PySTOP

Introduction

The STOP toolbox is designed for optimization problems on the Stiefel manifold, which could be expressed as $$ \begin{aligned} \min_{X \in \mathbb{R}^{n\times p}} ~ &f(X)\ \text{s. t.}~& X^\top X = I_p, \end{aligned} $$ where $I_p$ refers to the $p$-th order identity matrix, $X$ is a matrix with $n$ rows and $p$ columns. The feasible set of this optimization problem $$ \mathcal{S}_{n,p} := \left{X \in \mathbb{R}^{n\times p}: X^\top X = I_p \right}, $$ can be regarded as a Riemannian manifold in $\mathbb{R}^{n\times p}$, and we also call it as Stiefel manifold.

This document describes the python version of the STOP package (PySTOP). Currently, PySTOP only involves the SLPG algorithm, which could handle smooth, $\ell_1$-norm regularized and $\ell_{2,1}$-nom regularized optimization problems on the Stiefel manifold. The package is

Installation

The source code of PySTOP package can be found from the website. Besides, it supports direct installation from pip:

pip install pystop

Example

Problem formulation

In this section, we consider the following nonlinear eigenvalue problem $$ \min_{X \in \mathcal{S}_{n, p}} ~ \frac{1}{2}\mathrm{tr}(X^\top L X) + \frac{\alpha}{4} \rho^\top L^{\dagger} \rho, $$ where $\rho = \mathrm{Diag}(XX^\top)$, and $L^{\dagger}$ denotes the pseudo-inverse of the positive definite matrix $L$, i.e. $L^{\dagger}LL^{\dagger} = L^{\dagger}$, $LL^{\dagger}L = L$. Here we uses $\mathrm{Diag}(M)$ to denote the vector that is composed of diagonal entries of the square matrix $M$, while $\mathrm{diag}(v)$ refers to a diagonal matrix with $v$ to be its diagonal entries. Then the cost function and its Euclidean gradient can be expressed as $$ \begin{aligned} f(X) ={}& \frac{1}{2}\mathrm{tr}(X^\top L X) + \frac{\alpha}{4} \rho^\top L^{\dagger} \rho,\ \nabla f(X) ={}& LX + \alpha \mathrm{diag}(L^{\dagger}\rho)X. \end{aligned} $$

In this example, we choose $L$ as a tri-diagonal matrix generated by L = gallery('tridiag',n,-1,2,-1). Noting that $L$ is full-rank, then we can conclude that $L^{\dagger} = L^{-1}$ in this case. We solve this simple optimization problem using solvers in STOP to illustrate the most basic usage of the STOP toolbox. For additional theory, readers are recommended to refer the papers in the about page.

# Import packages 
import numpy as np
import scipy as sp
from scipy.sparse import diags
from scipy.sparse.linalg import spsolve

# Import manifolds and solvers
from pystop.manifold import Stiefel
from pystop.solver import SLPG_smooth


# Set parameters
n = 1000
p = 10
alpha = 1
M = Stiefel(n,p)

# Defining objective function
L = diags(np.array([-1, 2, -1]), np.array([1, 0, -1]), shape = (n,n)).tocsc()
def obj_fun(X):
    LX = L@X
    rho = np.sum(X * X, 1)
    Lrho = spsolve(L, rho)
    fval = 0.5*np.sum(X* LX) + (alpha /4) * np.sum(rho * Lrho)
    grad = LX + alpha * Lrho[: ,np.newaxis] * X
    return  fval, grad

# Execute the solver
X, out_dict = SLPG_smooth(obj_fun, M)

Let us review the code step by step. First, we specify the dimension of the problem and specify the Stiefel manifold. In pySTOP package, we need to specify the dimension of the Stiefel manifold before executing the solver. The Stiefel manifold should be specified as the STOP manifold class, for example,

# Set parameters
n = 1000
p = 10
alpha = 1
# Specify the Stiefel manifold
M = Stiefel(n,p)

Here pystop.manifold.stiefel is a build-in function to specify the Stiefel manifold and hence provides essential tools for the algorithm.

Then we generate the data (matrix $L$) for the optimization problem by the following code,

L = diags(np.array([-1, 2, -1]), np.array([1, 0, -1]), shape = (n,n)).tocsc()

Here we utilize SciPy.sparse to create a sparse representation of $L$ . Therefore, in each step we could use the scipy.sparse.linalg.spsolve function to compute .

Then we specify the cost function and its gradient in the following function

# Defin objective function
def obj_fun(X):
    LX = L@X
    rho = np.sum(X * X, 1)
    Lrho = spsolve(L, rho)
    fval = 0.5*np.sum(X* LX) + (alpha /4) * np.sum(rho * Lrho)
    grad = LX + alpha * Lrho[: ,np.newaxis] * X
    return fval, grad

Currently, in STOP toolbox, we require the function return the function value and its gradient simultaneously. Usually, computing the function value and gradient simultaneously is much faster than compute them separately, even when cache techniques are involved. To achieve a better performance, we strongly suggest to compute the function value and gradient in a single function.

Then we call a solver to solve the nonlinear eigenvalue problem,

# Execute the solver
X, out_dict = SLPG_smooth(obj_fun, M)

Solvers

The PySTOP solver classes provide the solvers for optimization. Once we specify the Stiefel manifold and define the objective function, the PySTOP solver can be executed by

X, out_dict = name_of_solver(obj_fun, M)

Here X is the final output of the problem, and out_dict is a dictionary that contains the log information.

It is worth mentioning that SLPG solver supports customized nonsmooth regularization terms, interested reader could refer to the description page for details.

Defining the objective function

Usually, computing the function value and gradient simultaneously is much faster than compute them separately, even when cache techniques are involved. Therefore, to achieve a better numerical performance, the existing solvers in PySTOP package requires an integrated call for function value and gradient of the objective function, i.e.

# Defin objective function
def obj_fun(X):
    '''
    	fval refers to the function value of f
    	grad refers to the gradient of f
    '''
    return fval, grad

Currently, PySTOP package does not involve build-in autodiff packages. However, we provides several useful functions in pystop.utility to help run the solvers only with specified objective function $f(X)$.

If you already know how to use NumPy, then it could be easy to use autograd package to generate the function value and gradient simultaneously. Just import autograd.numpy and setup the objective function by the build-in function provided in autograd.numpy to perform the computation. Once the function value is specified as obj(), we could apply the pystop.utility.fun_autodiff to generate the function that returns fval and grad simultaneously.

However, it is worth mentioning that autograd.numpy package only supports a subset of the standard NumPy package. Besides, the Autograd package does not support SciPy package. To achieve better numerical performance, we suggest the users to specify the obj_fun function manually.

import autograd.numpy as anp
import numpy as np

n = 1000
p = 30
Z = np.random.randn(n, p)

def obj_fun(X):
    return  anp.sum((X-Z) **2 )


from pystop.utility import fun_autodiff

obj_grad, obj_fungrad = fun_autodiff(obj_fun)


from pystop.manifold import Stiefel
from pystop.solver import SLPG_smooth

M = Stiefel(n,p)
X, out_dict = SLPG_smooth(obj_fungrad, M)