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Step function optimization with EXACT autoconvolution computation.
For a step function f = sum_i h_i * 1_{[a_i, b_i)}, the autoconvolution
(f*f)(t) = sum_{i,j} h_i * h_j * overlap(t, [a_i,b_i), [a_j,b_j))
where overlap is the length of intersection of [a_i,b_i) and [t-b_j, t-a_j).
This is piecewise linear in t, so we can find the exact max at breakpoints.
"""
import numpy as np
from scipy.optimize import minimize, differential_evolution
import time
def autoconv_step(heights, widths):
"""Compute exact autoconvolution of a step function at all breakpoints.
f(x) = heights[i] for x in [cum_widths[i], cum_widths[i+1])
where cum_widths[0] = 0.
Returns: (breakpoints, autoconv_values) - the autoconvolution evaluated at breakpoints.
"""
K = len(heights)
# Compute interval endpoints
endpoints = np.zeros(K + 1)
endpoints[1:] = np.cumsum(widths)
# Collect all breakpoints: t = a_i + a_j, a_i + b_j, b_i + a_j, b_i + b_j
bp_set = set()
for i in range(K):
for j in range(K):
ai, bi = endpoints[i], endpoints[i+1]
aj, bj = endpoints[j], endpoints[j+1]
# Overlap of [ai, bi) and [t-bj, t-aj) is nonzero when:
# t-bj < bi and t-aj > ai, i.e., ai+aj < t < bi+bj
# Breakpoints where the overlap function changes slope:
for bp in [ai+aj, ai+bj, bi+aj, bi+bj]:
bp_set.add(bp)
breakpoints = np.array(sorted(bp_set))
# Also add midpoints between consecutive breakpoints (for quadratic terms)
# Actually for piecewise constant f, autoconv is piecewise LINEAR, so
# max is at a breakpoint. But let's also check midpoints to be safe.
midpoints = (breakpoints[:-1] + breakpoints[1:]) / 2
all_t = np.concatenate([breakpoints, midpoints])
all_t = np.sort(all_t)
# Evaluate autoconvolution at each t
values = np.zeros(len(all_t))
for idx, t in enumerate(all_t):
val = 0.0
for i in range(K):
for j in range(K):
ai, bi = endpoints[i], endpoints[i+1]
aj, bj = endpoints[j], endpoints[j+1]
# Overlap of [ai, bi) and [t-bj, t-aj)
lo = max(ai, t - bj)
hi = min(bi, t - aj)
if hi > lo:
val += heights[i] * heights[j] * (hi - lo)
values[idx] = val
return all_t, values
def compute_c1_step(heights, widths):
"""Compute C1 for a step function."""
t_vals, conv_vals = autoconv_step(heights, widths)
integral = np.sum(heights * widths)
if integral < 1e-15:
return 1e10
return float(np.max(conv_vals) / integral**2)
def optimize_step_scipy(K, n_trials=100, method='de'):
"""Optimize step function with K equal-width steps."""
width = 0.5 / K
widths = np.full(K, width)
best_c1 = np.inf
best_heights = None
if method == 'de':
# Differential evolution on heights
bounds = [(0.0, 10.0)] * K
def obj(h):
h = np.maximum(h, 0.0)
return compute_c1_step(h, widths)
result = differential_evolution(obj, bounds, maxiter=1000,
seed=42, tol=1e-12,
popsize=30, mutation=(0.5, 1.5),
recombination=0.9)
best_heights = np.maximum(result.x, 0.0)
best_c1 = result.fun
else:
# Multi-start L-BFGS-B
for trial in range(n_trials):
np.random.seed(trial)
h0 = np.abs(np.random.randn(K)) + 0.5
def obj(h):
h = np.maximum(h, 0.0)
return compute_c1_step(h, widths)
try:
result = minimize(obj, h0, method='Nelder-Mead',
options={'maxiter': 10000, 'xatol': 1e-12, 'fatol': 1e-12})
c1 = result.fun
if c1 < best_c1:
best_c1 = c1
best_heights = np.maximum(result.x, 0.0)
except:
pass
return best_heights, best_c1
def optimize_step_variable_width(K, n_trials=50):
"""Optimize both heights and widths of step function."""
best_c1 = np.inf
best_h = None
best_w = None
for trial in range(n_trials):
np.random.seed(trial * 137 + 42)
# Parameterize widths via softmax: w_i = exp(p_i) / sum(exp(p_j)) * 0.5
# Heights: h_i = q_i^2
p0 = np.random.randn(K) * 0.5
q0 = np.abs(np.random.randn(K)) + 0.5
x0 = np.concatenate([p0, q0])
def obj(x):
p = x[:K]
q = x[K:]
# Softmax for widths
ep = np.exp(p - np.max(p))
widths = ep / np.sum(ep) * 0.5
heights = q ** 2
return compute_c1_step(heights, widths)
try:
result = minimize(obj, x0, method='Nelder-Mead',
options={'maxiter': 50000, 'xatol': 1e-12, 'fatol': 1e-12})
if result.fun < best_c1:
best_c1 = result.fun
x = result.x
p = x[:K]
ep = np.exp(p - np.max(p))
best_w = ep / np.sum(ep) * 0.5
best_h = x[K:] ** 2
print(f" Trial {trial}: C1 = {best_c1:.10f}")
except:
pass
return best_h, best_w, best_c1
# Main search
print("=== Step Function Optimization ===\n")
# Equal-width steps
print("Equal-width steps (DE):")
for K in [5, 8, 10, 15, 20, 30, 50]:
t0 = time.time()
h, c1 = optimize_step_scipy(K, method='de')
elapsed = time.time() - t0
print(f" K={K:3d}: C1 = {c1:.10f} ({elapsed:.1f}s)")
if c1 < 1.51:
print(f" Heights: {h}")
# Variable-width steps
print("\nVariable-width steps:")
for K in [5, 8, 10, 15, 20]:
t0 = time.time()
h, w, c1 = optimize_step_variable_width(K, n_trials=30)
elapsed = time.time() - t0
print(f" K={K:3d}: C1 = {c1:.10f} ({elapsed:.1f}s)")
if c1 < 1.51:
print(f" Heights: {h}")
print(f" Widths: {w}")
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