ml_linalg 2.0.0 ml_linalg: ^2.0.0 copied to clipboard
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 example
- Vectors sum
- Vector 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
- Vector operations example
- A couple of words about the underlying vector architecture
- 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
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]
Vector 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:linalg/ml_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:linalg/ml_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 = 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 = 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 = 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],
//];