EP-0140: API Symmetry Restoration — Inverse Transforms & Recomposition¶

def recompose_lu(P: SparseMatrix, L: SparseMatrix, U: SparseMatrix) -> SparseMatrix:
    """Reconstruct A from LU factorization: A = P^T @ L @ U"""

def recompose_qr(Q: SparseMatrix, R: SparseMatrix) -> SparseMatrix:
    """Reconstruct A from QR factorization: A = Q @ R"""

def recompose_svd(U: SparseMatrix, S: SparseVector, V_T: SparseMatrix) -> SparseMatrix:
    """Reconstruct A from SVD: A = U @ diag(S) @ V_T"""

def recompose_cholesky(L: SparseMatrix) -> SparseMatrix:
    """Reconstruct A from Cholesky: A = L @ L^T"""

def iwalsh_hadamard(signal: SparseVector, n: int | None = None) -> SparseVector:
    """Inverse Walsh-Hadamard transform: X_k = (1/N) * WHT(x)_k"""

def iz_transform(X: Callable, signal_length: int, radius: float = 1.0) -> SparseVector:
    """Inverse Z-transform via contour integration approximation."""

def deconvolve(signal: SparseVector, kernel: SparseVector) -> SparseVector:
    """Spectral deconvolution: recover f from g = f * kernel via DFT division."""

def unpermute_tensor(tensor: SparseTensor, axes: tuple[int, ...], original_axes: tuple[int, ...]) -> SparseTensor:
    """Inverse axis permutation restoring original tensor index order."""

def coboundary(complex: SparseChainComplex, k: int) -> SparseMatrix:
    """Coboundary operator d^k = D_{k+1}^T : C^k -> C^{k+1}"""

def cohomology_rank(complex: SparseChainComplex, k: int) -> int:
    """Compute k-th cohomology rank: dim(ker d^k) - dim(im d^{k-1})"""