''L''
2(''X'', ''μ'').
This can be shown by mimicking the linear algebraic argument for the matricial case above. ''VT''
''f''''V''* is the unique positive square root of ''M*M'', as given by the
Borel functional calculus for
self-adjoint operators. The reason why ''U'' need not be unitary is because, unlike the finite-dimensional case, given an isometry ''U''
1 with nontrivial kernel, a suitable ''U''
2 may not be found such that
:
is a unitary operator.
As for matrices, the singular value factorization is equivalent to the
polar decomposition for operators: we can simply write
:
and notice that ''U V*'' is still a partial isometry while ''VT''
''f''''V''* is positive.
Singular values and compact operators
The notion of singular values and left/right-singular vectors can be extended to
compact operator on Hilbert space as they have a discrete spectrum. If is compact, every non-zero in its spectrum is an eigenvalue. Furthermore, a compact self-adjoint operator can be diagonalized by its eigenvectors. If is compact, so is . Applying the diagonalization result, the unitary image of its positive square root has a set of orthonormal eigenvectors corresponding to strictly positive eigenvalues . For any ,
:
where the series converges in the norm topology on . Notice how this resembles the expression from the finite-dimensional case. are called the singular values of . (resp. ) can be considered the left-singular (resp. right-singular) vectors of .
Compact operators on a Hilbert space are the closure of
finite-rank operators in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is:
:Theorem. is compact if and only if is compact.
History
The singular value decomposition was originally developed by
differential geometers, who wished to determine whether a real
bilinear form could be made equal to another by independent orthogonal transformations of the two spaces it acts on.
Eugenio Beltrami and
Camille Jordan discovered independently, in 1873 and 1874 respectively, that the singular values of the bilinear forms, represented as a matrix, form a
complete set of
invariants for bilinear forms under orthogonal substitutions.
James Joseph Sylvester also arrived at the singular value decomposition for real square matrices in 1889, apparently independently of both Beltrami and Jordan. Sylvester called the singular values the ''canonical multipliers'' of the matrix ''A''. The fourth mathematician to discover the singular value decomposition independently is
Autonne in 1915, who arrived at it via the
polar decomposition. The first proof of the singular value decomposition for rectangular and complex matrices seems to be by
Carl Eckart
Carl Henry Eckart (May 4, 1902 – October 23, 1973) was an American physicist, physical oceanographer, geophysicist, and administrator. He co-developed the Wigner–Eckart theorem and is also known for the Eckart conditions in quantum mechanics ...
and
Gale J. Young in 1936; they saw it as a generalization of the
principal axis transformation for
Hermitian matrices.
In 1907,
Erhard Schmidt defined an analog of singular values for
integral operators (which are compact, under some weak technical assumptions); it seems he was unaware of the parallel work on singular values of finite matrices. This theory was further developed by
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at ...
in 1910, who is the first to call the numbers
''singular values'' (or in French, ''valeurs singulières'').
Practical methods for computing the SVD date back to
Kogbetliantz in 1954–1955 and
Hestenes in 1958, resembling closely the
Jacobi eigenvalue algorithm, which uses plane rotations or
Givens rotations. However, these were replaced by the method of
Gene Golub
Gene Howard Golub (February 29, 1932 – November 16, 2007), was an American numerical analyst who taught at Stanford University as Fletcher Jones Professor of Computer Science and held a courtesy appointment in electrical engineering.
Person ...
and
William Kahan published in 1965, which uses
Householder transformations or reflections. In 1970, Golub and Christian Reinsch
published a variant of the Golub/Kahan algorithm that is still the one most-used today.
See also
Notes
References
*
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*
*
*
* Halldor, Bjornsson and Venegas, Silvia A. (1997)
"A manual for EOF and SVD analyses of climate data" McGill University, CCGCR Report No. 97-1, Montréal, Québec, 52pp.
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External links
{{DEFAULTSORT:Singular Value Decomposition
Linear algebra
Numerical linear algebra
Matrix theory
Matrix decompositions
Functional analysis