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SIMD-based linear algebra with dart for machine learning purposes

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Linear algebra with Dart for machine learning

Table of contents

Vectors #

A couple of words about the underlying vector architecture

All vector operations are supported by SIMD (single instruction, multiple data) computation architecture. Actually, the main purpose of the library - connect such a powerful computation way with the pure math. So the library contains a high performance SIMD vector class, based on Float32x4 - Float32x4VectorInternal. As you can see from the class name, you cannot use it directly in your project. To create an instance of the vector, just import Float32x4Vector and instantiate a vector via this 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. It is also possible to implement Float64x2-based version of the vector using existing codebase, but so far there is no need to do so.

Vector operations examples

At the present moment most common vector operations are implemented:

Vectors sum
  import 'package:ml_linalg/linalg.dart';

  final vector1 = Float32x4Vector.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4Vector.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 = Float32x4Vector.from([4.0, 5.0, 6.0, 7.0, 8.0]);
  final vector2 = Float32x4Vector.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 = Float32x4Vector.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4Vector.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 = Float32x4Vector.from([6.0, 12.0, 24.0, 48.0, 96.0]);
  final vector2 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.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 = Float32x4Vector.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4Vector.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 = Float32x4Vector.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Float32x4Vector.from([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.distanceTo(vector2, Norm.manhattan);
  print(result); // 5.0

Vectorized non-mathematical methods

It is also needed to operate with vectors in non-mathematical way, for instance, to create a new vector applying some map function to an existing one. To do it effectively, one can use the following methods:

Vectorized map
  import 'package:ml_linalg/linalg.dart';

  final vector = Float32x4Vector.from([1.0, 2.0, 3.0, 4.0, 5.0]);
  final result = vector.vectorizedMap((Float32x4 element, int offsetStart, int offsetEnd) {
    // offsetStart - start index for the current vectorized element, e.g. if `element` is second in the inner collection,
    // the offsetStart will be 4 (because Float32x4 contains 4 elements)
    // offsetEnd - end index for the current vectorized element, e.g. if `element` is second in the inner collection,
    // the offsetStart will be 7
    return element.scale(2.0);
  });
  print(result); // [2.0, 4.0, 6.0, 8.0, 10.0]

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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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 = Float32x4Matrix.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],
  //];

Vectorized non-mathematical matrix methods #

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 = Float32x4Matrix.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]
*/

Contacts #

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SIMD-based linear algebra with dart for machine learning purposes

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