sparkmllib矩阵向量

Spark MLlib底层的向量、矩阵运算使用了Breeze库，Breeze库提供了Vector/Matrix的实现以及相应计算的接口（Linalg）。但是在MLlib里面同时也提供了Vector和Linalg等的实现。 
使用需导入：

import breeze.linalg._
import breeze.numerics._

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Breeze创建函数

val m1 = DenseMatrix.zeros[Double](2,3)

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DenseMatrix[Double] = 
0.0 0.0 0.0 
0.0 0.0 0.0

val v1 = DenseVector.zeros[Double](3)

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DenseVector(0.0, 0.0, 0.0)

val v2 = DenseVector.ones[Double](3)

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DenseVector(1.0, 1.0, 1.0)

val v3 = DenseVector.fill(3){5.0}

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DenseVector(5.0, 5.0, 5.0)

val v4 = DenseVector.range(1,10,2)

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DenseVector(1, 3, 5, 7, 9)

val m2 = DenseMatrix.eye[Double](3)

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DenseMatrix[Double] = 
1.0 0.0 0.0 
0.0 1.0 0.0 
0.0 0.0 1.0

val v6 = diag(DenseVector(1.0,2.0,3.0))

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DenseMatrix[Double] = 
1.0 0.0 0.0 
0.0 2.0 0.0 
0.0 0.0 3.0

val v8 = DenseVector(1,2,3,4)

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DenseVector(1, 2, 3, 4)

val v9 = DenseVector(1,2,3,4).t

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Transpose(DenseVector(1, 2, 3, 4))

val v10 = DenseVector.tabulate(3){i => 2*i}

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DenseVector(0, 2, 4)

val m4 = DenseMatrix.tabulate(3, 2){case (i, j) => i+j}

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DenseMatrix[Int] = 
0 1 
1 2 
2 3

val v11 = new DenseVector(Array(1, 2, 3, 4))

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DenseVector(1, 2, 3, 4)

val m5 = new DenseMatrix(2, 3, Array(11, 12, 13, 21, 22, 23))

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DenseMatrix[Int] = 
11 13 22 
12 21 23

val v12 = DenseVector.rand(4)

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DenseVector(0.7517657487447951, 0.8171495400874123, 0.8923542318540489, 0.174311259949119)

val m6 = DenseMatrix.rand(2, 3)

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DenseMatrix[Double] = 
0.5349430131148125 0.8822136832272578 0.7946323804433382 
0.41097756311601086 0.3181490074596882 0.34195102205697414

Breeze元素访问

val a = DenseVector(1,2,3,4,5,6,7,8,9,10)

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DenseVector(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

a(1 to 4)

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DenseVector(2, 3, 4, 5)

a(5 to 0 by -1)

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DenseVector(6, 5, 4, 3, 2, 1)

a(1 to -1)

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DenseVector(2, 3, 4, 5, 6, 7, 8, 9, 10)

a( -1 )

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Int = 10

val m = DenseMatrix((1.0,2.0,3.0), (3.0,4.0,5.0))

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DenseMatrix[Double] = 
1.0 2.0 3.0 
3.0 4.0 5.0

m(0,1)

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Double = 2.0

m(::,1)

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DenseVector(2.0, 4.0)

Breeze元素操作

val m = DenseMatrix((1.0,2.0,3.0), (3.0,4.0,5.0))

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DenseMatrix[Double] = 
1.0 2.0 3.0 
3.0 4.0 5.0

m.reshape(3, 2) //从列开始计数

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DenseMatrix[Double] = 
1.0 4.0 
3.0 3.0 
2.0 5.0

m.toDenseVector

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DenseVector(1.0, 3.0, 2.0, 4.0, 3.0, 5.0)

val m = DenseMatrix((1.0,2.0,3.0), (4.0,5.0,6.0) , (7.0,8.0,9.0))

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DenseMatrix[Double] = 
1.0 2.0 3.0 
4.0 5.0 6.0 
7.0 8.0 9.0

lowerTriangular(m)

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DenseMatrix[Double] = 
1.0 0.0 0.0 
4.0 5.0 0.0 
7.0 8.0 9.0

upperTriangular(m)

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DenseMatrix[Double] = 
1.0 2.0 3.0 
0.0 5.0 6.0 
0.0 0.0 9.0

m.copy

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linalg.DenseMatrix[Double] = 
1.0 2.0 3.0 
4.0 5.0 6.0 
7.0 8.0 9.0

diag(m)

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DenseVector(1.0, 5.0, 9.0)

m(::, 2) := 5.0

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DenseVector(5.0, 5.0, 5.0)

m

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DenseMatrix[Double] = 
1.0 2.0 5.0 
4.0 5.0 5.0 
7.0 8.0 5.0

m(1 to 2,1 to 2) := 5.0

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DenseMatrix[Double] = 
5.0 5.0 
5.0 5.0

val a = DenseVector(1,2,3,4,5,6,7,8,9,10)

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DenseVector(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

a(1 to 4) := 5

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DenseVector(5, 5, 5, 5)

a(1 to 4) := DenseVector(1,2,3,4)

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DenseVector(1, 2, 3, 4)

val a1 = DenseMatrix((1.0,2.0,3.0), (4.0,5.0,6.0))
val a2 = DenseMatrix((1.0,1.0,1.0), (2.0,2.0,2.0))
DenseMatrix.vertcat(a1,a2)

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DenseMatrix[Double] = 
1.0 2.0 3.0 
4.0 5.0 6.0 
1.0 1.0 1.0 
2.0 2.0 2.0

DenseMatrix.horzcat(a1,a2)

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DenseMatrix[Double] = 
1.0 2.0 3.0 1.0 1.0 1.0 
4.0 5.0 6.0 2.0 2.0 2.0

val b1 = DenseVector(1,2,3,4)
val b2 = DenseVector(1,1,1,1)
DenseVector.vertcat(b1,b2)

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DenseVector(1, 2, 3, 4, 1, 1, 1, 1)

Breeze数值计算函数

val a = DenseMatrix((1.0,2.0,3.0), (4.0,5.0,6.0))
val b = DenseMatrix((1.0,1.0,1.0), (2.0,2.0,2.0))
a + b

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DenseMatrix[Double] = 
2.0 3.0 4.0 
6.0 7.0 8.0

a :* b

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DenseMatrix[Double] = 
1.0 2.0 3.0 
8.0 10.0 12.0

a :/ b

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DenseMatrix[Double] = 
1.0 2.0 3.0 
2.0 2.5 3.0

a :< b

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DenseMatrix[Boolean] = 
false false false 
false false false

a :== b

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DenseMatrix[Boolean] = 
true false false 
false false false

a :+= 1.0

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DenseMatrix[Double] = 
2.0 3.0 4.0 
5.0 6.0 7.0

a :*= 2.0

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DenseMatrix[Double] = 
4.0 6.0 8.0 
10.0 12.0 14.0

max(a)

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Double = 14.0

argmax(a)

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(Int, Int) = (1,2)

DenseVector(1, 2, 3, 4) dot DenseVector(1, 1, 1, 1)//点积

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Int = 10

Breeze求和函数

val a = DenseMatrix((1.0,2.0,3.0), (4.0,5.0,6.0) , (7.0,8.0,9.0))
sum(a)

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Double = 45.0

sum(a, Axis._0)//每列求和

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DenseMatrix[Double] = 12.0 15.0 18.0

sum(a, Axis._1)//按行求和
trace(a) //对角线求和  15

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accumulate(DenseVector(1, 2, 3, 4)) //累计和 1+2 、1+2+3

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DenseVector(1, 3, 6, 10)

Breeze布尔函数

val a = DenseVector(true, false, true)
val b = DenseVector(false, true, true)
a :& b
a :| b
!a

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DenseVector(false, false, true)

val a = DenseVector(1.0, 0.0, -2.0)
any(a) //任一元素非0，true
all(a) //所有元素非0，false

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Breeze线性代数函数

a \ b //线性求解
a.t //转置
det(a) //求特征值
inv(a) //求逆
pinv(a) //求伪逆
norm(a) //求范数
eigSym(a)//特征值和特征向量
val (er, ei, _) = eig(a) (实部与虚部分开) //特征值
eig(a)._3//特征向量
val svd.SVD(u,s,v) = svd(a)//奇异值分解
rank(a)//求矩阵的秩
a.length//矩阵长度
a.rows//矩阵行数
a.cols//矩阵列数

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DenseMatrix((1.0,2.0,3.0), (4.0,5.0,6.0) , (7.0,8.0,9.0))
DenseMatrix((1.0,1.0,1.0), (1.0,1.0,1.0) , (1.0,1.0,1.0))
a \ b
a.t

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DenseMatrix[Double] = 
1.0 4.0 7.0 
2.0 5.0 8.0 
3.0 6.0 9.0

Breeze取整函数

round(a)//四舍五入
ceil(a)
floor(a)
signum(a)//符号函数
abs(a)

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val a = DenseVector(1.2, 0.6, -2.3)
signum(a)

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DenseVector(1.0, 1.0, -1.0)

Breeze其它函数

Breeze三角函数包括：

sin, sinh, asin, asinh
cos, cosh, acos, acosh
tan, tanh, atan, atanh
atan2
sinc(x) ，即sin(x)/x
sincpi(x) ，即 sinc(x * Pi)

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Breeze对数和指数函数 
Breeze对数和指数函数包括：

log, exp log10
log1p, expm1
sqrt, sbrt
pow

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BLAS介绍(一个线性代数库)

BLAS按照功能被分为三个级别： 
Level 1：矢量-矢量运算，比如点积（ddot)，加法和数乘 (daxpy)， 绝对值的和（dasum)，等等； 
Level 2：矩阵-矢量运算，最重要的函数是一般的矩阵向量乘法(dgemv)； 
Level 3：矩阵-矩阵运算，最重要的函数是一般的矩阵乘法 (dgemm)； 
每一种函数操作都区分不同数据类型（单精度、双精度、复数）