data/varipeps/utils/svd.py

from functools import partial

import numpy as np
import jax
import jax.numpy as jnp
from jax.lax import scan
from jax.lax.linalg import svd as lax_svd
from jax import jit, custom_jvp, lax

from jax._src.numpy.util import promote_dtypes_inexact, check_arraylike

from varipeps import varipeps_config

from .extensions import _svd_only_u_vt as _svd_only_u_vt_lib

from typing import Tuple


def _T(x):
    return jnp.swapaxes(x, -1, -2)


def _H(x):
    return jnp.conj(_T(x))


@partial(custom_jvp, nondiff_argnums=(1,))
def svd_wrapper(a, use_qr=False):
    check_arraylike("jnp.linalg.svd", a)
    (a,) = promote_dtypes_inexact(jnp.asarray(a))

    if use_qr:
        result = lax_svd(
            a,
            full_matrices=False,
            compute_uv=True,
            algorithm=lax.linalg.SvdAlgorithm.QR,
        )
    else:
        result = lax_svd(a, full_matrices=False, compute_uv=True)

        result = lax.cond(
            jnp.isnan(jnp.sum(result[1])),
            lambda matrix, _: lax_svd(
                matrix,
                full_matrices=False,
                compute_uv=True,
                algorithm=lax.linalg.SvdAlgorithm.QR,
            ),
            lambda _, res: res,
            a,
            result,
        )

    return result


def _svd_jvp_rule_impl(primals, tangents, only_u_or_vt=None, use_qr=False):
    (A,) = primals
    (dA,) = tangents

    if use_qr and only_u_or_vt is not None:
        U, s, Vt = _svd_only_u_vt_impl(A, u_or_vt=2, use_qr=True)
    else:
        U, s, Vt = svd_wrapper(A, use_qr=use_qr)

    Ut, V = _H(U), _H(Vt)
    s_dim = s[..., None, :]
    dS = Ut @ dA @ V
    ds = jnp.real(jnp.diagonal(dS, 0, -2, -1))

    s_sums = s_dim + _T(s_dim)
    s_sums = jnp.where(s_sums > 0, s_sums, 1)
    s_diffs = s_dim - _T(s_dim)

    if varipeps_config.svd_ad_use_lorentz_broadening:
        F = s_diffs / (s_diffs**2 + varipeps_config.svd_ad_lorentz_broadening_eps)
    else:
        s_diffs = jnp.where(jnp.abs(s_diffs / s[0]) >= 1e-12, s_diffs, 0)
        s_diffs_zeros = jnp.ones((), dtype=A.dtype) * (
            s_diffs == 0.0
        )  # is 1. where s_diffs is 0. and is 0. everywhere else
        s_diffs_zeros = lax.expand_dims(s_diffs_zeros, range(s_diffs.ndim - 2))
        F = 1 / (s_diffs + s_diffs_zeros) - s_diffs_zeros

    if only_u_or_vt is None or only_u_or_vt == "U":
        dSS = dS * (s_dim / s_sums).astype(A.dtype)  # dS.dot(s_j / (s_i + s_j))
    if only_u_or_vt is None or only_u_or_vt == "Vt":
        SdS = (_T(s_dim) / s_sums).astype(A.dtype) * dS  # (s_i / (s_i + s_j)).dot(dS)

    s_zeros = (s == 0).astype(s.dtype)
    s_inv = 1 / (s + s_zeros) - s_zeros
    s_inv_mat = jnp.vectorize(jnp.diag, signature="(k)->(k,k)")(s_inv)
    dUdV_diag = 0.5 * (dS - _H(dS)) * s_inv_mat.astype(A.dtype)

    if only_u_or_vt is None:
        dU = U @ (F.astype(A.dtype) * (dSS + _H(dSS)) + 0.5 * dUdV_diag)
        dV = V @ (F.astype(A.dtype) * (SdS + _H(SdS)) + 0.5 * dUdV_diag)
    elif only_u_or_vt == "U":
        dU = U @ (F.astype(A.dtype) * (dSS + _H(dSS)) + dUdV_diag)
    elif only_u_or_vt == "Vt":
        dV = V @ (F.astype(A.dtype) * (SdS + _H(SdS)) + dUdV_diag)

    m, n = A.shape[-2:]
    if m > n and (only_u_or_vt is None or only_u_or_vt == "U"):
        dAV = dA @ V
        dU = dU + (dAV - U @ (Ut @ dAV)) * s_inv.astype(A.dtype)
    if n > m and (only_u_or_vt is None or only_u_or_vt == "Vt"):
        dAHU = _H(dA) @ U
        dV = dV + (dAHU - V @ (Vt @ dAHU)) * s_inv.astype(A.dtype)

    if only_u_or_vt is None:
        return (U, s, Vt), (dU, ds, _H(dV))
    elif only_u_or_vt == "U":
        return (U, s), (dU, ds)
    elif only_u_or_vt == "Vt":
        return (s, Vt), (ds, _H(dV))


@svd_wrapper.defjvp
def _svd_jvp_rule(use_qr, primals, tangents):
    return _svd_jvp_rule_impl(primals, tangents, use_qr=use_qr)


jax.ffi.register_ffi_target(
    "svd_only_u_vt_f32", _svd_only_u_vt_lib.svd_only_u_vt_f32(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_f64", _svd_only_u_vt_lib.svd_only_u_vt_f64(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_c64", _svd_only_u_vt_lib.svd_only_u_vt_c64(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_c128", _svd_only_u_vt_lib.svd_only_u_vt_c128(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_qr_f32", _svd_only_u_vt_lib.svd_only_u_vt_qr_f32(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_qr_f64", _svd_only_u_vt_lib.svd_only_u_vt_qr_f64(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_qr_c64", _svd_only_u_vt_lib.svd_only_u_vt_qr_c64(), platform="cpu"
)
jax.ffi.register_ffi_target(
    "svd_only_u_vt_qr_c128", _svd_only_u_vt_lib.svd_only_u_vt_qr_c128(), platform="cpu"
)


def _svd_only_u_vt_impl(a, u_or_vt, use_qr=True):
    suffix = "_qr" if use_qr else ""

    if a.dtype == jnp.float32:
        fn = f"svd_only_u_vt{suffix}_f32"
        real_dtype = jnp.float32
    elif a.dtype == jnp.float64:
        fn = f"svd_only_u_vt{suffix}_f64"
        real_dtype = jnp.float64
    elif a.dtype == jnp.complex64:
        fn = f"svd_only_u_vt{suffix}_c64"
        real_dtype = jnp.float32
    elif a.dtype == jnp.complex128:
        fn = f"svd_only_u_vt{suffix}_c128"
        real_dtype = jnp.float64
    else:
        raise ValueError("Unsupported dtype")

    m, n = a.shape

    return_only = None
    if m > n and u_or_vt == 0:
        u_or_vt = 2
        return_only = "U"
    elif m < n and u_or_vt == 1:
        u_or_vt = 2
        return_only = "Vt"

    min_dim = min(m, n)

    if use_qr:
        if u_or_vt == 2:
            u_vt_buffer_shape = jax.ShapeDtypeStruct((min_dim, min_dim), a.dtype)
        else:
            u_vt_buffer_shape = jax.ShapeDtypeStruct((0, 0), a.dtype)

        call = jax.ffi.ffi_call(
            fn,
            (
                jax.ShapeDtypeStruct((m, n), a.dtype),
                jax.ShapeDtypeStruct((min_dim,), real_dtype),
                u_vt_buffer_shape,
                jax.ShapeDtypeStruct((1,), jnp.int32),
            ),
            vmap_method="sequential",
            input_layouts=((1, 0),),
            output_layouts=((1, 0), None, (1, 0), None),
            input_output_aliases={0: 0},
        )

        aout, S, u_vt_buffer, info = call(a, mode=np.int8(u_or_vt))

        if u_or_vt == 2:
            if m >= n:
                U = aout
                Vt = u_vt_buffer
            else:
                U = u_vt_buffer
                Vt = aout
        else:
            result = aout[:min_dim, :min_dim]
    else:
        call = jax.ffi.ffi_call(
            fn,
            (
                jax.ShapeDtypeStruct((m, n), a.dtype),
                jax.ShapeDtypeStruct((min_dim,), real_dtype),
                jax.ShapeDtypeStruct((min_dim, min_dim), a.dtype),
                jax.ShapeDtypeStruct((1,), jnp.int32),
            ),
            vmap_method="sequential",
            input_layouts=((1, 0),),
            output_layouts=((1, 0), None, (1, 0), None),
            input_output_aliases={0: 0},
        )

        aout, S, u_vt_buffer, info = call(a, mode=np.int8(u_or_vt))

        if u_or_vt == 0:
            if m == n:
                result = aout
            else:
                result = u_vt_buffer
        elif u_or_vt == 1:
            result = u_vt_buffer
        elif u_or_vt == 2:
            if m >= n:
                U = aout
                Vt = u_vt_buffer
            else:
                U = u_vt_buffer
                Vt = aout

    if u_or_vt == 2:
        U, S, Vt = jax.lax.cond(
            info[0] != 0,
            lambda u, s, r: (u * jnp.nan, s * jnp.nan, r * jnp.nan),
            lambda u, s, r: (u, s, r),
            U,
            S,
            Vt,
        )

        if return_only == "U":
            return S, U
        elif return_only == "Vt":
            return S, Vt

        return U, S, Vt

    S, result = jax.lax.cond(
        info[0] != 0,
        lambda s, r: (s * jnp.nan, r * jnp.nan),
        lambda s, r: (s, r),
        S,
        result,
    )

    return S, result


@partial(custom_jvp, nondiff_argnums=(1,))
def svd_only_u(a, use_qr=True):
    S, U = _svd_only_u_vt_impl(a, 0, use_qr)

    if not use_qr:
        S, U = lax.cond(
            jnp.isnan(jnp.sum(S)),
            lambda matrix, s, u: _svd_only_u_vt_impl(matrix, 0, True),
            lambda matrix, s, u: (s, u),
            a,
            S,
            U,
        )

    return U, S


@svd_only_u.defjvp
def _svd_only_u_jvp_rule(use_qr, primals, tangents):
    return _svd_jvp_rule_impl(primals, tangents, only_u_or_vt="U", use_qr=use_qr)


@partial(custom_jvp, nondiff_argnums=(1,))
def svd_only_vt(a, use_qr=True):
    S, Vt = _svd_only_u_vt_impl(a, 1, use_qr)

    if not use_qr:
        S, Vt = lax.cond(
            jnp.isnan(jnp.sum(S)),
            lambda matrix, s, v: _svd_only_u_vt_impl(matrix, 1, True),
            lambda matrix, s, v: (s, v),
            a,
            S,
            Vt,
        )

    return S, Vt


@svd_only_vt.defjvp
def _svd_only_vt_jvp_rule(use_qr, primals, tangents):
    return _svd_jvp_rule_impl(primals, tangents, only_u_or_vt="Vt", use_qr=use_qr)


@partial(jit, inline=True, static_argnums=(1, 2))
def gauge_fixed_svd(
    matrix: jnp.ndarray,
    only_u_or_vh=None,
    use_qr=False,
) -> Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray]:
    """
    Calculate the gauge-fixed (also called sign-fixed) SVD. To this end, each
    singular vector are rotate in the way that the first element bigger than
    some numerical stability threshold (config parameter eps) is ensured to be
    along the positive real axis.

    Args:
      matrix (:obj:`jnp.ndarray`):
        Matrix to calculate SVD for.
    Keyword args:
      only_u_or_vh (:obj:`str`):
        Flag if only U or Uh should be calculated. If `None` (default), calculate
        the full SVD, if `'U'` only calculate U, if `'Vh'` only calculate Vh.
    Returns:
      :obj:`tuple`\\ (:obj:`jnp.ndarray`, :obj:`jnp.ndarray`, :obj:`jnp.ndarray`):
        Tuple with sign-fixed U, S and Vh of the SVD.
    """
    if any(d.platform == "gpu" for d in jax.devices()):
        U, S, Vh = svd_wrapper(matrix, use_qr=use_qr)
        if only_u_or_vh is None:
            gauge_unitary = U
        elif only_u_or_vh == "U":
            gauge_unitary = U
        elif only_u_or_vh == "Vh":
            gauge_unitary = Vh.T.conj()
        else:
            raise ValueError("Invalid value for parameter 'only_u_or_vh'.")
    else:
        if only_u_or_vh is None:
            U, S, Vh = svd_wrapper(matrix, use_qr=use_qr)
            gauge_unitary = U
        elif only_u_or_vh == "U":
            U, S = svd_only_u(matrix, use_qr=use_qr)
            gauge_unitary = U
        elif only_u_or_vh == "Vh":
            S, Vh = svd_only_vt(matrix, use_qr=use_qr)
            gauge_unitary = Vh.T.conj()
        else:
            raise ValueError("Invalid value for parameter 'only_u_or_vh'.")

    # Fix the gauge of the SVD
    abs_gauge_unitary = jnp.abs(gauge_unitary)
    max_per_vector = jnp.max(abs_gauge_unitary, axis=0)
    normalized_gauge_unitary = abs_gauge_unitary / max_per_vector[jnp.newaxis, :]

    def phase_f(carry, x):
        x_row, normalized_x_row = x

        already_found, last_step_result = carry

        cond = normalized_x_row >= varipeps_config.svd_sign_fix_eps

        result = jnp.where(
            already_found, last_step_result, jnp.where(cond, x_row, last_step_result)
        )

        return (jnp.logical_or(already_found, cond), result), None

    phases, _ = scan(
        phase_f,
        (jnp.zeros(gauge_unitary.shape[1], dtype=bool), gauge_unitary[0, :]),
        (gauge_unitary, normalized_gauge_unitary),
    )
    phases = phases[1]
    phases /= jnp.abs(phases)

    if only_u_or_vh is None or only_u_or_vh == "U":
        U = U * phases.conj()[jnp.newaxis, :]
    if only_u_or_vh is None or only_u_or_vh == "Vh":
        Vh = Vh * phases[:, jnp.newaxis]

    if only_u_or_vh == "U":
        return U, S

    if only_u_or_vh == "Vh":
        return S, Vh

    return U, S, Vh