Efficient Object-Oriented Kronecker Product Manipulation

Version (55.3 KB) by Matt J
A class for efficient manipulation of N-fold Kronecker products in terms of their operands only.
Updated 5 Aug 2010

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This is a class for efficiently representing and manipulating N-fold Kronecker products of matrices (or of objects that behave like matrices) in terms of their operands only.

Given matrices {A,B,C,D,...} and a scalar s, an object M of this class can be used to represent

Matrix = s * A kron B kron C kron D kron ... (Eq. 1)

where "A kron B" denotes kron(A,B), the Kronecker product of A and B. Internally, however, M stores the operands {s,A,B,C,D,...} separately, which is typically far more byte-compact than numerically expanding out the RHS of Eq. 1.

Furthermore, many mathematical manipulations of Kronecker products are more efficient when done in terms of {s,A,B,C,D,...} separately than when done with the explicit numerical form of M as a matrix. The class overloads a number of methods and math operators in a way that exploits the Kronecker product structure accordingly.

Among these methods/operators are: mtimes (*), times (.*) , tranpose (.') , ctranpose (') , rdivide (./), ldivide (.\), mldivide (\), mrdivide (/), inv, pinv, power, mpower, norm, sum, cond, eig, svd, abs, nnz, orth, chol, lu, qr, full, sparse, ...

Some restrictions apply to these overloads. In particular, bi-operand math operations involving two KronProd objects, e.g. M1*M2, typically require the operands of each KronProd to be of compatible sizes. However, I find these restrictions to be satisfied often in applications.

Consult "help KronProd/methodname" for more info on each method. Optionally also, read krontest.m for demonstrations of their use.


A primary application of this class is to efficiently perform separable tensorial operations, i.e., where a linear transform is applied to all columns of an array, then all rows, and so on.

The following example is of a separable transformation of a 3D array X that transforms all of its columns via multiplication with a non-square matrix A, then transforms all rows by multiplication with B, then finally transforms all 3rd-dimensional axes by multiplication with C. Two approaches to this are compared. The first approach uses kron(). The second uses the KronProd class. Other operations are also shown for illustration purposes.

Notice the orders of magnitude reduction both in CPU time and in memory consumption, using the KronProd object.

m=25; n=15; p=40;
mm=16; nn=n; pp=10;
A=rand(mm,m); B=pi*eye(n); C=rand(pp,p);
s=4; % a scalar

%METHOD I: based on kron()

Matrix = s*kron(C,kron(B,A));

y1 = Matrix*X(:); %The tensorial transformation

z1 = Matrix.'*y1(:);

w1 = Matrix.'\z1;

%Elapsed time is 78.729007 seconds.

%METHOD II: based on KronProd object

Object = KronProd({A,pi,C},[1 2 3],[m,n,p],s); %equivalent to Matrix above

y2 = Object*X;
% This operation could also have been implemented
% as y2=reshape( Object*X(:) , [mm,nn,pp]);

z2 = Object.'*y1;

w2 = Object.'\z1;

% Elapsed time is 0.003958 seconds.

PercentError=@(x,y) norm(x(:)-y(:),2)/norm(x(:),'inf')*100;

PercentError(y1,y2), % = 3.0393e-012

PercentError(size(y1),size(y2)), % = 0

PercentError(z1,z2), % = 1.3017e-012
PercentError(w1,w2), % = 4.3409e-011


>> whos Matrix Object

Name Size Bytes Class Attributes

Matrix 2400x15000 288000000 double
Object 2400x15000 8102 KronProd

EXAMPLE #2: As a more practical example, the KronProd class is very useful in conjunction with the following tool for signal interpolation/reconstruction:


An example involving 2D signal reconstruction using cubic B-splines is provided in the file Example2D.m at the above link.

Cite As

Matt J (2024). Efficient Object-Oriented Kronecker Product Manipulation (https://www.mathworks.com/matlabcentral/fileexchange/25969-efficient-object-oriented-kronecker-product-manipulation), MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
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Version Published Release Notes

In constructor KronProd(opset,opinds, ...) introduced default value for opinds of 1:length(opset).

This allows simple 1-argument contructor calls such as
KronProd({C,B,A}), which is equivalent to kron(A,kron(B,C))

At the suggestion of a user, neatened up the Description section

*Overloaded kron() for the KronProd class.
*Reimplemented the class definition using classdef file.
*loadobj now always does a construct-on-load.
*See UpdateNotes for more details.

*Overloaded cellfun() for the KronProd class

*Added simpler options for the domainsize parameter in the constructor syntax KronProd(opset,opdims,domainsizes,...)

*See UpdateNotes for details

Added additional example to Description section with link to a related FEX submission.

Bug fix in qr() method (see UpdateNotes.txt). Also added relevant test of the fix to krontest.m

*New class methods: times, rdivide ldivide, rank, orth, svd, pinv, qr, lu, chol, loadobj.

*Expansions of old methods: eig, mldivide, mtimes, mrdivide

*See UpdateNotes.txt in package for more

Minor typo fixes in description of package