+ +

linearalgebra.lib

+

A library of linear algebra functions. Its official prefix is la.

+

This library adds some new linear algebra functions:

+

determinant

+

minor

+

inverse

+

transpose

+

matMul matrix multiplication

+

identity

+

How does it work? An NxM matrix can be flattened into a bus si.bus(N*M). These buses can be passed to functions as long as N and sometimes M (if the matrix need not be square) are passed too.

+

Some things to think about going forward

+
Implications for ML in Faust
+

Next step of making a "Dense"/"Linear" layer from machine learning +Where in the libraries should ReLU go? +What about 3D tensors instead of 2D matrices? Image convolutions take place on 3D tensors shaped HxWxC.

+
Design of matMul
+

Currently the design is matMul(J, K, L, M, leftHandMat, rightHandMat) where leftHandMat is JxK and rightHandMat is LxM.

+

It would also be neat to have matMul(J, K, rightHandMat, L, M, leftHandMat).

+

Then a "packed" matrix could be consistently stored as a combination of a 2-channel "header" N, M and the values si.bus(N*M).

+

This would ultimately enable result = packedLeftHand : matMul(packedRightHand); for the equivalent numpy code: result = packedLeftHand @ packedRightHand;.

+

References

+ +
+

(la.)determinant

+

Calculates the determinant of a bus that represents +an NxN matrix.

+

Usage

+
si.bus(N*N) : determinant(N) : _
+
+

Where:

+
    +
  • N: the size of each axis of the matrix.
    • +
+
+

(la.)minor

+

This is a utility for finding the matrix minor when inverting a matrix. +It returns the determinant of the submatrix formed by deleting the row at +index ROW and column at index COL. +The following implementation doesn't work but looks simple.

+
minor(N, ROW, COL) = par(r, N, par(c, N, select2((ROW==r)||(COL==c),_,!))) : determinant(N-1);
+
+

Usage

+
si.bus(N*N) : minor(N, ROW, COL) : _
+
+

Where:

+
    +
  • N: the size of each axis of the matrix.
    • +
  • ROW: the selected position on 0th dimension of the matrix (0 <= ROW < N)
    • +
  • COL: the selected position on the 1st dimension of the matrix (0 <= COL < N)
    • +
+

References: +* https://en.wikipedia.org/wiki/Minor_(linear_algebra)#First_minor

+
+

(la.)inverse

+

This inverts a matrix. The incoming bus represents an NxN matrix. +Note, this is an unsafe operation since not all matrices are invertible.

+

Usage

+
si.bus(N*N) : inverse(N) : si.bus(N*N)
+
+

Where:

+
    +
  • N: the size of each axis of the matrix.
    • +
+
+

(la.)transpose

+

Transposes an NxM matrix stored in row-major order, resulting +in an MxN matrix stored in row-major order.

+

Usage

+
si.bus(N*M) : transpose(N, M) : si.bus(M*N)
+
+

Where:

+
    +
  • N: the number of rows in the input matrix
    • +
  • M: the number of columns in the input matrix
    • +
+
+

(la.)matMul

+

Multiply a JxK matrix (mat1) and an LxM matrix (mat2) to produce a JxM matrix. +Note that K==L. +Both matrices should use row-major order. +In terms of numpy, this function is mat1 @ mat2.

+

Usage

+
matMul(J, K, L, M, si.bus(J*K), si.bus(L*M)) : si.bus(J*M)
+
+

Where:

+
    +
  • J: the number of rows in mat1
    • +
  • K: the number of columns in mat1
    • +
  • L: the number of rows in mat2
    • +
  • M: the number of columns in mat2
    • +
+
+

(la.)identity

+

Creates an NxN identity matrix.

+

Usage

+
identity(N) : si.bus(N*N)
+
+

Where:

+
    +
  • N: The size of each axis of the identity matrix.
    • +