ppdiffusers/schedulers/scheduling_vq_diffusion.py

# Copyright (c) 2022 PaddlePaddle Authors. All Rights Reserved.
# Copyright 2022 Microsoft and The HuggingFace Team. All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

from dataclasses import dataclass
from typing import List, Optional, Tuple, Union

import numpy as np
import paddle
import paddle.nn.functional as F

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput
from .scheduling_utils import SchedulerMixin


def logaddexp(a, b):
    return paddle.log(a.exp() + b.exp())


# (TODO junnyu) paddle logsumexp may has bug
def logsumexp(x, axis=None, keepdim=False):
    return paddle.log(x.exp().sum(axis=axis, keepdim=keepdim))


@dataclass
class VQDiffusionSchedulerOutput(BaseOutput):
    """
    Output class for the scheduler's step function output.

    Args:
        prev_sample (`paddle.Tensor` of shape `(batch size, num latent pixels)`):
            Computed sample x_{t-1} of previous timestep. `prev_sample` should be used as next model input in the
            denoising loop.
    """

    prev_sample: paddle.Tensor


def index_to_log_onehot(x: paddle.Tensor, num_classes: int) -> paddle.Tensor:
    """
    Convert batch of vector of class indices into batch of log onehot vectors

    Args:
        x (`paddle.Tensor` of shape `(batch size, vector length)`):
            Batch of class indices

        num_classes (`int`):
            number of classes to be used for the onehot vectors

    Returns:
        `paddle.Tensor` of shape `(batch size, num classes, vector length)`:
            Log onehot vectors
    """
    x_onehot = F.one_hot(x, num_classes)
    x_onehot = x_onehot.transpose([0, 2, 1])
    log_x = paddle.log(x_onehot.cast("float32").clip(min=1e-30))
    return log_x


def gumbel_noised(logits: paddle.Tensor, generator: Optional[paddle.Generator]) -> paddle.Tensor:
    """
    Apply gumbel noise to `logits`
    """
    uniform = paddle.rand(logits.shape, generator=generator)
    gumbel_noise = -paddle.log(-paddle.log(uniform + 1e-30) + 1e-30)
    noised = gumbel_noise + logits
    return noised


def alpha_schedules(num_diffusion_timesteps: int, alpha_cum_start=0.99999, alpha_cum_end=0.000009):
    """
    Cumulative and non-cumulative alpha schedules.

    See section 4.1.
    """
    att = (
        np.arange(0, num_diffusion_timesteps) / (num_diffusion_timesteps - 1) * (alpha_cum_end - alpha_cum_start)
        + alpha_cum_start
    )
    att = np.concatenate(([1], att))
    at = att[1:] / att[:-1]
    att = np.concatenate((att[1:], [1]))
    return at, att


def gamma_schedules(num_diffusion_timesteps: int, gamma_cum_start=0.000009, gamma_cum_end=0.99999):
    """
    Cumulative and non-cumulative gamma schedules.

    See section 4.1.
    """
    ctt = (
        np.arange(0, num_diffusion_timesteps) / (num_diffusion_timesteps - 1) * (gamma_cum_end - gamma_cum_start)
        + gamma_cum_start
    )
    ctt = np.concatenate(([0], ctt))
    one_minus_ctt = 1 - ctt
    one_minus_ct = one_minus_ctt[1:] / one_minus_ctt[:-1]
    ct = 1 - one_minus_ct
    ctt = np.concatenate((ctt[1:], [0]))
    return ct, ctt


class VQDiffusionScheduler(SchedulerMixin, ConfigMixin):
    """
    The VQ-diffusion transformer outputs predicted probabilities of the initial unnoised image.

    The VQ-diffusion scheduler converts the transformer's output into a sample for the unnoised image at the previous
    diffusion timestep.

    [`~ConfigMixin`] takes care of storing all config attributes that are passed in the scheduler's `__init__`
    function, such as `num_train_timesteps`. They can be accessed via `scheduler.config.num_train_timesteps`.
    [`SchedulerMixin`] provides general loading and saving functionality via the [`SchedulerMixin.save_pretrained`] and
    [`~SchedulerMixin.from_pretrained`] functions.

    For more details, see the original paper: https://arxiv.org/abs/2111.14822

    Args:
        num_vec_classes (`int`):
            The number of classes of the vector embeddings of the latent pixels. Includes the class for the masked
            latent pixel.

        num_train_timesteps (`int`):
            Number of diffusion steps used to train the model.

        alpha_cum_start (`float`):
            The starting cumulative alpha value.

        alpha_cum_end (`float`):
            The ending cumulative alpha value.

        gamma_cum_start (`float`):
            The starting cumulative gamma value.

        gamma_cum_end (`float`):
            The ending cumulative gamma value.
    """

    order = 1

    @register_to_config
    def __init__(
        self,
        num_vec_classes: int,
        num_train_timesteps: int = 100,
        alpha_cum_start: float = 0.99999,
        alpha_cum_end: float = 0.000009,
        gamma_cum_start: float = 0.000009,
        gamma_cum_end: float = 0.99999,
    ):
        self.num_embed = num_vec_classes

        # By convention, the index for the mask class is the last class index
        self.mask_class = self.num_embed - 1

        at, att = alpha_schedules(num_train_timesteps, alpha_cum_start=alpha_cum_start, alpha_cum_end=alpha_cum_end)
        ct, ctt = gamma_schedules(num_train_timesteps, gamma_cum_start=gamma_cum_start, gamma_cum_end=gamma_cum_end)

        num_non_mask_classes = self.num_embed - 1
        bt = (1 - at - ct) / num_non_mask_classes
        btt = (1 - att - ctt) / num_non_mask_classes

        at = paddle.to_tensor(at.astype("float64"))
        bt = paddle.to_tensor(bt.astype("float64"))
        ct = paddle.to_tensor(ct.astype("float64"))
        log_at = paddle.log(at)
        log_bt = paddle.log(bt)
        log_ct = paddle.log(ct)

        att = paddle.to_tensor(att.astype("float64"))
        btt = paddle.to_tensor(btt.astype("float64"))
        ctt = paddle.to_tensor(ctt.astype("float64"))
        log_cumprod_at = paddle.log(att)
        log_cumprod_bt = paddle.log(btt)
        log_cumprod_ct = paddle.log(ctt)

        self.log_at = log_at.cast("float32")
        self.log_bt = log_bt.cast("float32")
        self.log_ct = log_ct.cast("float32")
        self.log_cumprod_at = log_cumprod_at.cast("float32")
        self.log_cumprod_bt = log_cumprod_bt.cast("float32")
        self.log_cumprod_ct = log_cumprod_ct.cast("float32")

        # setable values
        self.num_inference_steps = None
        self.timesteps = paddle.to_tensor(np.arange(0, num_train_timesteps)[::-1].copy())

    def set_timesteps(self, num_inference_steps: int):
        """
        Sets the discrete timesteps used for the diffusion chain. Supporting function to be run before inference.

        Args:
            num_inference_steps (`int`):
                the number of diffusion steps used when generating samples with a pre-trained model.
        """
        self.num_inference_steps = num_inference_steps
        timesteps = np.arange(0, self.num_inference_steps)[::-1].copy()
        self.timesteps = paddle.to_tensor(timesteps)

    def step(
        self,
        model_output: paddle.Tensor,
        timestep: paddle.Tensor,
        sample: paddle.Tensor,
        generator: Optional[Union[paddle.Generator, List[paddle.Generator]]] = None,
        return_dict: bool = True,
    ) -> Union[VQDiffusionSchedulerOutput, Tuple]:
        """
        Predict the sample at the previous timestep via the reverse transition distribution i.e. Equation (11). See the
        docstring for `self.q_posterior` for more in depth docs on how Equation (11) is computed.

        Args:
            log_p_x_0: (`paddle.Tensor` of shape `(batch size, num classes - 1, num latent pixels)`):
                The log probabilities for the predicted classes of the initial latent pixels. Does not include a
                prediction for the masked class as the initial unnoised image cannot be masked.

            t (`paddle.Tensor`):
                The timestep that determines which transition matrices are used.

            x_t: (`paddle.Tensor` of shape `(batch size, num latent pixels)`):
                The classes of each latent pixel at time `t`

            generator: (`paddle.Generator` or None):
                RNG for the noise applied to p(x_{t-1} | x_t) before it is sampled from.

            return_dict (`bool`):
                option for returning tuple rather than VQDiffusionSchedulerOutput class

        Returns:
            [`~schedulers.scheduling_utils.VQDiffusionSchedulerOutput`] or `tuple`:
            [`~schedulers.scheduling_utils.VQDiffusionSchedulerOutput`] if `return_dict` is True, otherwise a `tuple`.
            When returning a tuple, the first element is the sample tensor.
        """
        if timestep == 0:
            log_p_x_t_min_1 = model_output
        else:
            log_p_x_t_min_1 = self.q_posterior(model_output, sample, timestep)

        log_p_x_t_min_1 = gumbel_noised(log_p_x_t_min_1, generator)

        x_t_min_1 = log_p_x_t_min_1.argmax(axis=1)

        if not return_dict:
            return (x_t_min_1,)

        return VQDiffusionSchedulerOutput(prev_sample=x_t_min_1)

    def q_posterior(self, log_p_x_0, x_t, t):
        """
        Calculates the log probabilities for the predicted classes of the image at timestep `t-1`. I.e. Equation (11).

        Instead of directly computing equation (11), we use Equation (5) to restate Equation (11) in terms of only
        forward probabilities.

        Equation (11) stated in terms of forward probabilities via Equation (5):

        Where:
        - the sum is over x_0 = {C_0 ... C_{k-1}} (classes for x_0)

        p(x_{t-1} | x_t) = sum( q(x_t | x_{t-1}) * q(x_{t-1} | x_0) * p(x_0) / q(x_t | x_0) )

        Args:
            log_p_x_0: (`paddle.Tensor` of shape `(batch size, num classes - 1, num latent pixels)`):
                The log probabilities for the predicted classes of the initial latent pixels. Does not include a
                prediction for the masked class as the initial unnoised image cannot be masked.

            x_t: (`paddle.Tensor` of shape `(batch size, num latent pixels)`):
                The classes of each latent pixel at time `t`

            t (paddle.Tensor):
                The timestep that determines which transition matrix is used.

        Returns:
            `paddle.Tensor` of shape `(batch size, num classes, num latent pixels)`:
                The log probabilities for the predicted classes of the image at timestep `t-1`. I.e. Equation (11).
        """
        log_onehot_x_t = index_to_log_onehot(x_t, self.num_embed)

        log_q_x_t_given_x_0 = self.log_Q_t_transitioning_to_known_class(
            t=t, x_t=x_t, log_onehot_x_t=log_onehot_x_t, cumulative=True
        )

        log_q_t_given_x_t_min_1 = self.log_Q_t_transitioning_to_known_class(
            t=t, x_t=x_t, log_onehot_x_t=log_onehot_x_t, cumulative=False
        )

        # p_0(x_0=C_0 | x_t) / q(x_t | x_0=C_0)          ...      p_n(x_0=C_0 | x_t) / q(x_t | x_0=C_0)
        #               .                    .                                   .
        #               .                            .                           .
        #               .                                      .                 .
        # p_0(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1})  ...      p_n(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1})
        q = log_p_x_0 - log_q_x_t_given_x_0

        # sum_0 = p_0(x_0=C_0 | x_t) / q(x_t | x_0=C_0) + ... + p_0(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}), ... ,
        # sum_n = p_n(x_0=C_0 | x_t) / q(x_t | x_0=C_0) + ... + p_n(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1})
        q_log_sum_exp = logsumexp(q, axis=1, keepdim=True)

        # p_0(x_0=C_0 | x_t) / q(x_t | x_0=C_0) / sum_0          ...      p_n(x_0=C_0 | x_t) / q(x_t | x_0=C_0) / sum_n
        #                        .                             .                                   .
        #                        .                                     .                           .
        #                        .                                               .                 .
        # p_0(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}) / sum_0  ...      p_n(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}) / sum_n
        q = q - q_log_sum_exp

        # (p_0(x_0=C_0 | x_t) / q(x_t | x_0=C_0) / sum_0) * a_cumulative_{t-1} + b_cumulative_{t-1}          ...      (p_n(x_0=C_0 | x_t) / q(x_t | x_0=C_0) / sum_n) * a_cumulative_{t-1} + b_cumulative_{t-1}
        #                                         .                                                .                                              .
        #                                         .                                                        .                                      .
        #                                         .                                                                  .                            .
        # (p_0(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}) / sum_0) * a_cumulative_{t-1} + b_cumulative_{t-1}  ...      (p_n(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}) / sum_n) * a_cumulative_{t-1} + b_cumulative_{t-1}
        # c_cumulative_{t-1}                                                                                 ...      c_cumulative_{t-1}
        q = self.apply_cumulative_transitions(q, t - 1)

        # ((p_0(x_0=C_0 | x_t) / q(x_t | x_0=C_0) / sum_0) * a_cumulative_{t-1} + b_cumulative_{t-1}) * q(x_t | x_{t-1}=C_0) * sum_0              ...      ((p_n(x_0=C_0 | x_t) / q(x_t | x_0=C_0) / sum_n) * a_cumulative_{t-1} + b_cumulative_{t-1}) * q(x_t | x_{t-1}=C_0) * sum_n
        #                                                            .                                                                 .                                              .
        #                                                            .                                                                         .                                      .
        #                                                            .                                                                                   .                            .
        # ((p_0(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}) / sum_0) * a_cumulative_{t-1} + b_cumulative_{t-1}) * q(x_t | x_{t-1}=C_{k-1}) * sum_0  ...      ((p_n(x_0=C_{k-1} | x_t) / q(x_t | x_0=C_{k-1}) / sum_n) * a_cumulative_{t-1} + b_cumulative_{t-1}) * q(x_t | x_{t-1}=C_{k-1}) * sum_n
        # c_cumulative_{t-1} * q(x_t | x_{t-1}=C_k) * sum_0                                                                                       ...      c_cumulative_{t-1} * q(x_t | x_{t-1}=C_k) * sum_0
        log_p_x_t_min_1 = q + log_q_t_given_x_t_min_1 + q_log_sum_exp

        # For each column, there are two possible cases.
        #
        # Where:
        # - sum(p_n(x_0))) is summing over all classes for x_0
        # - C_i is the class transitioning from (not to be confused with c_t and c_cumulative_t being used for gamma's)
        # - C_j is the class transitioning to
        #
        # 1. x_t is masked i.e. x_t = c_k
        #
        # Simplifying the expression, the column vector is:
        #                                                      .
        #                                                      .
        #                                                      .
        # (c_t / c_cumulative_t) * (a_cumulative_{t-1} * p_n(x_0 = C_i | x_t) + b_cumulative_{t-1} * sum(p_n(x_0)))
        #                                                      .
        #                                                      .
        #                                                      .
        # (c_cumulative_{t-1} / c_cumulative_t) * sum(p_n(x_0))
        #
        # From equation (11) stated in terms of forward probabilities, the last row is trivially verified.
        #
        # For the other rows, we can state the equation as ...
        #
        # (c_t / c_cumulative_t) * [b_cumulative_{t-1} * p(x_0=c_0) + ... + (a_cumulative_{t-1} + b_cumulative_{t-1}) * p(x_0=C_i) + ... + b_cumulative_{k-1} * p(x_0=c_{k-1})]
        #
        # This verifies the other rows.
        #
        # 2. x_t is not masked
        #
        # Simplifying the expression, there are two cases for the rows of the column vector, where C_j = C_i and where C_j != C_i:
        #                                                      .
        #                                                      .
        #                                                      .
        # C_j != C_i:        b_t * ((b_cumulative_{t-1} / b_cumulative_t) * p_n(x_0 = c_0) + ... + ((a_cumulative_{t-1} + b_cumulative_{t-1}) / b_cumulative_t) * p_n(x_0 = C_i) + ... + (b_cumulative_{t-1} / (a_cumulative_t + b_cumulative_t)) * p_n(c_0=C_j) + ... + (b_cumulative_{t-1} / b_cumulative_t) * p_n(x_0 = c_{k-1}))
        #                                                      .
        #                                                      .
        #                                                      .
        # C_j = C_i: (a_t + b_t) * ((b_cumulative_{t-1} / b_cumulative_t) * p_n(x_0 = c_0) + ... + ((a_cumulative_{t-1} + b_cumulative_{t-1}) / (a_cumulative_t + b_cumulative_t)) * p_n(x_0 = C_i = C_j) + ... + (b_cumulative_{t-1} / b_cumulative_t) * p_n(x_0 = c_{k-1}))
        #                                                      .
        #                                                      .
        #                                                      .
        # 0
        #
        # The last row is trivially verified. The other rows can be verified by directly expanding equation (11) stated in terms of forward probabilities.
        return log_p_x_t_min_1

    def log_Q_t_transitioning_to_known_class(
        self, *, t: paddle.Tensor, x_t: paddle.Tensor, log_onehot_x_t: paddle.Tensor, cumulative: bool
    ):
        """
        Returns the log probabilities of the rows from the (cumulative or non-cumulative) transition matrix for each
        latent pixel in `x_t`.

        See equation (7) for the complete non-cumulative transition matrix. The complete cumulative transition matrix
        is the same structure except the parameters (alpha, beta, gamma) are the cumulative analogs.

        Args:
            t (paddle.Tensor):
                The timestep that determines which transition matrix is used.

            x_t (`paddle.Tensor` of shape `(batch size, num latent pixels)`):
                The classes of each latent pixel at time `t`.

            log_onehot_x_t (`paddle.Tensor` of shape `(batch size, num classes, num latent pixels)`):
                The log one-hot vectors of `x_t`

            cumulative (`bool`):
                If cumulative is `False`, we use the single step transition matrix `t-1`->`t`. If cumulative is `True`,
                we use the cumulative transition matrix `0`->`t`.

        Returns:
            `paddle.Tensor` of shape `(batch size, num classes - 1, num latent pixels)`:
                Each _column_ of the returned matrix is a _row_ of log probabilities of the complete probability
                transition matrix.

                When non cumulative, returns `self.num_classes - 1` rows because the initial latent pixel cannot be
                masked.

                Where:
                - `q_n` is the probability distribution for the forward process of the `n`th latent pixel.
                - C_0 is a class of a latent pixel embedding
                - C_k is the class of the masked latent pixel

                non-cumulative result (omitting logarithms):
                ```
                q_0(x_t | x_{t-1} = C_0) ... q_n(x_t | x_{t-1} = C_0)
                          .      .                     .
                          .               .            .
                          .                      .     .
                q_0(x_t | x_{t-1} = C_k) ... q_n(x_t | x_{t-1} = C_k)
                ```

                cumulative result (omitting logarithms):
                ```
                q_0_cumulative(x_t | x_0 = C_0)    ...  q_n_cumulative(x_t | x_0 = C_0)
                          .               .                          .
                          .                        .                 .
                          .                               .          .
                q_0_cumulative(x_t | x_0 = C_{k-1}) ... q_n_cumulative(x_t | x_0 = C_{k-1})
                ```
        """
        if cumulative:
            a = self.log_cumprod_at[t]
            b = self.log_cumprod_bt[t]
            c = self.log_cumprod_ct[t]
        else:
            a = self.log_at[t]
            b = self.log_bt[t]
            c = self.log_ct[t]

        if not cumulative:
            # The values in the onehot vector can also be used as the logprobs for transitioning
            # from masked latent pixels. If we are not calculating the cumulative transitions,
            # we need to save these vectors to be re-appended to the final matrix so the values
            # aren't overwritten.
            #
            # `P(x_t!=mask|x_{t-1=mask}) = 0` and 0 will be the value of the last row of the onehot vector
            # if x_t is not masked
            #
            # `P(x_t=mask|x_{t-1=mask}) = 1` and 1 will be the value of the last row of the onehot vector
            # if x_t is masked
            log_onehot_x_t_transitioning_from_masked = log_onehot_x_t[:, -1, :].unsqueeze(1)

        # `index_to_log_onehot` will add onehot vectors for masked pixels,
        # so the default one hot matrix has one too many rows. See the doc string
        # for an explanation of the dimensionality of the returned matrix.
        log_onehot_x_t = log_onehot_x_t[:, :-1, :]

        # this is a cheeky trick to produce the transition probabilities using log one-hot vectors.
        #
        # Don't worry about what values this sets in the columns that mark transitions
        # to masked latent pixels. They are overwrote later with the `mask_class_mask`.
        #
        # Looking at the below logspace formula in non-logspace, each value will evaluate to either
        # `1 * a + b = a + b` where `log_Q_t` has the one hot value in the column
        # or
        # `0 * a + b = b` where `log_Q_t` has the 0 values in the column.
        #
        # See equation 7 for more details.
        log_Q_t = logaddexp(log_onehot_x_t + a, b)

        # The whole column of each masked pixel is `c`
        mask_class_mask = x_t == self.mask_class
        mask_class_mask = mask_class_mask.unsqueeze(1).expand([-1, self.num_embed - 1, -1])
        log_Q_t[mask_class_mask] = c

        if not cumulative:
            log_Q_t = paddle.concat((log_Q_t, log_onehot_x_t_transitioning_from_masked), axis=1)

        return log_Q_t

    def apply_cumulative_transitions(self, q, t):
        bsz = q.shape[0]
        a = self.log_cumprod_at[t]
        b = self.log_cumprod_bt[t]
        c = self.log_cumprod_ct[t]

        num_latent_pixels = q.shape[2]
        c = c.expand([bsz, 1, num_latent_pixels])

        q = logaddexp(q + a, b)
        q = paddle.concat((q, c), axis=1)

        return q