FastGaussTransform

FastGaussTransform [/AERR=aprxErr, /WDTH=h, /OUTW=locWave, /OUT1={x1,nx,x2 }, /OUT2={x1,nx,x2,y1,ny,y2 }, /OUT3={x1,nx,x2,y1,ny,y2,z1,nz,z2 }, /Q, /RX=rx, /RY=ry, /TET=nTerms, /Z] srcLocationsWave, srcWeightsWave

The FastGaussTransform operation implements an efficient algorithm for evaluating the discrete Gauss transform as given by

\displaystyle G\left(y_{j}\right)=\sum_{i=0}^{N-1} q_{i} \exp \left(-\frac{\left\|y_{j}-x_{i}\right\|^{2}}{h}\right),

where G is an M-dimensional vector, y is an N-dimensional vector representing the observation position, {q_i} are the M-dimensional weights, {x_i} are N-dimensional vectors representing source locations and h is the Gaussian width. The wave M_FGT contains the output in the current data folder.

Flags


/AERR=aprxErr	Sets the approximate error, which determines how many terms of the Taylor expansion of the Gaussian are used by the calculation. Default value is 1e-5.
/WDTH=h	Sets the Gaussian width. Default value is 1.
/OUTW=locWave	Specifies the locations at which the output is computed. locWave must have the same number of columns as srcLocationsWave. The other /OUT flags are mutually exclusive; you should use only one at any time.
/OUT1={x1,nx,x2 }
/OUT2={x1,nx,x2,y1,ny,y2 }
/OUT3={x1,nx,x2,y1,ny,y2,z1,nz,z2 }
	Specifies gridded output of the required dimension. In each case you set the starting and ending values together with the number of intervals in that dimension. You cannot specify an output that does not match the dimensions of the input source.
/Q	Does not print results in the history area.
/RX=rx	Sets the maximum radius of any cluster. The clustering algorithm terminates when the maximum radius is less than rx. Without /RX, the maximum radius is the same as the maximum radius encountered.
/RY=ry	Sets the upper bound for the distance between an observation point and a cluster center for which the cluster contributes to the transform value. The default value of ry is 4 times the Gaussian width as specified by the /WDTH flag.
/TET=nTerms	Sets the number of terms in the Taylor expansion. Use /TET to set the number of terms and bypass the default error estimate, which is estimated from the approximate error value (/AERR).
/Z	No error reporting.

Details

The discrete Gauss transform can be computed as a direct sum. An exact calculation is practical only for moderate number of sources and observation points and for low spatial dimensionality. With increasing dimensionality and increasing number of sources it is more efficient to take advantage of some properties of the Gaussian function. The FastGaussianTransform operation does so in two ways: It first arranges the sources in N-dimensional spatial clusters so that it is not necessary to compute the contributions of all source points that belong to remote clusters (see FPClustering). The second component of the algorithm is an approximation that factorizes the sum into a factor that depends only on source points and a factor that depends only on observation points. The factor that depends only on source points is computed only once while the factor that depends on observation points is evaluated once for each observation point.

The trade-off between computation efficiency and accuracy can be adjusted using multiple parameters. By default, the operation calculates the number of terms it needs to use in the Taylor expansion of the Gaussian. You can modify the default approximate error value using /AERR or you can directly set the number of terms in the expansion using /TET.

FastGaussianTransform supports calculations in dimensions that may exceed the maximum allowed wave dimensionality. Therefore, srcLocationsWave must be a 2D, real-valued single- or double-precision wave in which each row corresponds to a single source position and columns represent the components in each dimension (e.g., a triplet wave would represent 3D source locations). srcWeightsWave must have the same number of rows as srcLocationsWave and it must be a real-valued single- or double-precision wave. In most applications srcWeightsWave will have a single column so that the output G will be scalar. However, if srcWeightsWave has multiple columns than G is a vector. This can be handy if you need to test multiple sets of coefficients at one time. If you specify observation points using /OUTW then locWave must have the same number of columns as srcLocationsWave (the number of rows in the output is arbitrary). The operation does not support wave scaling.

References

Yang, C., R. Duraiswami, and L. Davis, Efficient Kernel Machines Using the Improved Fast Gauss Transform, Advances in Neural Information Processing Systems, 16, 2004.