Quick start

This example factors a nonnegative matrix \(A\in\mathbf{R}^{m\times n}\) as \(XY\), where \(X\in\mathbf{R}^{m\times k}\) and \(Y\in\mathbf{R}^{k\times n}\):

\[\begin{split} \begin{array}{ll} \text{minimize} & \|XY-A\|_F^2 \\ \text{subject to} & X_{ij} \geq 0,\quad i=1,\ldots,m,\quad j=1,\ldots,k\\ & Y_{ij} \geq 0,\quad i=1,\ldots,k,\quad j=1,\ldots,n. \end{array} \end{split}\]

The objective is convex in \(X\) for fixed \(Y\) and convex in \(Y\) for fixed \(X\).

import cvxpy as cp
import numpy as np

import dbcp

rng = np.random.default_rng(10015)
m, n, k = 5, 10, 3
A = rng.random((m, k)) @ rng.random((k, n))

X = cp.Variable((m, k), name="X")
Y = cp.Variable((k, n), name="Y")
X.value = rng.random(X.shape)
Y.value = rng.random(Y.shape)

problem = dbcp.BiconvexProblem(
    cp.Minimize(cp.sum_squares(X @ Y - A)),
    [X],
    [Y],
    [X >= 0, Y >= 0],
)

assert problem.is_dbcp()
value = problem.solve()

print(problem.status)
print("objective =", value)
print("reconstruction error =", np.linalg.norm(X.value @ Y.value - A, "fro") ** 2)

The [X] and [Y] arguments supply the x_var and y_var variable groups, while the last argument encodes the nonnegativity constraints. During the x-subproblem, DBCP optimizes X while holding Y fixed; during the y-subproblem it does the reverse. The original CVXPY variables receive the final numerical values.

Because unset variables are initialized randomly, different starting points can produce different factorizations. Assign X.value and Y.value before solve() when a specific warm start is desired.