Efficient Inversion of a Large Binary Matrix Mod 2 in Python

Below is a summary of several recommendations, along with a sketch of an approach that packs rows into machine‐words so that many bits are processed in “one shot.” The idea is to avoid Python’s per‐bit (or per‐entry) looping overhead by:

• Representing binary data densely: Instead of using boolean arrays (which are 1 byte per entry), pack rows into 64‑bit or 32‑bit integers. This immediately gives you an 8×–or–more speedup.
• Using bit‑level operations: With packed representations, you can perform row XORs using bit‑wise operations that run at the C level.
• Leveraging specialized libraries: Consider libraries like M4RI (for “Method of the Four Russians”) that are designed specifically for dense GF(2) algebra. This library is written in C and can be wrapped for Python usage.
• Compiling with Numba/Cython: If you must stick with Python, compile the inner loop (or use Cython/Numba) so that the elimination loop isn’t executed in interpreted Python.

Below is an outline of how you might pack the matrix and perform elimination. Note that the complete “block‐Gaussian elimination” requires careful handling of boundaries; this sample is only to illustrate the idea.

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import numpy as np

def pack_bits(matrix, word_size=64):
    """
    Pack a boolean matrix into an array where each row is represented 
    by an array of uint64 values. 
    """
    n, m = matrix.shape
    # Pad the number of columns to a multiple of word_size.
    pad = (-m) % word_size
    if pad:
        matrix = np.hstack([matrix, np.zeros((n, pad), dtype=bool)])
    m_packed = matrix.shape[1] // word_size
    # Allocate the packed matrix.
    packed = np.empty((n, m_packed), dtype=np.uint64)
    # Pack by processing word_size bits at a time.
    for i in range(m_packed):
        # Each chunk of 64 bits is read as bits and converted to a single integer.
        chunk = matrix[:, i*word_size:(i+1)*word_size]
        # Using bit shifts to pack; you can also use np.packbits then combine into uint64.
        # Here is a simple loop over bits in the word.
        word_vals = np.zeros(n, dtype=np.uint64)
        for bit in range(word_size):
            word_vals |= (chunk[:, bit].astype(np.uint64) << (word_size - 1 - bit))
        packed[:, i] = word_vals
    return packed, matrix.shape[1]  # number of bits per row now

def binary_inv_mod2(A):
    """
    Invert an (n x n) dense binary matrix mod 2.
    This routine does Gaussian elimination on the packed representation.
    It also computes the rank.
    """
    n = A.shape[0]
    # Form the augmented matrix [A | I]
    A_bool = A.astype(bool)
    I = np.eye(n, dtype=bool)
    augmented = np.hstack([A_bool, I])
    
    # Pack the augmented matrix.
    packed_aug, total_bits = pack_bits(augmented)
    words = total_bits // 64  # number of uint64 values per row
    
    # Create a working copy as uint64.
    M = packed_aug.copy()
    
    rank = 0
    for col in range(n):
        word_index = col // 64
        bit_index   = col % 64
        pivot = 1 << (63 - bit_index)  # use big-endian bit order
        
        # Find a pivot row (starting at rank) that has a '1' bit in column col.
        pivot_rows = np.nonzero(M[rank:, word_index] & pivot)[0]
        if pivot_rows.size == 0:
            continue  # No pivot in this column, matrix is singular.
        
        i_pivot = pivot_rows[0] + rank
        
        # Swap rows if needed.
        if i_pivot != rank:
            M[[rank, i_pivot]] = M[[i_pivot, rank]]
        
        # Eliminate the pivot bit in all other rows.
        for i in range(n):
            if i != rank and (M[i, word_index] & pivot):
                M[i, :] ^= M[rank, :]
        
        rank += 1
    
    # Optionally, extract the inverse when the rank equals n.
    if rank < n:
        print("Matrix is singular. Rank =", rank)
        return None
    else:
        # Inverse is contained in the [n:2n] columns of the augmented part.
        # You would need to unpack M back to boolean format.
        inv_bool = unpack_bits(M, n, total_bits)[:, n:2*n]
        return inv_bool

def unpack_bits(packed, n_cols, total_bits, word_size=64):
    """
    Unpack a packed matrix (each row in packed) back into a boolean array.
    n_cols is the number of rows, and total_bits is the number of bits per row.
    This routine assumes that total_bits is a multiple of word_size.
    """
    n, n_words = packed.shape
    # Prepare an empty boolean array.
    unpacked = np.empty((n, total_bits), dtype=bool)
    for i in range(n):
        bits = []
        for word in packed[i]:
            # Convert each uint64 word to bits.
            b = [(word >> (word_size - 1 - j)) & 1 for j in range(word_size)]
            bits.extend(b)
        unpacked[i, :] = bits[:total_bits]
    return unpacked

# Example usage:
if __name__ == '__main__':
    n = 512
    # Create a random invertible matrix over GF(2). For testing, you might want to ensure invertibility.
    A = np.random.randint(0, 2, size=(n, n), dtype=np.uint8)
    invA = binary_inv_mod2(A)
    if invA is not None:
        print("Inverse computed.")

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Explanation

1. Packing the Bits:
   The pack_bits() function packs each row of the boolean matrix into an array of 64‑bit unsigned integers. This turns 64 bits into one word, allowing vectorized XOR for 64 bits at a time.

2. Gaussian Elimination on Packed Data:
   In binary_inv_mod2(), the Gaussian elimination is performed on the packed rows. Each pivot search and row update (via XOR) happens on entire 64‑bit words at once.

3. Unpacking (if needed):
   Finally, if you need the inverse in a human‐readable boolean format, the unpack_bits() function converts the packed words back into boolean rows.

Caveats

• The implementation above is only a sketch. In practice, you would want to optimize the inner loops (for example, by rewriting the elimination loop in Numba or Cython).
• Make sure pivioting is done carefully so that word‐ and bit‑indices are correctly mapped.
• For matrices as large as 100,000 × 100,000, lower-level languages or specialized libraries like M4RI are recommended since even optimized Python might not be fast enough in pure Python.

Using these techniques should yield significant speedups when working with large binary matrices working mod 2.