Gradient-flow data generation for the kernel family exp(-|x-y|^q)

This package contains Python code to generate the point-flow data used for the portrait plot matrices.

What it computes

For ordered points on the unit interval, it studies the energy

E(x_1,...,x_k) = sum_{i<j} exp(-|x_j-x_i|^q).

The script computes a projected gradient-descent flow:

• starting from a uniform configuration shifted slightly into the interior
• taking small steps in the negative gradient direction
• clipping and sorting points back into the unit interval
• using simple backtracking if a step increases the energy

Included panel set

The helper generate_all_panels() builds the 2-column × 4-row portrait collection:

• k=9, q=0.1
• k=9, q=1
• k=9, q=q_odd
• k=9, q=2
• k=10, q=0.1
• k=10, q=1
• k=10, q=q_10
• k=10, q=2

where:

• q_odd = log(1/(-log((1+e^-1)/2))) / log(2)
• q_10 = 1.238523533

Requirements

Install NumPy:

pip install numpy

Usage

Run the script directly:

python gradient_flow_data_generation.py

It will:

• generate all panels
• print a small summary for the k=9, q=q_odd case
• write one CSV-like file called k9_qcrit_flow_data.csv

Main functions

• energy(x, q)
  Computes the discrete energy.

• gradient(x, q)
  Computes the gradient with respect to the point positions.

• gradient_flow(k, q, n_steps=400, dt=0.0025, margin=0.03)
  Generates one flow path.

• generate_all_panels()
  Generates the full 2×4 portrait data set.

• to_csv_like_table(flow)
  Exports one flow as plain CSV-like text.

• tikz_coordinates(flow, y_scale=1.0)
  Converts a flow into point trajectories suitable for TikZ.

• print_tikz_trajectory_block(flow, y_scale=1.0)
  Prints TikZ coordinate blocks for trajectory plotting.

Notes

• Endpoints are not fixed in this version.
• The initial condition is an equally spaced configuration inside [margin, 1-margin].
• Very small gaps are snapped to exact collisions for cleaner downstream plotting.

Files

• gradient_flow_data_generation.py
• README.md