ml_linalg 2.3.0

SIMD-based linear algebra with dart for machine learning purposes

Linear algebra with Dart for machine learning

Table of contents

• Vectors
  • A couple of words about the underlying vector architecture
    • Vector operations examples
      • Vectors sum
      • Vectors subtraction
      • Element wise vector by vector multiplication
      • Element wise vector by vector division
      • Euclidean norm
      • Manhattan norm
      • Mean value
      • Sum of all vector elements
      • Dot product
      • Sum of a vector and a scalar
      • Subtraction of a scalar from a vector
      • Multiplication (scaling) of a vector by a scalar
      • Division (scaling) of a vector by a scalar value
      • Euclidean distance between two vectors
      • Manhattan distance between two vectors
• Matrices
  • Matrix operations examples - Sum of a matrix and another matrix - Sum of a matrix and a scalar - Multiplication of a matrix and a vector - Multiplication of a matrix and another matrix - Multiplication of a matrix and a scalar - Element wise matrices subtraction - Matrix transposition - Matrix row wise reduce - Matrix column wise reduce - Submatrix
    • Matrix indexing

Vectors

A couple of words about the underlying vector architecture

All vector operations are supported by SIMD (single instruction, multiple data) computation architecture, so this library presents a high performance SIMD vector class, based on Float32x4 - Float32x4Vector. However, you cannot use it directly in your project. To create an instance of the vector, just import Float32x4VectorFactory and instantiate a vector via the factory. Most of operations in the vector are performed in four "threads". This kind of concurrency is reached by special 128-bit processor registers, which are used directly by program code. For better understanding of the topic please read the article.

Vector operations examples

At the present moment most common vector operations are implemented:

Vectors sum

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1 + vector2;
  print(result.toList()); // [3.0, 5.0, 7.0, 9.0, 11.0]

Vectors subtraction

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([4.0, 5.0, 6.0, 7.0, 8.0]);
  final vector2 = Float32x4VectorFactory.from([2.0, 3.0, 2.0, 3.0, 2.0]);
  final result = vector1 - vector2;
  print(result.toList()); // [2.0, 2.0, 4.0, 4.0, 6.0]

Element wise vector by vector multiplication

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1 * vector2;
  print(result.toList()); // [2.0, 6.0, 12.0, 20.0, 30.0]

Element wise vector by vector division

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([6.0, 12.0, 24.0, 48.0, 96.0]);
  final vector2 = Float32x4VectorFactory.from([3.0, 4.0, 6.0, 8.0, 12.0]);
  final result = vector1 / vector2;
  print(result.toList()); // [2.0, 3.0, 4.0, 6.0, 8.0]

Euclidean norm

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.norm();
  print(result); // sqrt(2^2 + 3^2 + 4^2 + 5^2 + 6^2) = sqrt(90) ~~ 9.48

Manhattan norm

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.norm(Norm.manhattan);
  print(result); // 2 + 3 + 4 + 5 + 6 = 20.0

Mean value

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.mean();
  print(result); // (2 + 3 + 4 + 5 + 6) / 5 = 4.0

Sum of all vector elements

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.sum();
  print(result); // 2 + 3 + 4 + 5 + 6 = 20.0 (equivalent to Manhattan norm)

Dot product of two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.dot(vector2);
  print(result); // 1.0 * 2.0 + 2.0 * 3.0 + 3.0 * 4.0 + 4.0 * 5.0 + 5.0 * 6.0 = 70.0

Sum of a vector and a scalar

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final scalar = 5.0;
  final result = vector1 + scalar;
  print(result.toList()); // [6.0, 7.0, 8.0, 9.0, 10.0]

Subtraction of a scalar from a vector

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final scalar = 5.0;
  final result = vector1 - scalar;
  print(result.toList()); // [-4.0, -3.0, -2.0, -1.0, 0.0]

Multiplication (scaling) of a vector by a scalar

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final scalar = 5.0;
  final result = vector1 * scalar;
  print(result.toList()); // [5.0, 10.0, 15.0, 20.0, 25.0]

Division (scaling) of a vector by a scalar value

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([25.0, 50.0, 75.0, 100.0, 125.0]);
  final scalar = 5.0;
  final result = vector1.scalarDiv(scalar);
  print(result.toList()); // [5.0, 10.0, 15.0, 20.0, 25.0]

Euclidean distance between two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.distanceTo(vector2);
  print(result); // ~~2.23

Manhattan distance between two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4VectorFactory.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4VectorFactory.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.distanceTo(vector2, Norm.manhattan);
  print(result); // 5.0

Matrices

Also, a class for matrix is available. It is based on Float32x4 and Float32x4Vector types.

Matrix operations examples

Sum of a matrix and another matrix

import 'package:ml_linalg/linalg.dart';

final matrix1 = Float32x4MatrixFactory.from([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
final matrix2 = Float32x4MatrixFactory.from([
  [10.0, 20.0, 30.0, 40.0],
  [-5.0, 16.0, 2.0, 18.0],
  [2.0, -1.0, -2.0, -7.0],
]);
print(matrix1 + matrix2);
// [
//  [11.0, 22.0, 33.0, 44.0],
//  [0.0, 22.0, 9.0, 26.0],
//  [11.0, -1.0, -4.0, -10.0],
// ];

Sum of a matrix and a scalar

import 'package:ml_linalg/linalg.dart';

final matrix = Float32x4MatrixFactory.from([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
print(matrix + 7);
//  [
//    [8.0, 9.0, 10.0, 11.0],
//    [12.0, 13.0, 14.0, 15.0],
//    [16.0, 7.0, 5.0, 4.0],
//  ];

Multiplication of a matrix and a vector

  import 'package:ml_linalg/linalg.dart';

  final matrix = Float32x4MatrixFactory.from([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
    [9.0, .0, -2.0, -3.0],
  ]);
  final vector = Float32x4Vector.from([2.0, 3.0, 4.0, 5.0]);
  final result = matrix * vector;
  print(result); 
  // a vector-column [
  //  [40],
  //  [96],
  //  [-5],
  //]

Multiplication of a matrix and another matrix

  import 'package:ml_linalg/linalg.dart';

  final matrix1 = Float32x4MatrixFactory.from([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
    [9.0, .0, -2.0, -3.0],
  ]);
  final matrix2 = Float32x4MatrixFactory.from([
    [1.0, 2.0],
    [5.0, 6.0],
    [9.0, .0],
    [-9.0, 1.0],
  ]);
  final result = matrix1 * matrix2;
  print(result);
  //[
  // [2.0, 18.0],
  // [26.0, 54.0],
  // [18.0, 15.0],
  //]

Multiplication of a matrix and a scalar

import 'package:ml_linalg/linalg.dart';

final matrix = Float32x4MatrixFactory.from([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
print(matrix * 3);
// [
//   [3.0, 6.0, 9.0, 12.0],
//   [15.0, 18.0, 21.0, 24.0],
//   [27.0, .0, -6.0, -9.0],
// ];

Element wise matrices subtraction

import 'package:ml_linalg/linalg.dart';

final matrix1 = Float32x4MatrixFactory.from([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
final matrix2 = Float32x4MatrixFactory.from([
  [10.0, 20.0, 30.0, 40.0],
  [-5.0, 16.0, 2.0, 18.0],
  [2.0, -1.0, -2.0, -7.0],
]);
print(matrix1 - matrix2);
// [
//   [-9.0, -18.0, -27.0, -36.0],
//   [10.0, -10.0, 5.0, -10.0],
//   [7.0, 1.0, .0, 4.0],
// ];

Matrix transposition

  import 'package:ml_linalg/linalg.dart';
  
  final matrix = Float32x4MatrixFactory.from([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
    [9.0, .0, -2.0, -3.0],
  ]);
  final result = matrix.transpose();
  print(result);
  //[
  // [1.0, 5.0, 9.0],
  // [2.0, 6.0, .0],
  // [3.0, 7.0, -2.0],
  // [4.0, 8.0, -3.0],
  //]

Matrix row wise reduce

  import 'package:ml_linalg/linalg.dart';

  final matrix = Float32x4MatrixFactory.from([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
  ]); 
  final reduced = matrix.reduceRows((combine, row) => combine + row);
  print(reduced); // [6.0, 8.0, 10.0, 12.0]

Matrix column wise reduce

  import 'package:ml_linalg/linalg.dart';

  final matrix = Float32x4MatrixFactory.from([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 17.0, 18.0],
    [21.0, 22.0, 23.0, 24.0],
  ]);
  final result = matrix.reduceColumns((combine, vector) => combine + vector);
  print(result); // [50, 66, 90]

Submatrix (taking a lower dimension matrix of the current matrix)

  import 'package:ml_linalg/linalg.dart';

  final matrix = Float32x4MatrixFactory.from([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 17.0, 18.0],
    [21.0, 22.0, 23.0, 24.0],
    [24.0, 32.0, 53.0, 74.0],
  ]);
  final submatrix = matrix.submatrix(rows: Range(0, 2));
  print(submatrix);
  // [
  //  [11.0, 12.0, 13.0, 14.0],
  //  [15.0, 16.0, 17.0, 18.0],
  //];

Matrix indexing

The library's matrix interface offers pick method, that supposed to return a new matrix, consisting of different segments of a source matrix (like in Pandas dataframe in Python, e.g. loc method). It's possible to build a new matrix from certain columns and vectors and they should not be necessarily subsequent: for example, it is needed to create a matrix from rows 1, 3, 5 and columns 1 and 3. To do so, it's needed to access the matrix this way:

final matrix = Float32x4MatrixFactory.from([
//| 1 |         | 3 |                
  [4.0,   8.0,   12.0,   16.0,  34.0], // 1 Range(0, 1)
  [20.0,  24.0,  28.0,   32.0,  23.0],
  [36.0,  .0,   -8.0,   -12.0,  12.0], // 3 Range(2, 3)
  [16.0,  1.0,  -18.0,   3.0,   11.0],
  [112.0, 10.0,  34.0,   2.0,   10.0], // 5 Range(4, 5)
]);
final result = matrix.pick(
  rowRanges: [Range(0, 1), Range(2, 3), Range(4, 5)],
  columnRanges: [Range(0, 1), Range(2, 3)],
);
print(result);
/*
  [4.0,   12.0],
  [36.0,  -8.0],
  [112.0, 34.0]
*/