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Deep Dive: The Kronecker-Weyl Probe

The Kronecker-Weyl Probe uses a mathematically rigorous Low-Discrepancy Sequence to sample the search space.

This section explains the mathematics behind how ArqonHPO samples the search space to maximize information gain while avoiding the pitfalls of random sampling and rigid grids.

The Problem

  1. Random Sampling (Monte Carlo):

    • Issue: It "clumps". You often get points very close to each other (wasted effort) and large empty voids (missed information).
    • Result: Inefficient coverage of the landscape.

  2. Grid Sampling:

    • Issue: It suffers from the "Curse of Dimensionality". The number of points needed grows exponentially (10^d).
    • Issue: It aliases. If the underlying function has a period that matches the grid, you miss the structure entirely.

  3. Legacy p/1000 Heuristic (DEPRECATED):

    • Issue: primes[i] / 1000 produces collisions and striping artifacts.
    • Result: Wasted budget on duplicate regions.

The Solution: Kronecker Sequence with Prime Square Root Slopes

ArqonHPO v2 uses the PrimeSqrtSlopesRotProbe—a Kronecker/Weyl sequence with irrational slopes derived from prime square roots.

The Math

For the i-th sample in dimension d:

sample[i][d] = fract( i × √prime[d] + shift[d] )

Where:

  • √prime[d]: The square root of the d-th prime (2, 3, 5, 7, 11...). Irrational slopes prevent collisions.
  • shift[d]: Optional Cranley-Patterson shift for QMC randomization.
  • fract(x): The fractional part of x (wraps to [0, 1)).

Key Properties

Property Description
Anytime Quality of first K samples does NOT depend on total N
Collision-Free √prime slopes are mutually irrational—no aliasing
Deterministic Same (seed, index) always produces same point
Shardable Stateless: workers can generate disjoint ranges independently

Robustness Hedge

A configurable random_spice_ratio (default 10%) of uniform random points hedges against multimodal fragility.

Periodic Dimension Support

For dimensions marked as Scale::Periodic (angles, phases), the probe uses toroidal topology:

  • wrap01(x): Wrap to [0, 1)
  • diff01(a, b): Shortest signed distance in circular space
  • circular_mean01(values): Mean via sin/cos averaging

Visual Proof

The Kronecker sequence creates a Low-Discrepancy Lattice. It looks random (no obvious repeating pattern) but fills space uniformly.

Probe Lattice Pattern

Comparison of Kronecker Probe (Blue) vs Uniform Random (Red). Note how Blue covers uniformly without clumping or gaps.

Why It Matters

High-quality probe data is critical for the Classifier phase:

  • Uniform coverage avoids misclassifying structured landscapes as chaotic.
  • Anytime property allows early stopping without quality degradation.
  • Sharding enables parallel probing on expensive objectives.

Constitution Reference

Per Constitution v1.1.0 Section II.12:

  • Kronecker/Weyl sequences are REQUIRED.
  • The legacy p/1000 heuristic is BANNED.
  • Cranley-Patterson shifts are the approved randomization mechanism.