 # Eigen decomposition of sparse matrix

Documentation Help Center. The computational complexity of sparse operations is proportional to nnzthe number of nonzero elements in the matrix. The complexity of fairly complicated operations, such as the solution of sparse linear equations, involves factors like ordering and fill-in, which are discussed in the previous section.

In general, however, the computer time required for a sparse matrix operation is proportional to the number of arithmetic operations on nonzero quantities.

Functions that accept a matrix and return a scalar or constant-size vector always produce output in full storage format. For example, the size function always returns a full vector, whether its input is full or sparse. Functions that accept scalars or vectors and return matrices, such as zerosonesrandand eyealways return full results.

This is necessary to avoid introducing sparsity unexpectedly. The sparse analog of zeros m,n is simply sparse m,n. The sparse analogs of rand and eye are sprand and speyerespectively. There is no sparse analog for the function ones. Unary functions that accept a matrix and return a matrix or vector preserve the storage class of the operand. If S is a sparse matrix, then chol S is also a sparse matrix, and diag S is a sparse vector. Columnwise functions such as max and sum also return sparse vectors, even though these vectors can be entirely nonzero.

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Important exceptions to this rule are the sparse and full functions. Binary operators yield sparse results if both operands are sparse, and full results if both are full. For mixed operands, the result is full unless the operation preserves sparsity. In some cases, the result might be sparse even though the matrix has few zero elements. Matrix concatenation using either the cat function or square brackets produces sparse results for mixed operands.

A permutation of the rows and columns of a sparse matrix S can be represented in two ways:. A permutation vector pwhich is a full vector containing a permutation of 1:nacts on the rows of S as S p,:or on the columns as S :,p.

You can now try some permutations using the permutation vector p and the permutation matrix P. If P is a sparse matrix, then both representations use storage proportional to n and you can apply either to S in time proportional to nnz S.

The vector representation is slightly more compact and efficient, so the various sparse matrix permutation routines all return full row vectors with the exception of the pivoting permutation in LU triangular factorization, which returns a matrix compatible with the full LU factorization. Reordering the columns of a matrix can often make its LU or QR factors sparser. Reordering the rows and columns can often make its Cholesky factors sparser. The simplest such reordering is to sort the columns by nonzero count.

This is sometimes a good reordering for matrices with very irregular structures, especially if there is great variation in the nonzero counts of rows or columns. The colperm computes a permutation that orders the columns of a matrix by the number of nonzeros in each column from smallest to largest. The reverse Cuthill-McKee ordering is intended to reduce the profile or bandwidth of the matrix. It is not guaranteed to find the smallest possible bandwidth, but it usually does. This ordering is useful for matrices that come from one-dimensional problems or problems that are in some sense long and thin.

The degree of a node in a graph is the number of connections to that node. This is the same as the number of off-diagonal nonzero elements in the corresponding row of the adjacency matrix.

The approximate minimum degree algorithm generates an ordering based on how these degrees are altered during Gaussian elimination or Cholesky factorization.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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The sparsity pattern arises out of the constraints on the contact of two subdomains in a finite element problem. The coefficients on the left are positive whereas the coefficients on the right are negative. This is due to a sign convention. I ordered the node indices of each subdomain using a nodal connectivity matrix from the finite element mesh and fed that into METIS which made my conjugate gradient solves MUCH faster!

Although not important for this question, I am including my procedure for interfacing between Eigen and SuiteSparse, just in case it does matter. From my research on this website and other sources, the QR decomposition seems to be the best way to compute the nullspace of a sparse matrix.

I think my code is optimized for my platform of choice Windows on a six-core CPU laptop. However, my real question is if there is a better way of approaching this problem. Is there a linear algebra trick I am missing? Update : I used umfpack to calculate the LU decomposition first, and then calculate the QR decomposition. The nullspace is computed in a fraction of the time including the LU decompositionbut the resulting sparsity pattern is quite bizarre which appears to slow down subsequent operations.

My numerical results appear to be unaffected. Maybe a different node ordering strategy would fix this. Compared to the QR decomposition, the LU decomposition is instantaneous. Sign up to join this community. The best answers are voted up and rise to the top.

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I wonder how long it will take someone to delete the pleasantries Update : I used umfpack to calculate the LU decomposition first, and then calculate the QR decomposition.

Charlie S. Charlie S Charlie S 8 8 bronze badges. Note that the QR factorization without column permutation isn't reliable in detecting the rank of a matrix or giving you the null space if the matrix is rank deficient.

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Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta.Sparse Approximation also known as Sparse Representation theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in image processingsignal processingmachine learningmedical imagingand more.

Put formally, we solve. This problem is known to be NP-Hard with a reduction to NP-complete subset selection problems in combinatorial optimization. While the above posed problem is indeed NP-Hard, its solution can often be found using approximation algorithms.

This is known as the Basis Pursuit BP algorithm, which can be handled using any linear programming solver. An alternative approximation method is a greedy technique, such as the Matching Pursuit MPwhich finds the location of the non-zeros one at a time.

Just as in the noiseless case, these two problems are NP-Hard in general, but can be approximated using pursuit algorithms. Similarly, matching pursuit can be used for approximating the solution of the above problems, finding the locations of the non-zeros one at a time until the error threshold is met.

Structured sparsity : In the original version of the problem, any of the atoms in the dictionary can be picked. In the structured block sparsity model, instead of picking atoms individually, groups of them are to be picked.

These groups can be overlapping and of varying size. In this case, the pursuit task aims to recover a set of sparse representations that best describe the data while forcing them to share the same or close-by support. Two cases of interest that have been extensively studied are tree-based structure, and more generally, a Boltzmann distributed support.

As already mentioned above, there are various approximation also referred to as pursuit algorithms that have been developed for addressing the sparse representation problem:. Sparse approximation ideas and algorithms have been extensively used in signal processingimage processingmachine learningmedical imagingarray processingdata miningand more.

In most of these applications, the unknown signal of interest is modeled as a sparse combination of a few atoms from a given dictionary, and this is used as the regularization of the problem. The use of sparsity-inspired models has led to state-of-the-art results in a wide set of applications.

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Proceedings of the National Academy of Sciences. Bibcode : PNAS.The number of eigenvalues and eigenvectors desired. It is not possible to compute all eigenvectors of a matrix.

M must represent a real, symmetric matrix if A is real, and must represent a complex, hermitian matrix if A is complex. For best results, the data type of M should be the same as that of A.

This is done internally via a sparse LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Find eigenvalues near sigma using shift-invert mode. When sigma! If small eigenvalues are desired, consider using shift-invert mode for better performance.

Relative accuracy for eigenvalues stopping criterion The default value of 0 implies machine precision. An array of k eigenvectors. When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as eigenvalues and eigenvectors attributes of the exception object. Lehoucq, D. Sorensen, and C. If sigma is None, M is positive definite If sigma is specified, M is positive semi-definite.

See also eigsh eigenvalues and eigenvectors for symmetric matrix A svds singular value decomposition for a matrix A. Previous topic scipy. Last updated on Jul 11, Created using Sphinx 3.In order to find these solutions, it requires only left-multiplication by the matrix in question.

This operation is performed through a reverse-communication interface. The result of this structure is that ARPACK is able to find eigenvalues and eigenvectors of any linear function mapping a vector to a vector. This is accomplished through the keyword which.

The following values of which are available:. A better approach is to use shift-invert mode. We now have a symmetric matrix Xwith which to test the routines. First, compute a standard eigenvalue decomposition using eigh :. As the dimension of X grows, this routine becomes very slow. The results are as expected. There are a few ways this problem can be addressed.

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We could increase the tolerance tol to lead to faster convergence:. This works, but we lose the precision in the results. Another option is to increase the maximum number of iterations maxiter from to As mentioned above, this mode involves transforming the eigenvalue problem to an equivalent problem with different eigenvalues.

We get the results we were hoping for, with much less computational time. The user need not worry about the details. The shift-invert mode provides more than just a fast way to obtain a few small eigenvalues. Say, you desire to find internal eigenvalues and eigenvectors, e. Note that the shift-invert mode requires the internal solution of a matrix inverse. This is taken care of automatically by eigsh and eigsbut the operation can also be specified by the user.

See the docstring of scipy. LinearOperator instead. The eigenvalues and eigenvectors obtained with eigsh are compared to those obtained by using eigh when applied to the dense matrix:. In this case, we have created a quick and easy Diagonal operator. The external library PyLops provides similar capabilities in the Diagonal operator, as well as several other operators. Finally, we consider a linear operator that mimics the application of a first-derivative stencil.

## Sparse approximation

In this case, the operator is equivalent to a real nonsymmetric matrix. Once again, we compare the estimated eigenvalues and eigenvectors with those from a dense matrix that applies the same first derivative to an input signal:. Note that the eigenvalues of this operator are all imaginary. Again, a more advanced implementation of the first-derivative operator is available in the PyLops library under the name of FirstDerivative operator.

Linear Algebra scipy. Compressed Sparse Graph Routines scipy. T create a symmetric matrix. Nself. Last updated on Jul 11, Created using Sphinx 3.This module provides efficient implementations of all the basic linear algebra operations for sparse, symmetric, positive-definite matrices as, for instance, commonly arise in least squares problems.

See logdetlog. However, you should be aware that for least squares problems, the Cholesky method is usually faster but somewhat less numerically stable than QR- or SVD-based techniques. All usage of this module starts by calling one of four functions, all of which return a Factor object, documented below. Most users will want one of the cholesky functions, which perform a fill-reduction analysis and decomposition together:.

Only the lower triangular part of A is used. Note that if you are solving a conventional least-squares problem, you will need to transpose your matrix before calling this function, and therefore it will be somewhat more efficient to construct your matrix in CSR format so that its transpose will be in CSC format.

What is a determinant?

However, some users may want to break the fill-reduction analysis and actual decomposition into separate steps, and instead begin with one of the analyze functions, which perform only fill-reduction:.

Computes the optimal fill-reducing permutation for the symmetric matrix A, but does not factor it i. This function ignores the actual contents of the matrix A. All it cares about are 1 which entries are non-zero, and 2 whether A has real or complex type.

A Factor object representing the analysis. Many operations on this object will fail, because it does not yet hold a full decomposition. Use Factor. Each Factor fixes:. Given a Factor object, you can:. Note that no fill-reduction analysis is done; whatever permutation was chosen by the initial call to analyze will be used regardless of the pattern of non-zeros in C.

The L matrix returned by this method and the one returned by L are different! All methods in this section accept both sparse and dense matrices or vectors band return either a sparse or dense x accordingly. If f is a factor, then f. Consider using logdet instead, for improved numerical stability.

In particular, determinants are often prone to problems with underflow or overflow. Computes the log-determinant of the matrix A, with the same API as numpy. This returns a tuple sign, logdetwhere sign is always the number 1. For most purposes, it is better to use solve instead of computing the inverse explicitly. That is, the following two pieces of code produce identical results:.

Sometimes, though, you really do need the inverse explicitly e. Copies the current Factor.

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If you pass some other sort of matrix, then the wrapper code will convert it for you before passing it to CHOLMOD, and issue a warning of type CholmodTypeConversionWarning to let you know that your efficiency is not as high as it might be.In linear algebraeigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical formwhereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

The above equation is called the eigenvalue equation or the eigenvalue problem. The set of solutions, that is, the eigenvalues, is called the spectrum of A. We can factor p as. If the field of scalars is algebraically closedthe algebraic multiplicities sum to N :.

The total number of linearly independent eigenvectors, N vcan be calculated by summing the geometric multiplicities. The eigenvectors can be indexed by eigenvalues, using a double index, with v ij being the j th eigenvector for the i th eigenvalue. Then A can be factorized as.

Note that only diagonalizable matrices can be factorized in this way. The n eigenvectors q i are usually normalized, but they need not be. A non-normalized set of n eigenvectors, v i can also be used as the columns of Q. The above equation can be decomposed into two simultaneous equations :. Factoring out the eigenvalues x and y :. Since B is non-singular, it is essential that u is non-zero. Thus the matrix B required for the eigendecomposition of A is. If a matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by. When eigendecomposition is used on a matrix of measured, real datathe inverse may be less valid when all eigenvalues are used unmodified in the form above.

This is because as eigenvalues become relatively small, their contribution to the inversion is large. Those near zero or at the "noise" of the measurement system will have undue influence and could hamper solutions detection using the inverse. Two mitigations have been proposed: truncating small or zero eigenvalues, and extending the lowest reliable eigenvalue to those below it. The first mitigation method is similar to a sparse sample of the original matrix, removing components that are not considered valuable.

However, if the solution or detection process is near the noise level, truncating may remove components that influence the desired solution. The second mitigation extends the eigenvalue so that lower values have much less influence over inversion, but do still contribute, such that solutions near the noise will still be found.