Search results
Results From The WOW.Com Content Network
Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension =, and bilinear interpolation, which operates with dimension =, to dimension =.
Isomap is one representative of isometric mapping methods, and extends metric multidimensional scaling (MDS) by incorporating the geodesic distances imposed by a weighted graph. To be specific, the classical scaling of metric MDS performs low-dimensional embedding based on the pairwise distance between data points, which is generally measured ...
A bilinear map is a function: such that for all , the map (,) is a linear map from to , and for all , the map (,) is a linear map from to . In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
The locations of the points in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution from the distribution , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}}
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer , a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form.
A vector v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if Ax = 0 ⇔ x T A = 0. The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate. Suppose W is a subspace.
Marching squares takes a similar approach to the 3D marching cubes algorithm: Process each cell in the grid independently. Calculate a cell index using comparisons of the contour level(s) with the data values at the cell corners. Use a pre-built lookup table, keyed on the cell index, to describe the output geometry for the cell.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.