Add mu-trajectory plot for the projection onto an ellipse
Cleaned-up extraction of test_mu_ellipse from proj_ellipsoid: - quadric.py: minimal Ellipsoid class (standardization, KKT trajectory, extreme roots by exact bracketing, degenerate points, asymptotes). - plot_mu_trajectories.py: the 4-panel figure, typeset with LaTeX. - test_quadric.py: checks against brute-force sampling of the ellipse. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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__pycache__/
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.pytest_cache/
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figures/
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# mu-trajectories
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Visualize the KKT trajectory of the Euclidean projection of a point `x0` onto an ellipse
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`x'Ax + b'x + c = 0`:
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x(mu) = (I + mu A)^{-1} (x0 - mu b / 2)
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The figure shows the trajectory in the plane, `f(mu) = psi(x(mu))`, the distance
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`||x0 - x(mu)||` and `f'(mu)`, together with the nearest/farthest KKT points `x(mu*)`,
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`x(mu**)` and, when `x0` has a zero standardized coordinate, the degenerate point `x^d`.
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## Usage
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pip install -r requirements.txt
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python plot_mu_trajectories.py # degenerate case x0 = (0, -0.1)
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python plot_mu_trajectories.py --x0 0.3 0.2 --output figures/generic
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pytest
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The ellipse is set by `A`, `B`, `C` at the top of `plot_mu_trajectories.py`.
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Text is typeset with LaTeX (`text.usetex`) when `latex` is on the `PATH`, with the `cm-super` and
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`type1cm` packages; otherwise matplotlib's Computer Modern math font is used.
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## Files
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- `quadric.py`: the `Ellipsoid` class (standardization, trajectory, extreme roots, degenerate points).
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- `plot_mu_trajectories.py`: the figure.
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- `test_quadric.py`: checks against brute-force sampling of the ellipse.
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#!/usr/bin/env python3
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"""Plot the KKT trajectory mu -> x(mu) of the projection of x0 onto an ellipse.
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Panels: the trajectory in the plane, f(mu) = psi(x(mu)), the distance ||x0 - x(mu)|| and
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f'(mu). Colors encode mu, from mu -> -inf (blue) to mu -> +inf (red).
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"""
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import argparse
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import shutil
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from pathlib import Path
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import matplotlib.pyplot as plt
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import numpy as np
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from matplotlib import colormaps
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from matplotlib.cm import ScalarMappable
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from matplotlib.collections import LineCollection
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from matplotlib.colors import Normalize
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from matplotlib.lines import Line2D
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from quadric import TOL, Ellipsoid
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A = np.array([[2.0, 0.0], [0.0, 1.0]])
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B = np.zeros(2)
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C = -1.0
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N_PER_BRANCH = 266 # mu samples between consecutive poles
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EPS = 0.01 # gap left on each side of a pole
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MU_HALF_WIDTH = 3 # mu is sampled in [m - MU_HALF_WIDTH, m + MU_HALF_WIDTH], m the mean of the poles
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N_DOTTED_OUTER = 2 # points drawn dotted at the ends of the outer branches
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COLORMAP = 'RdYlBu_r'
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ELLIPSE_COLOR = 'royalblue'
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STAR_STYLE = dict(color='purple', marker='v', s=12, zorder=2)
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STAR_STAR_STYLE = dict(color='purple', marker='s', s=12, zorder=2)
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POLE_STYLE = dict(linestyle='dashed', color='grey')
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TEXT_STYLE = dict(ha='center', va='center', fontsize=12, clip_on=True)
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POLE_LABELS = [r'$\frac{-1}{\lambda_2}$', r'$\frac{-1}{\lambda_1}$']
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DIST_D_LABEL = r'$||\mathbf{x}^0 - \mathbf{x}^\mathrm{d}||_2$'
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def colored_line(ax, x, y, idx, cmap, norm, n_dotted=0):
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"""Draw y(x) colored by idx, with the first and last n_dotted points joined by dotted segments."""
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points = np.column_stack((x, y))
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n = len(points)
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if n_dotted == 0:
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parts = [(0, n, 'solid')]
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else:
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parts = [(0, n_dotted, 'dotted'),
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(n_dotted - 1, n - n_dotted + 1, 'solid'),
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(n - n_dotted, n, 'dotted')]
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for start, stop, linestyle in parts:
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p = points[start:stop]
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lc = LineCollection(np.stack((p[:-1], p[1:]), axis=1), cmap=cmap, norm=norm, linestyle=linestyle)
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lc.set_array(idx[start:stop - 1])
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ax.add_collection(lc)
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def plot_trajectories(Q, x0):
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assert Q.dim == 2, 'only 2D ellipses can be plotted'
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active = np.abs(Q.to_standardized(x0)) > TOL
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degenerate = not active.all()
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# Q.eig is positive and decreasing, so reversing it sorts the poles in increasing order.
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poles = -1 / Q.eig[::-1]
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pole_active = active[::-1]
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mu_min = poles.mean() - MU_HALF_WIDTH
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mu_max = poles.mean() + MU_HALF_WIDTH
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mu = np.concatenate([np.linspace(lo, hi, N_PER_BRANCH) for lo, hi in
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[(mu_min, poles[0] - EPS), (poles[0] + EPS, poles[1] - EPS), (poles[1] + EPS, mu_max)]])
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idx = np.arange(len(mu))
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# A pole where x0 has a zero standardized component vanishes and joins its neighbouring branches.
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branches = np.split(idx, [k * N_PER_BRANCH for k in (1, 2) if pole_active[k - 1]])
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cmap = colormaps[COLORMAP]
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norm = Normalize(0, len(mu) - 1)
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xlim = (mu_min, mu_max)
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ticks = [mu_min, poles[0], poles[1], 0, mu_max]
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tick_labels = [r'$\mu \to -\infty$', *POLE_LABELS, r'$0$', r'$\mu \to \infty$']
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roots = Q.extreme_roots(x0)
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markers = [] if roots is None else list(zip(roots, (STAR_STYLE, STAR_STAR_STYLE)))
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fig, axs = plt.subplots(2, 2, figsize=(10, 7))
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ax_x, ax_f, ax_dist, ax_df = axs.flat
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panels = [
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(ax_f, lambda m: Q.f(m, x0), r'$f(\mu)$', (-1, 1), 5),
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(ax_dist, lambda m: np.linalg.norm(Q.x(m, x0) - x0), r'$||\mathbf{x}^0 - \mathbf{x}(\mu)||_2$', (0, 2), 6),
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(ax_df, lambda m: Q.df(m, x0), r'$f^\prime(\mu)$', (-2, 2), 50),
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]
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for ax, fun, ylabel, ylim, n_dotted_middle in panels:
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values = np.array([fun(m) for m in mu])
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for i, branch in enumerate(branches):
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n_dotted = 0 if degenerate else (n_dotted_middle if i == 1 else N_DOTTED_OUTER)
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colored_line(ax, mu[branch], values[branch], branch, cmap, norm, n_dotted)
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for pole in poles[pole_active]:
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ax.axvline(pole, **POLE_STYLE)
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for root, style in markers:
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ax.scatter(root, fun(root), **style)
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ax.set(xlabel=r'$\mu$', ylabel=ylabel, xlim=xlim, ylim=ylim)
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ax.set_xticks(ticks, tick_labels)
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ax_f.axhline(0, color='red', alpha=0.5, zorder=1)
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for ax in (ax_f, ax_df):
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ax.yaxis.set_ticks_position('right')
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xs = np.array([Q.x(m, x0) for m in mu])
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ax_x.plot(*Q.boundary().T, color=ELLIPSE_COLOR, zorder=1)
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ax_x.scatter(*Q.d, color=ELLIPSE_COLOR)
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ax_x.scatter(xs[:, 0], xs[:, 1], c=idx, cmap=cmap, norm=norm, s=4)
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ax_x.scatter(*x0, color='red', marker='x')
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for root, style in markers:
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ax_x.scatter(*Q.x(root, x0), **style)
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if not degenerate:
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for k, label in zip((1, 2), POLE_LABELS):
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for i in (k * N_PER_BRANCH - 1, k * N_PER_BRANCH):
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ax_x.text(*xs[i], label, **TEXT_STYLE)
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for k in range(Q.dim):
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point, direction = Q.pole_line(x0, k)
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ax_x.axline(point, point + direction, linestyle='dotted', color='grey')
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for k in np.flatnonzero(~active):
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point, direction = Q.pole_line(x0, k)
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xd = Q.degenerate_point(x0, k)
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if xd is None:
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ax_x.plot(*np.column_stack((point - direction, point + direction)), color='grey', zorder=0)
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continue
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ax_x.plot(*np.column_stack((point, xd)), color='grey', zorder=0)
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ax_x.scatter(*xd, color='green', zorder=2)
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dist = np.linalg.norm(xd - x0)
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ax_dist.axhline(dist + 0.1, color='green')
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ax_dist.text(mu_max + (poles[1] - mu_max) * 2 / 3, dist + 0.2, DIST_D_LABEL, color='green')
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legend = [
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Line2D([], [], color=ELLIPSE_COLOR, lw=2, label=r'$\mathcal{Q}(\Psi)$'),
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Line2D([], [], linestyle='none', marker='v', color='purple', markersize=6, label=r'$\mathbf{x}(\mu^*)$'),
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Line2D([], [], linestyle='none', marker='s', color='purple', markersize=6, label=r'$\mathbf{x}(\mu^{**})$'),
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Line2D([], [], linestyle='none', marker='x', color='red', markersize=8, label=r'$\mathbf{x}^0 = \mathbf{x}(0)$'),
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]
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if degenerate:
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legend.append(Line2D([], [], linestyle='none', marker='o', color='green', markersize=8,
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label=r'$\mathbf{x}^\mathrm{d}$'))
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ax_x.legend(handles=legend, loc='lower left', fontsize=9)
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ax_x.set(xlabel=r'$x_1$', ylabel=r'$x_2$', xlim=(-2, 2), ylim=(-2, 2), aspect='equal')
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fig.tight_layout(pad=2.1)
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colorbar = fig.colorbar(ScalarMappable(norm, cmap), ax=ax_x, ticks=[0, len(mu) - 1])
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colorbar.ax.set_yticklabels([r'$\mu \to -\infty$', r'$\mu \to \infty$'])
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return fig
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def use_latex_fonts():
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"""Typeset text with LaTeX when available, otherwise with matplotlib's Computer Modern."""
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plt.rcParams.update({'font.family': 'serif', 'font.size': 12})
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if shutil.which('latex'):
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plt.rcParams.update({'text.usetex': True, 'text.latex.preamble': r'\usepackage{amsmath}'})
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else:
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plt.rcParams.update({'mathtext.fontset': 'cm', 'font.serif': ['cmr10'],
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'axes.formatter.use_mathtext': True})
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def main():
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use_latex_fonts()
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parser = argparse.ArgumentParser(description=__doc__.splitlines()[0])
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parser.add_argument('--x0', type=float, nargs=2, default=[0.0, -0.1], metavar=('X1', 'X2'))
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parser.add_argument('--output', type=Path, default=Path('figures/x_trajectories_mu'),
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help='output path, without extension (a .png and a .pdf are written)')
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parser.add_argument('--show', action='store_true')
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args = parser.parse_args()
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fig = plot_trajectories(Ellipsoid(A, B, C), np.array(args.x0))
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args.output.parent.mkdir(parents=True, exist_ok=True)
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for extension in ('.png', '.pdf'):
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fig.savefig(args.output.with_name(args.output.name + extension))
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if args.show:
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plt.show()
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if __name__ == '__main__':
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main()
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"""Ellipsoid x'Ax + b'x + c = 0 and the KKT trajectory of the projection onto it.
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The KKT conditions of min ||x - x0|| subject to x on the ellipsoid give
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x(mu) = (I + mu A)^{-1} (x0 - mu b / 2),
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and the projections are the points x(mu) with f(mu) = psi(x(mu)) = 0.
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With d = -A^{-1} b / 2, gamma = c + b'd + d'Ad < 0 and A = V diag(eig) V', the standardized
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coordinates u = V'(x - d) / sqrt(-gamma) turn the ellipsoid into sum_i eig_i u_i^2 = 1 and the
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trajectory into u_i(mu) = u0_i / (1 + mu eig_i), which has a pole at mu = -1/eig_i when u0_i != 0.
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"""
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import numpy as np
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from scipy.optimize import brentq
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TOL = 1e-6
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class Ellipsoid:
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def __init__(self, A, b, c):
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A = np.asarray(A, dtype=float)
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b = np.asarray(b, dtype=float)
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d = np.linalg.solve(A, -b / 2)
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gamma = c + b @ d + d @ A @ d
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if gamma > 0: # same set, written with gamma < 0
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A, b, c, gamma = -A, -b, -c, -gamma
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eig, V = np.linalg.eigh(A)
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order = np.argsort(-eig)
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self.eig = eig[order]
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self.V = V[:, order]
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if not np.all(self.eig > 0):
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raise ValueError('the quadric is not a non-empty ellipsoid')
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self.A, self.b, self.c, self.d = A, b, c, d
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self.dim = len(b)
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self.scale = np.sqrt(-gamma)
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def psi(self, x):
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return x @ self.A @ x + self.b @ x + self.c
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def to_standardized(self, x):
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return self.V.T @ (x - self.d) / self.scale
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def to_non_standardized(self, u):
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return self.d + self.scale * (self.V @ u)
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def _solve(self, mu, y):
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"""Return (I + mu A)^{-1} y."""
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return self.V @ ((self.V.T @ y) / (1 + mu * self.eig))
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def x(self, mu, x0):
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return self._solve(mu, x0 - mu * self.b / 2)
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def f(self, mu, x0):
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return self.psi(self.x(mu, x0))
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def df(self, mu, x0):
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x = self.x(mu, x0)
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dx = -self._solve(mu, self.A @ x + self.b / 2)
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return (2 * self.A @ x + self.b) @ dx
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def extreme_roots(self, x0):
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"""Return (mu*, mu**), the roots of f right of the rightmost and left of the leftmost active pole.
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x(mu*) and x(mu**) are the nearest and farthest non-degenerate KKT points; a degenerate
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point (see degenerate_point) may be nearer than x(mu*). Return None if x0 is the centre.
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"""
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u0 = self.to_standardized(x0)
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active = np.abs(u0) > TOL
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if not active.any():
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return None
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eig, u0 = self.eig[active], u0[active]
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def secular(mu):
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return np.sum(eig * (u0 / (1 + mu * eig))**2) - 1
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mu_star = brentq(secular, *_bracket(secular, -1 / eig.max(), +1))
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mu_star_star = brentq(secular, *_bracket(secular, -1 / eig.min(), -1))
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return mu_star, mu_star_star
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def _pole_point(self, u0, k):
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"""Standardized limit of u(mu) as mu -> -1/eig_k, with the eig_k components set to 0."""
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on_pole = np.abs(self.eig - self.eig[k]) < TOL
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u = np.zeros(self.dim)
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np.divide(u0, 1 - self.eig / self.eig[k], out=u, where=~on_pole)
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return u, on_pole
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def pole_line(self, x0, k):
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"""Return (point, direction) of the line {x : u_i = u0_i / (1 - eig_i/eig_k) for eig_i != eig_k}.
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It is the asymptote of x(mu) at mu = -1/eig_k when u0_k != 0, and the line containing
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the degenerate points otherwise.
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"""
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u, _ = self._pole_point(self.to_standardized(x0), k)
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return self.to_non_standardized(u), self.V[:, k]
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def degenerate_point(self, x0, k):
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"""Return the KKT point at mu = -1/eig_k when u0 vanishes on that eigenspace, or None."""
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u0 = self.to_standardized(x0)
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u, on_pole = self._pole_point(u0, k)
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if np.any(np.abs(u0[on_pole]) > TOL):
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return None
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r = (1 - self.eig @ u**2) / self.eig[k]
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if r <= 0:
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return None
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u[np.argmax(on_pole)] = np.sqrt(r)
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return self.to_non_standardized(u)
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def boundary(self, n=1000):
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"""Return n points of a 2D ellipse as an (n, 2) array."""
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assert self.dim == 2, 'only 2D ellipses can be sampled'
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t = np.linspace(-np.pi, np.pi, n)
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u = np.column_stack((np.cos(t), np.sin(t))) / np.sqrt(self.eig)
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return self.d + self.scale * u @ self.V.T
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def _bracket(f, pole, direction):
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"""Return an interval on the `direction` side of `pole` where f changes sign.
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Assumes f -> +inf next to the pole and f -> -1 far from it.
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"""
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near = far = 1.0
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while f(pole + direction * near) <= 0:
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near /= 2
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while f(pole + direction * far) >= 0:
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far *= 2
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return sorted((pole + direction * near, pole + direction * far))
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numpy
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scipy
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matplotlib>=3.6
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pytest
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import numpy as np
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import pytest
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from quadric import Ellipsoid
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AXIS_ELLIPSE = Ellipsoid([[2, 0], [0, 1]], [0, 0], -1)
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ROTATED_ELLIPSE = Ellipsoid([[3, 1], [1, 2]], [1, -1], -4)
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def boundary_distances(Q, x0):
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return np.linalg.norm(Q.boundary(200_000) - x0, axis=1)
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@pytest.mark.parametrize('x0', [[0.3, 0.2], [2.0, -1.5], [-0.1, 0.4]])
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def test_extreme_roots_give_nearest_and_farthest_points(x0):
|
||||
Q, x0 = ROTATED_ELLIPSE, np.array(x0)
|
||||
mu_star, mu_star_star = Q.extreme_roots(x0)
|
||||
x_star, x_star_star = Q.x(mu_star, x0), Q.x(mu_star_star, x0)
|
||||
|
||||
assert abs(Q.psi(x_star)) < 1e-8
|
||||
assert abs(Q.psi(x_star_star)) < 1e-8
|
||||
distances = boundary_distances(Q, x0)
|
||||
assert np.linalg.norm(x_star - x0) == pytest.approx(distances.min(), abs=1e-6)
|
||||
assert np.linalg.norm(x_star_star - x0) == pytest.approx(distances.max(), abs=1e-6)
|
||||
|
||||
|
||||
def test_trajectory_satisfies_kkt_conditions():
|
||||
Q, x0, mu = ROTATED_ELLIPSE, np.array([0.3, 0.2]), 0.7
|
||||
x = Q.x(mu, x0)
|
||||
np.testing.assert_allclose(x - x0 + mu * (Q.A @ x + Q.b / 2), 0, atol=1e-12)
|
||||
|
||||
|
||||
def test_df_matches_finite_difference():
|
||||
Q, x0, mu, h = ROTATED_ELLIPSE, np.array([0.3, 0.2]), 0.7, 1e-6
|
||||
finite_difference = (Q.f(mu + h, x0) - Q.f(mu - h, x0)) / (2 * h)
|
||||
assert Q.df(mu, x0) == pytest.approx(finite_difference, rel=1e-6)
|
||||
|
||||
|
||||
def test_degenerate_point_is_the_projection():
|
||||
Q, x0 = AXIS_ELLIPSE, np.array([0.0, -0.1])
|
||||
xd = Q.degenerate_point(x0, 0)
|
||||
assert abs(Q.psi(xd)) < 1e-12
|
||||
assert np.linalg.norm(xd - x0) == pytest.approx(boundary_distances(Q, x0).min(), abs=1e-6)
|
||||
assert Q.degenerate_point(x0, 1) is None
|
||||
|
||||
|
||||
def test_pole_line_is_the_asymptote():
|
||||
Q, x0 = AXIS_ELLIPSE, np.array([0.3, 0.2])
|
||||
point, direction = Q.pole_line(x0, 1)
|
||||
x = Q.x(-1 / Q.eig[1] + 1e-9, x0)
|
||||
residual = (x - point) - ((x - point) @ direction) * direction
|
||||
np.testing.assert_allclose(residual, 0, atol=1e-6)
|
||||
|
||||
|
||||
def test_sign_convention_does_not_matter():
|
||||
flipped = Ellipsoid(-ROTATED_ELLIPSE.A, -ROTATED_ELLIPSE.b, -ROTATED_ELLIPSE.c)
|
||||
x0 = np.array([0.3, 0.2])
|
||||
np.testing.assert_allclose(flipped.extreme_roots(x0), ROTATED_ELLIPSE.extreme_roots(x0))
|
||||
Reference in new issue
Block a user