PCA 保留了 99% 的方差，为什么分类线索还是全丢了？

"""Exact rectangle PCA and finite-label collision bound; no fitted classifier."""
from collections import Counter, defaultdict
from itertools import product
import math

CHECKS = []

def positive(value):
    if isinstance(value, bool) or not isinstance(value, (int, float)) or not 1e-6 <= value <= 1e6:
        raise ValueError('scale must be finite and in [1e-6, 1e6]')
    return float(value)

def rectangle(horizontal, vertical):
    a, b = positive(horizontal), positive(vertical)
    return [(sx*a, sy*b, int(sy > 0)) for sx, sy in product([-1, 1], repeat=2)]

def summarize(horizontal, vertical, mode='pca'):
    rows = rectangle(horizontal, vertical)
    if mode not in ('pca', 'x', 'y', 'both'):
        raise ValueError('projection mode')
    n = len(rows)
    means = [sum(row[j] for row in rows)/n for j in range(2)]
    centered = [(x-means[0], y-means[1], label) for x, y, label in rows]
    vx = sum(x*x for x, y, label in centered)/n
    vy = sum(y*y for x, y, label in centered)/n
    covariance = sum(x*y for x, y, label in centered)/n
    axis = ('x' if vx >= vy else 'y') if mode == 'pca' else mode
    # This exact axis shortcut is valid ONLY for this symmetric rectangle.
    projected = [(x,) if axis == 'x' else (y,) if axis == 'y' else (x, y) for x, y, label in centered]
    groups = defaultdict(Counter)
    for point, row in zip(projected, centered):
        groups[point][row[2]] += 1
    ceiling = sum(max(counts.values()) for counts in groups.values())/n
    kept = vx if axis == 'x' else vy if axis == 'y' else vx+vy
    error = vy if axis == 'x' else vx if axis == 'y' else 0.
    return dict(axis=axis, vx=vx, vy=vy, covariance=covariance, ratio=kept/(vx+vy),
                mse=error, ceiling=ceiling, tie=vx == vy, projected=projected, groups=groups)

def check(name, ok):
    if not ok:
        raise AssertionError(name)
    CHECKS.append(name)

def main():
    cases = [('pca_original', 10., 1., 'pca'), ('keep_small_axis', 10., 1., 'y'),
             ('keep_both', 10., 1., 'both'), ('scale_horizontal_down', 0.5, 1., 'pca'),
             ('equal_variances', 1., 1., 'pca')]
    for name, a, b, mode in cases:
        result = summarize(a, b, mode)
        check(name + ' diagonal covariance', result['covariance'] == 0.)
        check(name + ' variance decomposition', math.isclose(result['mse']+(result['vx']+result['vy'])*result['ratio'], result['vx']+result['vy']))
        # Exhaust all deterministic label assignments to distinct projected points.
        groups = list(result['groups'].values())
        best = max(sum(counts[label] for counts, label in zip(groups, labels))/4 for labels in product([0,1], repeat=len(groups)))
        check(name + ' exhaustive classification ceiling', best == result['ceiling'])
        print(f"{name}: axis={result['axis']}; retained={result['ratio']:.6f}; reconstruction_mse={result['mse']:.2f}; finite_label_ceiling={result['ceiling']:.2f}; tie={result['tie']}")
    original = summarize(10, 1)
    check('99 percent variance is not label information', original['ratio'] > .99 and original['ceiling'] == .5)
    check('small axis keeps labels', summarize(10, 1, 'y')['ceiling'] == 1.)
    check('PCA ignores labels by construction', original['axis'] == 'x')
    for a, b in [(2., 1.), (1., 3.), (.5, 4.)]:
        result = summarize(a, b)
        # Grid directions independently verify the diagonal quadratic maximum.
        for degree in range(181):
            angle = math.radians(degree)
            variance = a*a*math.cos(angle)**2 + b*b*math.sin(angle)**2
            check('PCA direction dominates angle grid', variance <= max(a*a, b*b)+1e-10)
    for args in [(0,1), (-1,1), (True,1), (float('nan'),1), (1,float('inf')), (1,1e7), (1e-300,1e-300)]:
        try:
            summarize(*args)
        except ValueError:
            check('reject invalid scales', True)
        else:
            raise AssertionError('invalid input accepted')
    try:
        summarize(1, 1, 'unknown')
    except ValueError:
        check('reject invalid mode', True)
    else:
        raise AssertionError('invalid mode accepted')
    print(f'PASS checks={len(CHECKS)}; covariance_divisor=N; reference=label_assignment_enumeration+angle_grid')

if __name__ == '__main__':
    main()