# mypy: allow-untyped-defs
import math
from typing import Optional

import torch
from torch import Tensor
from torch.distributions import constraints
from torch.distributions.distribution import Distribution
from torch.distributions.multivariate_normal import _batch_mahalanobis, _batch_mv
from torch.distributions.utils import _standard_normal, lazy_property
from torch.types import _size


__all__ = ["LowRankMultivariateNormal"]


def _batch_capacitance_tril(W, D):
    r"""
    Computes Cholesky of :math:`I + W.T @ inv(D) @ W` for a batch of matrices :math:`W`
    and a batch of vectors :math:`D`.
    """
    m = W.size(-1)
    Wt_Dinv = W.mT / D.unsqueeze(-2)
    K = torch.matmul(Wt_Dinv, W).contiguous()
    K.view(-1, m * m)[:, :: m + 1] += 1  # add identity matrix to K
    return torch.linalg.cholesky(K)


def _batch_lowrank_logdet(W, D, capacitance_tril):
    r"""
    Uses "matrix determinant lemma"::
        log|W @ W.T + D| = log|C| + log|D|,
    where :math:`C` is the capacitance matrix :math:`I + W.T @ inv(D) @ W`, to compute
    the log determinant.
    """
    return 2 * capacitance_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) + D.log().sum(
        -1
    )


def _batch_lowrank_mahalanobis(W, D, x, capacitance_tril):
    r"""
    Uses "Woodbury matrix identity"::
        inv(W @ W.T + D) = inv(D) - inv(D) @ W @ inv(C) @ W.T @ inv(D),
    where :math:`C` is the capacitance matrix :math:`I + W.T @ inv(D) @ W`, to compute the squared
    Mahalanobis distance :math:`x.T @ inv(W @ W.T + D) @ x`.
    """
    Wt_Dinv = W.mT / D.unsqueeze(-2)
    Wt_Dinv_x = _batch_mv(Wt_Dinv, x)
    mahalanobis_term1 = (x.pow(2) / D).sum(-1)
    mahalanobis_term2 = _batch_mahalanobis(capacitance_tril, Wt_Dinv_x)
    return mahalanobis_term1 - mahalanobis_term2


class LowRankMultivariateNormal(Distribution):
    r"""
    Creates a multivariate normal distribution with covariance matrix having a low-rank form
    parameterized by :attr:`cov_factor` and :attr:`cov_diag`::

        covariance_matrix = cov_factor @ cov_factor.T + cov_diag

    Example:
        >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> m = LowRankMultivariateNormal(
        ...     torch.zeros(2), torch.tensor([[1.0], [0.0]]), torch.ones(2)
        ... )
        >>> m.sample()  # normally distributed with mean=`[0,0]`, cov_factor=`[[1],[0]]`, cov_diag=`[1,1]`
        tensor([-0.2102, -0.5429])

    Args:
        loc (Tensor): mean of the distribution with shape `batch_shape + event_shape`
        cov_factor (Tensor): factor part of low-rank form of covariance matrix with shape
            `batch_shape + event_shape + (rank,)`
        cov_diag (Tensor): diagonal part of low-rank form of covariance matrix with shape
            `batch_shape + event_shape`

    Note:
        The computation for determinant and inverse of covariance matrix is avoided when
        `cov_factor.shape[1] << cov_factor.shape[0]` thanks to `Woodbury matrix identity
        <https://en.wikipedia.org/wiki/Woodbury_matrix_identity>`_ and
        `matrix determinant lemma <https://en.wikipedia.org/wiki/Matrix_determinant_lemma>`_.
        Thanks to these formulas, we just need to compute the determinant and inverse of
        the small size "capacitance" matrix::

            capacitance = I + cov_factor.T @ inv(cov_diag) @ cov_factor
    """

    arg_constraints = {
        "loc": constraints.real_vector,
        "cov_factor": constraints.independent(constraints.real, 2),
        "cov_diag": constraints.independent(constraints.positive, 1),
    }
    support = constraints.real_vector
    has_rsample = True

    def __init__(
        self,
        loc: Tensor,
        cov_factor: Tensor,
        cov_diag: Tensor,
        validate_args: Optional[bool] = None,
    ) -> None:
        if loc.dim() < 1:
            raise ValueError("loc must be at least one-dimensional.")
        event_shape = loc.shape[-1:]
        if cov_factor.dim() < 2:
            raise ValueError(
                "cov_factor must be at least two-dimensional, "
                "with optional leading batch dimensions"
            )
        if cov_factor.shape[-2:-1] != event_shape:
            raise ValueError(
                f"cov_factor must be a batch of matrices with shape {event_shape[0]} x m"
            )
        if cov_diag.shape[-1:] != event_shape:
            raise ValueError(
                f"cov_diag must be a batch of vectors with shape {event_shape}"
            )

        loc_ = loc.unsqueeze(-1)
        cov_diag_ = cov_diag.unsqueeze(-1)
        try:
            loc_, self.cov_factor, cov_diag_ = torch.broadcast_tensors(
                loc_, cov_factor, cov_diag_
            )
        except RuntimeError as e:
            raise ValueError(
                f"Incompatible batch shapes: loc {loc.shape}, cov_factor {cov_factor.shape}, cov_diag {cov_diag.shape}"
            ) from e
        self.loc = loc_[..., 0]
        self.cov_diag = cov_diag_[..., 0]
        batch_shape = self.loc.shape[:-1]

        self._unbroadcasted_cov_factor = cov_factor
        self._unbroadcasted_cov_diag = cov_diag
        self._capacitance_tril = _batch_capacitance_tril(cov_factor, cov_diag)
        super().__init__(batch_shape, event_shape, validate_args=validate_args)

    def expand(self, batch_shape, _instance=None):
        new = self._get_checked_instance(LowRankMultivariateNormal, _instance)
        batch_shape = torch.Size(batch_shape)
        loc_shape = batch_shape + self.event_shape
        new.loc = self.loc.expand(loc_shape)
        new.cov_diag = self.cov_diag.expand(loc_shape)
        new.cov_factor = self.cov_factor.expand(loc_shape + self.cov_factor.shape[-1:])
        new._unbroadcasted_cov_factor = self._unbroadcasted_cov_factor
        new._unbroadcasted_cov_diag = self._unbroadcasted_cov_diag
        new._capacitance_tril = self._capacitance_tril
        super(LowRankMultivariateNormal, new).__init__(
            batch_shape, self.event_shape, validate_args=False
        )
        new._validate_args = self._validate_args
        return new

    @property
    def mean(self) -> Tensor:
        return self.loc

    @property
    def mode(self) -> Tensor:
        return self.loc

    @lazy_property
    def variance(self) -> Tensor:  # type: ignore[override]
        return (
            self._unbroadcasted_cov_factor.pow(2).sum(-1) + self._unbroadcasted_cov_diag
        ).expand(self._batch_shape + self._event_shape)

    @lazy_property
    def scale_tril(self) -> Tensor:
        # The following identity is used to increase the numerically computation stability
        # for Cholesky decomposition (see http://www.gaussianprocess.org/gpml/, Section 3.4.3):
        #     W @ W.T + D = D1/2 @ (I + D-1/2 @ W @ W.T @ D-1/2) @ D1/2
        # The matrix "I + D-1/2 @ W @ W.T @ D-1/2" has eigenvalues bounded from below by 1,
        # hence it is well-conditioned and safe to take Cholesky decomposition.
        n = self._event_shape[0]
        cov_diag_sqrt_unsqueeze = self._unbroadcasted_cov_diag.sqrt().unsqueeze(-1)
        Dinvsqrt_W = self._unbroadcasted_cov_factor / cov_diag_sqrt_unsqueeze
        K = torch.matmul(Dinvsqrt_W, Dinvsqrt_W.mT).contiguous()
        K.view(-1, n * n)[:, :: n + 1] += 1  # add identity matrix to K
        scale_tril = cov_diag_sqrt_unsqueeze * torch.linalg.cholesky(K)
        return scale_tril.expand(
            self._batch_shape + self._event_shape + self._event_shape
        )

    @lazy_property
    def covariance_matrix(self) -> Tensor:
        covariance_matrix = torch.matmul(
            self._unbroadcasted_cov_factor, self._unbroadcasted_cov_factor.mT
        ) + torch.diag_embed(self._unbroadcasted_cov_diag)
        return covariance_matrix.expand(
            self._batch_shape + self._event_shape + self._event_shape
        )

    @lazy_property
    def precision_matrix(self) -> Tensor:
        # We use "Woodbury matrix identity" to take advantage of low rank form::
        #     inv(W @ W.T + D) = inv(D) - inv(D) @ W @ inv(C) @ W.T @ inv(D)
        # where :math:`C` is the capacitance matrix.
        Wt_Dinv = (
            self._unbroadcasted_cov_factor.mT
            / self._unbroadcasted_cov_diag.unsqueeze(-2)
        )
        A = torch.linalg.solve_triangular(self._capacitance_tril, Wt_Dinv, upper=False)
        precision_matrix = (
            torch.diag_embed(self._unbroadcasted_cov_diag.reciprocal()) - A.mT @ A
        )
        return precision_matrix.expand(
            self._batch_shape + self._event_shape + self._event_shape
        )

    def rsample(self, sample_shape: _size = torch.Size()) -> Tensor:
        shape = self._extended_shape(sample_shape)
        W_shape = shape[:-1] + self.cov_factor.shape[-1:]
        eps_W = _standard_normal(W_shape, dtype=self.loc.dtype, device=self.loc.device)
        eps_D = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
        return (
            self.loc
            + _batch_mv(self._unbroadcasted_cov_factor, eps_W)
            + self._unbroadcasted_cov_diag.sqrt() * eps_D
        )

    def log_prob(self, value):
        if self._validate_args:
            self._validate_sample(value)
        diff = value - self.loc
        M = _batch_lowrank_mahalanobis(
            self._unbroadcasted_cov_factor,
            self._unbroadcasted_cov_diag,
            diff,
            self._capacitance_tril,
        )
        log_det = _batch_lowrank_logdet(
            self._unbroadcasted_cov_factor,
            self._unbroadcasted_cov_diag,
            self._capacitance_tril,
        )
        return -0.5 * (self._event_shape[0] * math.log(2 * math.pi) + log_det + M)

    def entropy(self):
        log_det = _batch_lowrank_logdet(
            self._unbroadcasted_cov_factor,
            self._unbroadcasted_cov_diag,
            self._capacitance_tril,
        )
        H = 0.5 * (self._event_shape[0] * (1.0 + math.log(2 * math.pi)) + log_det)
        if len(self._batch_shape) == 0:
            return H
        else:
            return H.expand(self._batch_shape)
