### Parallel combining

By parallel combining sets of mappings are combined that operate from different input spaces into the same output space. The resulting mapping concatenates the output datasets. The combining operator is the vertical concatenation. So if we have three mappings `W1`

, `W2`

and `W3`

, they are combined by

W = [W1; W2; W3];

It can map a concatenation `A`

of three input datasets `A1`

,` A2`

and `A3`

, referring to the same objects, but defined for three different feature sets. They will be mapped into the same output space which is a concatenation of the output spaces of the three constituting mappings.

B = A*W = [A1 A2 A3]*W; B = [A1 A2 A3]*[W1;W2;W3]; B = [A1*W1 A2*W2 A3*W3] = [B1 B2 B3];

The dimension of the output space of `W`

and thereby the feature size of `B`

is equal to the sum of the output dimensions of `W1`

, `W2`

and `W3`

. Their input sizes should match the feature sizes of `A1`

,` A2`

and `A3`

.

Note that according to the standard Matlab rules the operation `W = [W1; W2; W3]`

is irregular as the sizes of `W1`

, `W2`

and `W3`

may differ and that the size of the second dimension of `W`

equals the sum of the sizes of `W1`

, `W2`

and `W3`

.

Parallel combining can also be applied to untrained mappings. In that case a set of untrained base mappings `U1`

, `U2`

, `...` are trained separately, resulting in `W1`

, `W2`

, `...`. These constitute a new, now trained, combined mapping.

% concatenation of untrained mappings U = [U1; U2; U3];

% training, results in W = [A*U1; A*U2; A*U3] = [W1; W2; W3] W = [A1 A2 A3]*U;

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