Merge potential next release v0.4 (#187) Breaking Changes
* First draft of the new n-dimensional arrays + NB use case * Improves default implementation of multiple Array methods * Refactors tree methods * Adds matrix decomposition routines * Adds matrix decomposition methods to ndarray and nalgebra bindings * Refactoring + linear regression now uses array2 * Ridge & Linear regression * LBFGS optimizer & logistic regression * LBFGS optimizer & logistic regression * Changes linear methods, metrics and model selection methods to new n-dimensional arrays * Switches KNN and clustering algorithms to new n-d array layer * Refactors distance metrics * Optimizes knn and clustering methods * Refactors metrics module * Switches decomposition methods to n-dimensional arrays * Linalg refactoring - cleanup rng merge (#172) * Remove legacy DenseMatrix and BaseMatrix implementation. Port the new Number, FloatNumber and Array implementation into module structure. * Exclude AUC metrics. Needs reimplementation * Improve developers walkthrough New traits system in place at `src/numbers` and `src/linalg` Co-authored-by: Lorenzo <tunedconsulting@gmail.com> * Provide SupervisedEstimator with a constructor to avoid explicit dynamical box allocation in 'cross_validate' and 'cross_validate_predict' as required by the use of 'dyn' as per Rust 2021 * Implement getters to use as_ref() in src/neighbors * Implement getters to use as_ref() in src/naive_bayes * Implement getters to use as_ref() in src/linear * Add Clone to src/naive_bayes * Change signature for cross_validate and other model_selection functions to abide to use of dyn in Rust 2021 * Implement ndarray-bindings. Remove FloatNumber from implementations * Drop nalgebra-bindings support (as decided in conf-call to go for ndarray) * Remove benches. Benches will have their own repo at smartcore-benches * Implement SVC * Implement SVC serialization. Move search parameters in dedicated module * Implement SVR. Definitely too slow * Fix compilation issues for wasm (#202) Co-authored-by: Luis Moreno <morenol@users.noreply.github.com> * Fix tests (#203) * Port linalg/traits/stats.rs * Improve methods naming * Improve Display for DenseMatrix Co-authored-by: Montana Low <montanalow@users.noreply.github.com> Co-authored-by: VolodymyrOrlov <volodymyr.orlov@gmail.com>
This commit is contained in:
Lorenzo
@@ -0,0 +1,206 @@
|
||||
//! # Cholesky Decomposition
|
||||
//!
|
||||
//! every positive definite matrix \\(A \in R^{n \times n}\\) can be factored as
|
||||
//!
|
||||
//! \\[A = R^TR\\]
|
||||
//!
|
||||
//! where \\(R\\) is upper triangular matrix with positive diagonal elements
|
||||
//!
|
||||
//! Example:
|
||||
//! ```
|
||||
//! use smartcore::linalg::basic::matrix::DenseMatrix;
|
||||
//! use smartcore::linalg::traits::cholesky::*;
|
||||
//!
|
||||
//! let A = DenseMatrix::from_2d_array(&[
|
||||
//! &[25., 15., -5.],
|
||||
//! &[15., 18., 0.],
|
||||
//! &[-5., 0., 11.]
|
||||
//! ]);
|
||||
//!
|
||||
//! let cholesky = A.cholesky().unwrap();
|
||||
//! let lower_triangular: DenseMatrix<f64> = cholesky.L();
|
||||
//! let upper_triangular: DenseMatrix<f64> = cholesky.U();
|
||||
//! ```
|
||||
//!
|
||||
//! ## References:
|
||||
//! * ["No bullshit guide to linear algebra", Ivan Savov, 2016, 7.6 Matrix decompositions](https://minireference.com/)
|
||||
//! * ["Numerical Recipes: The Art of Scientific Computing", Press W.H., Teukolsky S.A., Vetterling W.T, Flannery B.P, 3rd ed., 2.9 Cholesky Decomposition](http://numerical.recipes/)
|
||||
//!
|
||||
//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
||||
//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||||
#![allow(non_snake_case)]
|
||||
|
||||
use std::fmt::Debug;
|
||||
use std::marker::PhantomData;
|
||||
|
||||
use crate::error::{Failed, FailedError};
|
||||
use crate::linalg::basic::arrays::Array2;
|
||||
use crate::numbers::basenum::Number;
|
||||
use crate::numbers::realnum::RealNumber;
|
||||
|
||||
#[derive(Debug, Clone)]
|
||||
/// Results of Cholesky decomposition.
|
||||
pub struct Cholesky<T: Number + RealNumber, M: Array2<T>> {
|
||||
R: M,
|
||||
t: PhantomData<T>,
|
||||
}
|
||||
|
||||
impl<T: Number + RealNumber, M: Array2<T>> Cholesky<T, M> {
|
||||
pub(crate) fn new(R: M) -> Cholesky<T, M> {
|
||||
Cholesky { R, t: PhantomData }
|
||||
}
|
||||
|
||||
/// Get lower triangular matrix.
|
||||
pub fn L(&self) -> M {
|
||||
let (n, _) = self.R.shape();
|
||||
let mut R = M::zeros(n, n);
|
||||
|
||||
for i in 0..n {
|
||||
for j in 0..n {
|
||||
if j <= i {
|
||||
R.set((i, j), *self.R.get((i, j)));
|
||||
}
|
||||
}
|
||||
}
|
||||
R
|
||||
}
|
||||
|
||||
/// Get upper triangular matrix.
|
||||
pub fn U(&self) -> M {
|
||||
let (n, _) = self.R.shape();
|
||||
let mut R = M::zeros(n, n);
|
||||
|
||||
for i in 0..n {
|
||||
for j in 0..n {
|
||||
if j <= i {
|
||||
R.set((j, i), *self.R.get((i, j)));
|
||||
}
|
||||
}
|
||||
}
|
||||
R
|
||||
}
|
||||
|
||||
/// Solves Ax = b
|
||||
pub(crate) fn solve(&self, mut b: M) -> Result<M, Failed> {
|
||||
let (bn, m) = b.shape();
|
||||
let (rn, _) = self.R.shape();
|
||||
|
||||
if bn != rn {
|
||||
return Err(Failed::because(
|
||||
FailedError::SolutionFailed,
|
||||
"Can\'t solve Ax = b for x. FloatNumber of rows in b != number of rows in R.",
|
||||
));
|
||||
}
|
||||
|
||||
for k in 0..bn {
|
||||
for j in 0..m {
|
||||
for i in 0..k {
|
||||
b.sub_element_mut((k, j), *b.get((i, j)) * *self.R.get((k, i)));
|
||||
}
|
||||
b.div_element_mut((k, j), *self.R.get((k, k)));
|
||||
}
|
||||
}
|
||||
|
||||
for k in (0..bn).rev() {
|
||||
for j in 0..m {
|
||||
for i in k + 1..bn {
|
||||
b.sub_element_mut((k, j), *b.get((i, j)) * *self.R.get((i, k)));
|
||||
}
|
||||
b.div_element_mut((k, j), *self.R.get((k, k)));
|
||||
}
|
||||
}
|
||||
Ok(b)
|
||||
}
|
||||
}
|
||||
|
||||
/// Trait that implements Cholesky decomposition routine for any matrix.
|
||||
pub trait CholeskyDecomposable<T: Number + RealNumber>: Array2<T> {
|
||||
/// Compute the Cholesky decomposition of a matrix.
|
||||
fn cholesky(&self) -> Result<Cholesky<T, Self>, Failed> {
|
||||
self.clone().cholesky_mut()
|
||||
}
|
||||
|
||||
/// Compute the Cholesky decomposition of a matrix. The input matrix
|
||||
/// will be used for factorization.
|
||||
fn cholesky_mut(mut self) -> Result<Cholesky<T, Self>, Failed> {
|
||||
let (m, n) = self.shape();
|
||||
|
||||
if m != n {
|
||||
return Err(Failed::because(
|
||||
FailedError::DecompositionFailed,
|
||||
"Can\'t do Cholesky decomposition on a non-square matrix",
|
||||
));
|
||||
}
|
||||
|
||||
for j in 0..n {
|
||||
let mut d = T::zero();
|
||||
for k in 0..j {
|
||||
let mut s = T::zero();
|
||||
for i in 0..k {
|
||||
s += *self.get((k, i)) * *self.get((j, i));
|
||||
}
|
||||
s = (*self.get((j, k)) - s) / *self.get((k, k));
|
||||
self.set((j, k), s);
|
||||
d += s * s;
|
||||
}
|
||||
d = *self.get((j, j)) - d;
|
||||
|
||||
if d < T::zero() {
|
||||
return Err(Failed::because(
|
||||
FailedError::DecompositionFailed,
|
||||
"The matrix is not positive definite.",
|
||||
));
|
||||
}
|
||||
|
||||
self.set((j, j), d.sqrt());
|
||||
}
|
||||
|
||||
Ok(Cholesky::new(self))
|
||||
}
|
||||
|
||||
/// Solves Ax = b
|
||||
fn cholesky_solve_mut(self, b: Self) -> Result<Self, Failed> {
|
||||
self.cholesky_mut().and_then(|qr| qr.solve(b))
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
use crate::linalg::basic::matrix::DenseMatrix;
|
||||
use approx::relative_eq;
|
||||
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
||||
#[test]
|
||||
fn cholesky_decompose() {
|
||||
let a = DenseMatrix::from_2d_array(&[&[25., 15., -5.], &[15., 18., 0.], &[-5., 0., 11.]]);
|
||||
let l =
|
||||
DenseMatrix::from_2d_array(&[&[5.0, 0.0, 0.0], &[3.0, 3.0, 0.0], &[-1.0, 1.0, 3.0]]);
|
||||
let u =
|
||||
DenseMatrix::from_2d_array(&[&[5.0, 3.0, -1.0], &[0.0, 3.0, 1.0], &[0.0, 0.0, 3.0]]);
|
||||
let cholesky = a.cholesky().unwrap();
|
||||
|
||||
assert!(relative_eq!(cholesky.L().abs(), l.abs(), epsilon = 1e-4));
|
||||
assert!(relative_eq!(cholesky.U().abs(), u.abs(), epsilon = 1e-4));
|
||||
assert!(relative_eq!(
|
||||
cholesky.L().matmul(&cholesky.U()).abs(),
|
||||
a.abs(),
|
||||
epsilon = 1e-4
|
||||
));
|
||||
}
|
||||
|
||||
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
||||
#[test]
|
||||
fn cholesky_solve_mut() {
|
||||
let a = DenseMatrix::from_2d_array(&[&[25., 15., -5.], &[15., 18., 0.], &[-5., 0., 11.]]);
|
||||
let b = DenseMatrix::from_2d_array(&[&[40., 51., 28.]]);
|
||||
let expected = DenseMatrix::from_2d_array(&[&[1.0, 2.0, 3.0]]);
|
||||
|
||||
let cholesky = a.cholesky().unwrap();
|
||||
|
||||
assert!(relative_eq!(
|
||||
cholesky.solve(b.transpose()).unwrap().transpose(),
|
||||
expected,
|
||||
epsilon = 1e-4
|
||||
));
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user