Matrix-Matrix Multiplication

Suppose we want to multiply $m \times n$ matrix $A$ on $n \times p$ matrix $B$, we get an $m \times p$ matrix $C$


Linear Transformation

What is matrix-matrix multiplication in terms of Linear Transformations?

  • Let $A$ be an $m \times n$ matrix,
  • then there's a linear transformation $T_A \ : \ \mathbb R^n \to \mathbb R^m$: $T_A(\mathbf x) = A \mathbf x$ where $A \mathbf x$ is Matrix-Vector Multiplication
  • now let $B$ be an $n \times k$ matrix, then $T_B \ : \ \mathbb R^m \to \mathbb R^k$: $T_B(\mathbf y) = B \mathbf x$
  • what is $T_A \circ T_B$? It's $T_A \circ T_B \ : \ \mathbb R^k \to \mathbb R^m $
  • $(T_A \circ T_B)(\mathbf x) = T_A \big( T_B(\mathbf x) \big) = T_A \big( B \mathbf x \big) = A \, B \, \mathbf x$
  • $AB$ is matrix-matrix multiplication


Multiplication

We can see matrix by matrix multiplication from 5 different positions:

  • row by column multiplication
  • column at a time
  • row at a time
  • as sum of outer products
  • block multiplication

All of them are equivalent and lead to the same result


Row By Columns

This is usual dot product multiplication:

  • for each row of matrix $A$ we calculate a dot product with each column of matrix $B$
  • bad3a8b38db64a918543146979adcea0.png
  • $c_{ij} = (\text{row $i$ of $A$})^T \times (\text{col $j$ of $B$}) = \sum\limits_{k=1}^{m} c_{ik} b_{kj}$


Column at a Time

For each column $\mathbf{b}_j$ of $B$

  • we multiply each column of $A$ with $\mathbf{b}_j$ - like in column at a time for matrix by vector
  • fe8ffbfb9ede4ad18a868024f8e791a1.png
  • [math]\mathbf{c}_j = \begin{bmatrix} \mathop{a_1}\limits_|^| \ \mathop{a_2}\limits_|^| \ \cdots \ \mathop{a_n}\limits_|^| \end{bmatrix} \times \mathbf{b}_j[/math]
  • so each $\mathbf{c}_j$ is a combination of columns of $A$

Row at a Time

For each row $\mathbf{a}^T_i$ of $A$

  • multiply $\mathbf{a}^T_i$ with each rows of $B$ - like for left vector multiplication
  • 9acc1ab9d7784a96b3e42f72fe4f1882.png
  • $\mathbf{c}^T_i = \mathbf{a}^T_i \times B$
  • Note that we can see this as Column at a Time case, but transposed:
    • row at a time in $A\times B = C$ is the same as column at a time in $B^T \times A^T = C^T$


Sum of Outer Products

For $i$ from 1 to $n$,

  • multiply column of $A$ $\mathbf{a}_i$ by row of $B$ $\mathbf{b}^T_i$
  • it gives us a rank-1 matrix - an outer product
  • then sum over all $i$
  • c8c6b790cafc4240b41015c484fdb4f2.png
  • $C = AB = \sum\limits_{i=1}^n \mathbf{a}_i \mathbf{b}^T_i$


Block Multiplication

$AB = \left[ \begin{array}{c|c} A_1 & A_2 \\ \hline A_3 & A_4 \end{array} \right] \times \left[ \begin{array}{c|c} B_1 & B_2 \\ \hline B_3 & B_4 \end{array} \right] = \left[ \begin{array}{c|c} A_1B_1 + A_2B_3 & A_3B_1 + A_4B_3 \\ \hline A_1B2 + A_2B_1 & A_3B_1 + A_4B_3 \end{array} \right] = C$


Properties

Transposition

  • $(AB)^T = B^T A^T$
  • $(A_1 \cdot \ ... \ \cdot A_n)^T = A_n^T \cdot \ ... \ \cdot A_1^T$


Inverse Matrices

$(AB)^{-1} = B^{-1} A^{-1}$

  • inverse of product is product of inverses in the reversed order
  • check:
    • $AB \times B^{-1} A^{-1} = A \times (B B^{-1}) \times A^{-1} = A \times I \times A^{-1} A \times A^{-1} = I$
    • $B^{-1} A^{-1} \times AB = B^{-1} \times (A^{-1} A) \times B = B^{-1} \times B = I$


Row Space and Column Space

Let's show the following:

  • $C(A\, B) \subseteq C(A)$
  • $R(A \, B) \subseteq R(B)$
  • where $C(\cdot)$ is Column Space and $R(\cdot)$ is Row Space


Image of linear transformation

  • first, let's consider a linear transformation of an $m \times n$ matrix $A$
  • $T_A \ : \ \mathbb R^n \to \mathbb R^m$ : $T_A(\mathbf x) = A \mathbf x$
  • Column Space of $A$ is the image of $T_A$ in $\mathbb R^m$


$C(A\, B) \subseteq C(A)$

  • let $A$ be an $m \times n$ matrix and $B$ be an $k \times n$
  • $A \, B$ corresponds to linear transformation $T_A \circ T_B$
  • so need to show that $\text{image}(T_A \circ T_B) \subseteq \text{image}(T_A)$
  • 693fc99de6f7493d97e571e0a3b3e0c8.png


What about $R(A \, B) \subseteq R(B)$?

  • $R(A \, B) = C(B^T A^T) \subseteq C(B^T) = R(B)$


Implementation

Possible Ways to Implement

Suppose we want to implement the update $C = C + AB$:

  • let $A$ be $m \times r$ matrix and $B$ be $r \times n$
  • then $C$ is $m \times n$ matrix
  • $C$ can be initialized as an array filled with zeros - then $C$ will contain the results of $AB$
for (int i = 0; i < m; i++) {
    for (int j = 0; i < n; j++) {
        for (int k = 0; k < r; k++) {
            C[i, j] = C[i, j] + A[i, k] * B[k, j]
        }
    }
}


Note that here it is not important in which order we do the loop:

  • it can be ijk, ikj, etc, with $3! = 6$ possible ways
Order Inner Loop Outer Loops Inner Loop Access
ijk dot Vector-Matrix Mult $A$ row, $B$ col
jik dot Matrix-Vector Mult $A$ row, $B$ col
ikj saxpy Row gaxpy $B$ row, $C$ row
jki saxpy Col gaxpy $A$ col, $C$ col
kij saxpy row outer product $A$ row, $C$ row
kji saxpy col outer product $A$ col, $C$ col


Some details:

  • The way inner loop accesses the data is important for storing the data to make it faster - data should be contiguos in memory
  • dot is usual Vector-Vector multiplication
  • saxpy (scalar $a$ times $x$ plus $y$): $y \leftarrow ax + y$
  • gaxpy (generalized axpy): $y \leftarrow y + Ax$
  • outer product update: $A \leftarrow A + xy^T$


SQL and MapReduce

Suppose we have two sparce matrices $A$ and $B$

  • we store these matrices as (row_num, col_num, value) triples
  • it is easy to use SQL for this multiplication


Sparse matrix multiplication with SQL

select a.row_num, b.col_num, sum(a.value * b.value)
  from A a, B b
 where a.col_num = b.row_num
group by a.row_num, b.col_num;


MapReduce

  • We can translate this SQL expression into MapReduce
  • Link [1]


E.g. Apache Flink:

matrixA.join(matrixB).where(1).equalTo(0)
       .map(new ProjectJoinResultMapper()).groupBy(0, 1).sum(2)


Full code of Matrix Multiplication in Flink: [2]



Sources

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