SpinPress

An attempt at achieving data compression using inspiration from the Ising Model

📌 Overview

This compression scheme introduces a novel approach to data representation by leveraging the Ising model, a mathematical model from statistical mechanics, to encode and reconstruct 2D binary data arrays.

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The 2D Ising Model

We define a 2D lattice of size $( N \times N )$, which has spins $( s_{ij} )$, where each spin can take on a value:

$$[ s_{ij} \in {-1, +1} ]$$

Each spin interacts with its nearest neighbours (up, down, left, right), and the energy of a given configuration of spins is defined by the Ising Hamiltonian:

$$[ E = -J \sum_{\langle i,j \rangle} s_{ij} \cdot s_{i'j'} ]$$

Where:

• The sum is taken over all nearest neighbour pairs $( \langle i,j \rangle )$
• $( J )$ is the interaction constant (assumed $( J > 0 )$ for ferromagnetic coupling)
• $( s_{ij}, s_{i'j'} \in {-1, +1} )$

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Microstate Realisation

In our compression scheme, we define the energy of a configuration using the above Hamiltonian and try to find the ground state of the system.

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⚙️ How It Works

Compression Algorithm (Encoding)

Input:
A 2D binary array of size $( N \times N )$, where each element is a bit (0 or 1) representing meaningful data.

Steps:

1. Map to Ising Spins:
   Convert the binary array to Ising spins:

   • Bit 1 $( \rightarrow )$ Spin $( +1 )$
   • Bit 0 $( \rightarrow )$ Spin $( -1 )$

2. Select Minimal Seed Set:
   Identify a minimal subset of spin values and their positions such that:

   • If only these spins are known and fixed in value, with all others being random,
   • Then, under energy minimization to the ground state, the system will evolve into the original spin matrix.

3. Store the Seed Set:
   Save:

   • A list of coordinates of the necessary spins to fix: $( (i, j) )$
   • Corresponding spin values for these fixed spins: $( s_{ij} \in {-1, +1} )$

Output:
A compressed representation consisting of a small list of $((i, j, s_{ij}))$ entries.

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Decompression Algorithm (Decoding)

Input:

• The list of seed coordinates and spin values
• The lattice size $( N \times N )$
• Knowledge of the energy minimization rules (i.e., Ising Hamiltonian)

Steps:

1. Initialize Lattice:

   • Create an $( N \times N )$ spin lattice
   • Set the spins at the given coordinates to their seed values
   • Initialize all other spins randomly (e.g., sampled from $( {-1, +1} ))$

2. Find the Ground State:

   • Apply an energy minimization algorithm to evolve the lattice to the ground state whilst keeping the original set of spins fixed throughout.

3. Map Spins Back to Bits:

   • Convert each spin back to a bit:
     • Spin $( +1 \rightarrow 1 )$
     • Spin $( -1 \rightarrow 0 )$

Output:
The reconstructed binary array — the original uncompressed data.

Any knowledge on whether this problem is solvable as well as the theoretical limits of this scheme are welcomed. Please get in touch: toby.hallett9@gmail.com.