varpro

Nonlinear function fitting made simple. This library provides robust and fast least-squares fitting of a wide class of model functions to data. It uses the VarPro algorithm to achieve this, hence the name.

Introduction

This crate implements a powerful algorithm to fit model functions to data, but it is restricted to so called separable models. The lack of formulas on this site makes it hard to go into detail, but a brief overview is provided in the next sections. Refer to the documentation for all the meaty details including the math.

What are Separable Models?

Put simply, separable models are nonlinear functions which can be written as a linear combination of some nonlinear basis functions. A common use case for VarPro is e.g. fitting sums of exponentials, which is a notoriously ill-conditioned problem.

What is VarPro?

Variable Projection (VarPro) is an algorithm that exploits that its fitting problem can be separated into linear and nonlinear parameters. First, the linear parameters are eliminated using some clever linear algebra. Then, the fitting problem is rewritten so that it depends only on the nonlinear parameters. Finally, this reduced problem is solved by using a general purpose nonlinear minimization algorithm, such as Levenberg-Marquardt (LM).

When Should You Give it a Try?

VarPro can dramatically increase the robustness and speed of the fitting process compared to using a general purpose nonlinear least squares fitting algorithm. When

the model function you want to fit is a linear combination of nonlinear functions,
and you know the analytical derivatives of all those functions

then you should give it a whirl. Also consider the section on global fitting below, which provides another great use case for this crate.

Example Usage

The following example shows, how to use this crate to fit a double exponential decay with constant offset to a data vector y obtained at time points t. Refer to the documentation for a more in-depth guide.

use varpro::prelude::*;
use varpro::solvers::levmar::{LevMarProblemBuilder, LevMarSolver};
use nalgebra::{dvector,DVector};

// Define the exponential decay e^(-t/tau).
// Both of the nonlinear basis functions in this example
// are exponential decays.
fn exp_decay(t :&DVector<f64>, tau : f64) 
  -> DVector<f64> {
  t.map(|t|(-t/tau).exp())
}

// the partial derivative of the exponential
// decay with respect to the nonlinear parameter tau.
// d/dtau e^(-t/tau) = e^(-t/tau)*t/tau^2
fn exp_decay_dtau(t: &DVector<f64>,tau: f64) 
  -> DVector<f64> {
  t.map(|t| (-t / tau)
    .exp() * t / tau.powi(2))
}

// temporal (or spatial) coordintates of the observations
let t = dvector![0.,1.,2.,3.,4.,5.,6.,7.,8.,9.,10.];
// the observations we want to fit
let y = dvector![6.0,4.8,4.0,3.3,2.8,2.5,2.2,1.9,1.7,1.6,1.5];

// 1. create the model by giving only the nonlinear parameter names it depends on
let model = SeparableModelBuilder::<f64>::new(&["tau1", "tau2"])
  // provide the nonlinear basis functions and their derivatives.
  // In general, base functions can depend on more than just one parameter.
  // first function:
  .function(&["tau1"], exp_decay)
  .partial_deriv("tau1", exp_decay_dtau)
  // second function and derivatives with respect to all parameters
  // that it depends on (just one in this case)
  .function(&["tau2"], exp_decay)
  .partial_deriv("tau2", exp_decay_dtau)
  // a constant offset is added as an invariant basefunction
  // as a vector of ones. It is multiplied with its own linear coefficient,
  // creating a fittable offset
  .invariant_function(|v|DVector::from_element(v.len(),1.))
  // give the coordinates of the problem
  .independent_variable(t)
  // provide guesses only for the nonlinear parameters in the
  // order that they were given on construction.
  .initial_parameters(vec![2.5,5.5])
  .build()
  .unwrap();
// 2. Cast the fitting problem as a nonlinear least squares minimization problem
let problem = LevMarProblemBuilder::new(model)
  .observations(y)
  .build()
  .unwrap();
// 3. Solve the fitting problem
let fit_result = LevMarSolver::default()
    .fit(problem)
    .expect("fit must exit successfully");
// 4. obtain the nonlinear parameters after fitting
let alpha = fit_result.nonlinear_parameters();
// 5. obtain the linear parameters
let c = fit_result.linear_coefficients().unwrap();

For more in depth examples, please refer to the crate documentation.

Fit Statistics

Additionally to the fit member function, the LevMarSolver provides a fit_with_statistics function that calculates quite a bit of useful additional statistical information.

Global Fitting of Multiple Right Hand Sides

In the example above, we have passed a single column vector as the observations. The library also allows fitting multiple right hand sides, by constructing a problem via LevMarProblemBuilder::mrhs. When fitting multiple right hand sides, vapro will performa a global fit, in which the nonlinear parameters are optimized across all right hand sides, but linear coefficients of the fit are optimized for each right hand side individually.

This is another application where varpro really shines, since it can take advantage of the separable nature of the problem. It allows us to perform a global fit over thousands, or even tens of thousands of right hand sides in reasonable time (fractions of seconds to minutes), where conventional purely nonlinear solvers must perform much more work.

Maximum Performance and Advanced Use Cases

The example code above will already run many times faster than just using a nonlinear solver without the magic of varpro. But this crate offers an additional way to eek out the last bits of performance.

The SeparableNonlinearModel trait can be manually implemented to describe a model function. This often allows us to shave of the last hundreds of microseconds from the computation, e.g. by caching intermediate calculations. The crate documentation contains detailed examples.

This is not only useful for performance, but also for use cases that are difficult or impossible to accomodate using only the SeparableModelBuilder. The builder was created for ease of use and performance, but it has some limitations by design.

Acknowledgements

I am grateful to Professor Dianne P. O'Leary and Bert W. Rust ✝ who published the paper that enabled me to understand varpro and come up with this implementation. Professor O'Leary also graciously answered my questions on her paper and some implementation details.

References and Further Reading

(O'Leary2013) O’Leary, D.P., Rust, B.W. Variable projection for nonlinear least squares problems. Comput Optim Appl 54, 579–593 (2013). DOI: 10.1007/s10589-012-9492-9

attention: the O'Leary paper contains errors that are fixed (so I hope) in this blog article of mine.

(Golub2003) Golub, G. , Pereyra, V Separable nonlinear least squares: the variable projection method and its applications. Inverse Problems 19 R1 (2003) https://iopscience.iop.org/article/10.1088/0266-5611/19/2/201

geo-ant / varpro