## Mahout-Samsara’s In-Core Linear Algebra DSL Reference

#### Imports

The following imports are used to enable Mahout-Samsara’s Scala DSL bindings for in-core Linear Algebra:

import org.apache.mahout.math._
import scalabindings._
import RLikeOps._


#### Inline initalization

Dense vectors:

val densVec1: Vector = (1.0, 1.1, 1.2)
val denseVec2 = dvec(1, 0, 1,1 ,1,2)


Sparse vectors:

val sparseVec1: Vector = (5 -> 1.0) :: (10 -> 2.0) :: Nil
val sparseVec1 = svec((5 -> 1.0) :: (10 -> 2.0) :: Nil)

// to create a vector with specific cardinality
val sparseVec1 = svec((5 -> 1.0) :: (10 -> 2.0) :: Nil, cardinality = 20)


Inline matrix initialization, either sparse or dense, is always done row wise.

Dense matrices:

val A = dense((1, 2, 3), (3, 4, 5))


Sparse matrices:

val A = sparse(
(1, 3) :: Nil,
(0, 2) :: (1, 2.5) :: Nil
)


Diagonal matrix with constant diagonal elements:

diag(3.5, 10)


Diagonal matrix with main diagonal backed by a vector:

diagv((1, 2, 3, 4, 5))


Identity matrix:

eye(10)


####Slicing and Assigning

Getting a vector element:

val d = vec(5)


Setting a vector element:

vec(5) = 3.0


Getting a matrix element:

val d = m(3,5)


Setting a matrix element:

M(3,5) = 3.0


Getting a matrix row or column:

val rowVec = M(3, ::)
val colVec = M(::, 3)


Setting a matrix row or column via vector assignment:

M(3, ::) := (1, 2, 3)
M(::, 3) := (1, 2, 3)


Setting a subslices of a matrix row or column:

a(0, 0 to 1) = (3, 5)


Setting a subslices of a matrix row or column via vector assignment:

a(0, 0 to 1) := (3, 5)


Getting a matrix as from matrix contiguous block:

val B = A(2 to 3, 3 to 4)


Assigning a contiguous block to a matrix:

A(0 to 1, 1 to 2) = dense((3, 2), (3 ,3))


Assigning a contiguous block to a matrix using the matrix assignment operator:

A(o to 1, 1 to 2) := dense((3, 2), (3, 3))


Assignment operator used for copying between vectors or matrices:

vec1 := vec2
M1 := M2


Assignment operator using assignment through a functional literal for a matrix:

M := ((row, col, x) => if (row == col) 1 else 0


Assignment operator using assignment through a functional literal for a vector:

vec := ((index, x) => sqrt(x)


#### BLAS-like operations

Plus/minus either vector or numeric with assignment or not:

a + b
a - b
a + 5.0
a - 5.0


Hadamard (elementwise) product, either vector or matrix or numeric operands:

a * b
a * 0.5


Operations with assignment:

a += b
a -= b
a += 5.0
a -= 5.0
a *= b
a *= 5


Some nuanced rules:

1/x in R (where x is a vector or a matrix) is elementwise inverse. In scala it would be expressed as:

val xInv = 1 /: x


and R’s 5.0 - x would be:

val x1 = 5.0 -: x


note: All assignment operations, including :=, return the assignee just like in C++:

a -= b


assigns a - b to b (in-place) and returns b. Similarly for a /=: b or 1 /=: v

Dot product:

a dot b


Matrix and vector equivalency (or non-equivalency). Dangerous, exact equivalence is rarely useful, better to use norm comparisons with an allowance of small errors.

a === b
a !== b


Matrix multiply:

a %*% b


Optimized Right Multiply with a diagonal matrix:

diag(5, 5) :%*% b


Optimized Left Multiply with a diagonal matrix:

A %*%: diag(5, 5)


Second norm, of a vector or matrix:

a.norm


Transpose:

val Mt = M.t


note: Transposition is currently handled via view, i.e. updating a transposed matrix will be updating the original. Also computing something like $$\mathbf{X^\top}\mathbf{X}$$:

val XtX = X.t %*% X


will not therefore incur any additional data copying.

#### Decompositions

Matrix decompositions require an additional import:

import org.apache.mahout.math.decompositions._


All arguments in the following are matricies.

Cholesky decomposition

val ch = chol(M)


SVD

val (U, V, s) = svd(M)


EigenDecomposition

val (V, d) = eigen(M)


QR decomposition

val (Q, R) = qr(M)


Rank: Check for rank deficiency (runs rank-revealing QR)

M.isFullRank


In-core SSVD

Val (U, V, s) = ssvd(A, k = 50, p = 15, q = 1)


Solving linear equation systems and matrix inversion: fully similar to R semantics; there are three forms of invocation:

Solve $$\mathbf{AX}=\mathbf{B}$$:

solve(A, B)


Solve $$\mathbf{Ax}=\mathbf{b}$$:

solve(A, b)


Compute $$\mathbf{A^{-1}}$$:

solve(A)


#### Misc

Vector cardinality:

a.length


Matrix cardinality:

m.nrow
m.ncol


Means and sums:

m.colSums
m.colMeans
m.rowSums
m.rowMeans


Copy-By-Value:

val b = a cloned


#### Random Matrices

$$\mathcal{U}$$(0,1) random matrix view:

val incCoreA = Matrices.uniformView(m, n, seed)


$$\mathcal{U}$$(-1,1) random matrix view:

val incCoreA = Matrices.symmetricUniformView(m, n, seed)


$$\mathcal{N}$$(-1,1) random matrix view:

val incCoreA = Matrices.gaussianView(m, n, seed)


#### Iterators

Mahout-Math already exposes a number of iterators. Scala code just needs the following imports to enable implicit conversions to scala iterators.

import collection._
import JavaConversions._


Iterating over rows in a Matrix:

for (row <- m) {
... do something with row
}


For more information including information on Mahout-Samsara’s out-of-core Linear algebra bindings see: Mahout Scala Bindings and Mahout Spark Bindings for Linear Algebra Subroutines