modules/uni_pc/uni_pc.py

# https://github.com/comfyanonymous/ComfyUI/blob/v0.3.64/comfy/extra_samplers/uni_pc.py

import math

import torch
from tqdm.auto import trange


class NoiseScheduleVP:
    def __init__(
        self,
        schedule="discrete",
        betas=None,
        alphas_cumprod=None,
        continuous_beta_0=0.1,
        continuous_beta_1=20.0,
    ):

        if schedule not in ["discrete", "linear", "cosine"]:
            raise ValueError(f"Unsupported noise schedule {schedule}. The schedule needs to be 'discrete' or 'linear' or 'cosine'")

        self.schedule = schedule
        if schedule == "discrete":
            if betas is not None:
                log_alphas = 0.5 * torch.log(1 - betas).cumsum(dim=0)
            else:
                assert alphas_cumprod is not None
                log_alphas = 0.5 * torch.log(alphas_cumprod)
            self.total_N = len(log_alphas)
            self.T = 1.0
            self.t_array = torch.linspace(0.0, 1.0, self.total_N + 1)[1:].reshape((1, -1))
            self.log_alpha_array = log_alphas.reshape((1, -1))
        else:
            self.total_N = 1000
            self.beta_0 = continuous_beta_0
            self.beta_1 = continuous_beta_1
            self.cosine_s = 0.008
            self.cosine_beta_max = 999.0
            self.cosine_t_max = math.atan(self.cosine_beta_max * (1.0 + self.cosine_s) / math.pi) * 2.0 * (1.0 + self.cosine_s) / math.pi - self.cosine_s
            self.cosine_log_alpha_0 = math.log(math.cos(self.cosine_s / (1.0 + self.cosine_s) * math.pi / 2.0))
            self.schedule = schedule
            if schedule == "cosine":
                self.T = 0.9946
            else:
                self.T = 1.0

    def marginal_log_mean_coeff(self, t):
        """
        Compute log(alpha_t) of a given continuous-time label t in [0, T]
        """
        if self.schedule == "discrete":
            return interpolate_fn(t.reshape((-1, 1)), self.t_array.to(t.device), self.log_alpha_array.to(t.device)).reshape((-1))
        elif self.schedule == "linear":
            return -0.25 * t**2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
        elif self.schedule == "cosine":
            log_alpha_fn = lambda s: torch.log(torch.cos((s + self.cosine_s) / (1.0 + self.cosine_s) * math.pi / 2.0))
            log_alpha_t = log_alpha_fn(t) - self.cosine_log_alpha_0
            return log_alpha_t

    def marginal_alpha(self, t):
        """
        Compute alpha_t of a given continuous-time label t in [0, T]
        """
        return torch.exp(self.marginal_log_mean_coeff(t))

    def marginal_std(self, t):
        """
        Compute sigma_t of a given continuous-time label t in [0, T]
        """
        return torch.sqrt(1.0 - torch.exp(2.0 * self.marginal_log_mean_coeff(t)))

    def marginal_lambda(self, t):
        """
        Compute lambda_t = log(alpha_t) - log(sigma_t) of a given continuous-time label t in [0, T]
        """
        log_mean_coeff = self.marginal_log_mean_coeff(t)
        log_std = 0.5 * torch.log(1.0 - torch.exp(2.0 * log_mean_coeff))
        return log_mean_coeff - log_std

    def inverse_lambda(self, lamb):
        """
        Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t
        """
        if self.schedule == "linear":
            tmp = 2.0 * (self.beta_1 - self.beta_0) * torch.logaddexp(-2.0 * lamb, torch.zeros((1,)).to(lamb))
            Delta = self.beta_0**2 + tmp
            return tmp / (torch.sqrt(Delta) + self.beta_0) / (self.beta_1 - self.beta_0)
        elif self.schedule == "discrete":
            log_alpha = -0.5 * torch.logaddexp(torch.zeros((1,)).to(lamb.device), -2.0 * lamb)
            t = interpolate_fn(log_alpha.reshape((-1, 1)), torch.flip(self.log_alpha_array.to(lamb.device), [1]), torch.flip(self.t_array.to(lamb.device), [1]))
            return t.reshape((-1,))
        else:
            log_alpha = -0.5 * torch.logaddexp(-2.0 * lamb, torch.zeros((1,)).to(lamb))
            t_fn = lambda log_alpha_t: torch.arccos(torch.exp(log_alpha_t + self.cosine_log_alpha_0)) * 2.0 * (1.0 + self.cosine_s) / math.pi - self.cosine_s
            t = t_fn(log_alpha)
            return t


def model_wrapper(
    model,
    noise_schedule,
    model_type="noise",
    model_kwargs={},
    guidance_type="uncond",
    condition=None,
    unconditional_condition=None,
    guidance_scale=1.0,
    classifier_fn=None,
    classifier_kwargs={},
):

    def get_model_input_time(t_continuous):
        """
        Convert the continuous-time `t_continuous` (in [epsilon, T]) to the model input time.
        For discrete-time DPMs, we convert `t_continuous` in [1 / N, 1] to `t_input` in [0, 1000 * (N - 1) / N].
        For continuous-time DPMs, we just use `t_continuous`.
        """
        if noise_schedule.schedule == "discrete":
            return (t_continuous - 1.0 / noise_schedule.total_N) * 1000.0
        else:
            return t_continuous

    def noise_pred_fn(x, t_continuous, cond=None):
        if t_continuous.reshape((-1,)).shape[0] == 1:
            t_continuous = t_continuous.expand((x.shape[0]))
        t_input = get_model_input_time(t_continuous)
        output = model(x, t_input, **model_kwargs)
        if model_type == "noise":
            return output
        elif model_type == "x_start":
            alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
            dims = x.dim()
            return (x - expand_dims(alpha_t, dims) * output) / expand_dims(sigma_t, dims)
        elif model_type == "v":
            alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
            dims = x.dim()
            return expand_dims(alpha_t, dims) * output + expand_dims(sigma_t, dims) * x
        elif model_type == "score":
            sigma_t = noise_schedule.marginal_std(t_continuous)
            dims = x.dim()
            return -expand_dims(sigma_t, dims) * output

    def cond_grad_fn(x, t_input):
        """
        Compute the gradient of the classifier, i.e. nabla_{x} log p_t(cond | x_t)
        """
        with torch.enable_grad():
            x_in = x.detach().requires_grad_(True)
            log_prob = classifier_fn(x_in, t_input, condition, **classifier_kwargs)
            return torch.autograd.grad(log_prob.sum(), x_in)[0]

    def model_fn(x, t_continuous):
        """
        The noise prediction model function that is used for DPM-Solver
        """
        if t_continuous.reshape((-1,)).shape[0] == 1:
            t_continuous = t_continuous.expand((x.shape[0]))
        if guidance_type == "uncond":
            return noise_pred_fn(x, t_continuous)
        elif guidance_type == "classifier":
            assert classifier_fn is not None
            t_input = get_model_input_time(t_continuous)
            cond_grad = cond_grad_fn(x, t_input)
            sigma_t = noise_schedule.marginal_std(t_continuous)
            noise = noise_pred_fn(x, t_continuous)
            return noise - guidance_scale * expand_dims(sigma_t, dims=cond_grad.dim()) * cond_grad
        elif guidance_type == "classifier-free":
            if guidance_scale == 1.0 or unconditional_condition is None:
                return noise_pred_fn(x, t_continuous, cond=condition)
            else:
                x_in = torch.cat([x] * 2)
                t_in = torch.cat([t_continuous] * 2)
                c_in = torch.cat([unconditional_condition, condition])
                noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in).chunk(2)
                return noise_uncond + guidance_scale * (noise - noise_uncond)

    assert model_type in ["noise", "x_start", "v"]
    assert guidance_type in ["uncond", "classifier", "classifier-free"]
    return model_fn


class UniPC:
    def __init__(self, model_fn, noise_schedule, predict_x0=True, thresholding=False, max_val=1.0, variant="bh1"):
        """
        Construct a UniPC
        We support both data_prediction and noise_prediction
        """
        self.model = model_fn
        self.noise_schedule = noise_schedule
        self.variant = variant
        self.predict_x0 = predict_x0
        self.thresholding = thresholding
        self.max_val = max_val

    def dynamic_thresholding_fn(self, x0, t=None):
        """
        The dynamic thresholding method
        """
        dims = x0.dim()
        p = self.dynamic_thresholding_ratio
        s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1)
        s = expand_dims(torch.maximum(s, self.thresholding_max_val * torch.ones_like(s).to(s.device)), dims)
        x0 = torch.clamp(x0, -s, s) / s
        return x0

    def noise_prediction_fn(self, x, t):
        """
        Return the noise prediction model
        """
        return self.model(x, t)

    def data_prediction_fn(self, x, t):
        """
        Return the data prediction model (with thresholding)
        """
        noise = self.noise_prediction_fn(x, t)
        dims = x.dim()
        alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t)
        x0 = (x - expand_dims(sigma_t, dims) * noise) / expand_dims(alpha_t, dims)
        if self.thresholding:
            p = 0.995
            s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1)
            s = expand_dims(torch.maximum(s, self.max_val * torch.ones_like(s).to(s.device)), dims)
            x0 = torch.clamp(x0, -s, s) / s
        return x0

    def model_fn(self, x, t):
        """
        Convert the model to the noise prediction model or the data prediction model
        """
        if self.predict_x0:
            return self.data_prediction_fn(x, t)
        else:
            return self.noise_prediction_fn(x, t)

    def get_time_steps(self, skip_type, t_T, t_0, N, device):
        """Compute the intermediate time steps for sampling"""
        if skip_type == "logSNR":
            lambda_T = self.noise_schedule.marginal_lambda(torch.tensor(t_T).to(device))
            lambda_0 = self.noise_schedule.marginal_lambda(torch.tensor(t_0).to(device))
            logSNR_steps = torch.linspace(lambda_T.cpu().item(), lambda_0.cpu().item(), N + 1).to(device)
            return self.noise_schedule.inverse_lambda(logSNR_steps)
        elif skip_type == "time_uniform":
            return torch.linspace(t_T, t_0, N + 1).to(device)
        elif skip_type == "time_quadratic":
            t_order = 2
            t = torch.linspace(t_T ** (1.0 / t_order), t_0 ** (1.0 / t_order), N + 1).pow(t_order).to(device)
            return t
        else:
            raise ValueError(f"Unsupported skip_type {skip_type}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'")

    def get_orders_and_timesteps_for_singlestep_solver(self, steps, order, skip_type, t_T, t_0, device):
        """
        Get the order of each step for sampling by the singlestep DPM-Solver
        """
        if order == 3:
            K = steps // 3 + 1
            if steps % 3 == 0:
                orders = [3] * (K - 2) + [2, 1]
            elif steps % 3 == 1:
                orders = [3] * (K - 1) + [1]
            else:
                orders = [3] * (K - 1) + [2]
        elif order == 2:
            if steps % 2 == 0:
                K = steps // 2
                orders = [2] * K
            else:
                K = steps // 2 + 1
                orders = [2] * (K - 1) + [1]
        elif order == 1:
            K = steps
            orders = [1] * steps
        else:
            raise ValueError("'order' must be '1' or '2' or '3'.")
        if skip_type == "logSNR":
            timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, K, device)
        else:
            timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, steps, device)[torch.cumsum(torch.tensor([0] + orders), 0).to(device)]
        return timesteps_outer, orders

    def denoise_to_zero_fn(self, x, s):
        """
        Denoise at the final step, which is equivalent to solve the ODE from lambda_s to infty by first-order discretization
        """
        return self.data_prediction_fn(x, s)

    def multistep_uni_pc_update(self, x, model_prev_list, t_prev_list, t, order, **kwargs):
        if len(t.shape) == 0:
            t = t.view(-1)
        if "bh" in self.variant:
            return self.multistep_uni_pc_bh_update(x, model_prev_list, t_prev_list, t, order, **kwargs)
        else:
            assert self.variant == "vary_coeff"
            return self.multistep_uni_pc_vary_update(x, model_prev_list, t_prev_list, t, order, **kwargs)

    def multistep_uni_pc_vary_update(self, x, model_prev_list, t_prev_list, t, order, use_corrector=True):
        ns = self.noise_schedule
        assert order <= len(model_prev_list)

        t_prev_0 = t_prev_list[-1]
        lambda_prev_0 = ns.marginal_lambda(t_prev_0)
        lambda_t = ns.marginal_lambda(t)
        model_prev_0 = model_prev_list[-1]
        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
        log_alpha_t = ns.marginal_log_mean_coeff(t)
        alpha_t = torch.exp(log_alpha_t)

        h = lambda_t - lambda_prev_0

        rks = []
        D1s = []
        for i in range(1, order):
            t_prev_i = t_prev_list[-(i + 1)]
            model_prev_i = model_prev_list[-(i + 1)]
            lambda_prev_i = ns.marginal_lambda(t_prev_i)
            rk = (lambda_prev_i - lambda_prev_0) / h
            rks.append(rk)
            D1s.append((model_prev_i - model_prev_0) / rk)

        rks.append(1.0)
        rks = torch.tensor(rks, device=x.device)

        K = len(rks)
        C = []

        col = torch.ones_like(rks)
        for k in range(1, K + 1):
            C.append(col)
            col = col * rks / (k + 1)
        C = torch.stack(C, dim=1)

        if len(D1s) > 0:
            D1s = torch.stack(D1s, dim=1)
            C_inv_p = torch.linalg.inv(C[:-1, :-1])
            A_p = C_inv_p

        if use_corrector:
            C_inv = torch.linalg.inv(C)
            A_c = C_inv

        hh = -h if self.predict_x0 else h
        h_phi_1 = torch.expm1(hh)
        h_phi_ks = []
        factorial_k = 1
        h_phi_k = h_phi_1
        for k in range(1, K + 2):
            h_phi_ks.append(h_phi_k)
            h_phi_k = h_phi_k / hh - 1 / factorial_k
            factorial_k *= k + 1

        model_t = None
        if self.predict_x0:
            x_t_ = sigma_t / sigma_prev_0 * x - alpha_t * h_phi_1 * model_prev_0
            x_t = x_t_
            if len(D1s) > 0:
                for k in range(K - 1):
                    x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum("bkchw,k->bchw", D1s, A_p[k])
            if use_corrector:
                model_t = self.model_fn(x_t, t)
                D1_t = model_t - model_prev_0
                x_t = x_t_
                k = 0
                for k in range(K - 1):
                    x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum("bkchw,k->bchw", D1s, A_c[k][:-1])
                x_t = x_t - alpha_t * h_phi_ks[K] * (D1_t * A_c[k][-1])
        else:
            log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
            x_t_ = (torch.exp(log_alpha_t - log_alpha_prev_0)) * x - (sigma_t * h_phi_1) * model_prev_0
            x_t = x_t_
            if len(D1s) > 0:
                for k in range(K - 1):
                    x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum("bkchw,k->bchw", D1s, A_p[k])
            if use_corrector:
                model_t = self.model_fn(x_t, t)
                D1_t = model_t - model_prev_0
                x_t = x_t_
                k = 0
                for k in range(K - 1):
                    x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum("bkchw,k->bchw", D1s, A_c[k][:-1])
                x_t = x_t - sigma_t * h_phi_ks[K] * (D1_t * A_c[k][-1])
        return x_t, model_t

    def multistep_uni_pc_bh_update(self, x, model_prev_list, t_prev_list, t, order, x_t=None, use_corrector=True):
        ns = self.noise_schedule
        assert order <= len(model_prev_list)
        dims = x.dim()

        t_prev_0 = t_prev_list[-1]
        lambda_prev_0 = ns.marginal_lambda(t_prev_0)
        lambda_t = ns.marginal_lambda(t)
        model_prev_0 = model_prev_list[-1]
        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
        log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
        alpha_t = torch.exp(log_alpha_t)

        h = lambda_t - lambda_prev_0

        rks = []
        D1s = []
        for i in range(1, order):
            t_prev_i = t_prev_list[-(i + 1)]
            model_prev_i = model_prev_list[-(i + 1)]
            lambda_prev_i = ns.marginal_lambda(t_prev_i)
            rk = ((lambda_prev_i - lambda_prev_0) / h)[0]
            rks.append(rk)
            D1s.append((model_prev_i - model_prev_0) / rk)

        rks.append(1.0)
        rks = torch.tensor(rks, device=x.device)

        R = []
        b = []

        hh = -h[0] if self.predict_x0 else h[0]
        h_phi_1 = torch.expm1(hh)
        h_phi_k = h_phi_1 / hh - 1

        factorial_i = 1

        if self.variant == "bh1":
            B_h = hh
        elif self.variant == "bh2":
            B_h = torch.expm1(hh)
        else:
            raise NotImplementedError()

        for i in range(1, order + 1):
            R.append(torch.pow(rks, i - 1))
            b.append(h_phi_k * factorial_i / B_h)
            factorial_i *= i + 1
            h_phi_k = h_phi_k / hh - 1 / factorial_i

        R = torch.stack(R)
        b = torch.tensor(b, device=x.device)

        use_predictor = len(D1s) > 0 and x_t is None
        if len(D1s) > 0:
            D1s = torch.stack(D1s, dim=1)
            if x_t is None:
                if order == 2:
                    rhos_p = torch.tensor([0.5], device=b.device)
                else:
                    rhos_p = torch.linalg.solve(R[:-1, :-1], b[:-1])
        else:
            D1s = None

        if use_corrector:
            if order == 1:
                rhos_c = torch.tensor([0.5], device=b.device)
            else:
                rhos_c = torch.linalg.solve(R, b)

        model_t = None
        if self.predict_x0:
            x_t_ = expand_dims(sigma_t / sigma_prev_0, dims) * x - expand_dims(alpha_t * h_phi_1, dims) * model_prev_0

            if x_t is None:
                if use_predictor:
                    pred_res = torch.tensordot(D1s, rhos_p, dims=([1], [0]))
                else:
                    pred_res = 0
                x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * pred_res

            if use_corrector:
                model_t = self.model_fn(x_t, t)
                if D1s is not None:
                    corr_res = torch.tensordot(D1s, rhos_c[:-1], dims=([1], [0]))
                else:
                    corr_res = 0
                D1_t = model_t - model_prev_0
                x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * (corr_res + rhos_c[-1] * D1_t)
        else:
            x_t_ = expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x - expand_dims(sigma_t * h_phi_1, dims) * model_prev_0
            if x_t is None:
                if use_predictor:
                    pred_res = torch.tensordot(D1s, rhos_p, dims=([1], [0]))
                else:
                    pred_res = 0
                x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * pred_res

            if use_corrector:
                model_t = self.model_fn(x_t, t)
                if D1s is not None:
                    corr_res = torch.tensordot(D1s, rhos_c[:-1], dims=([1], [0]))
                else:
                    corr_res = 0
                D1_t = model_t - model_prev_0
                x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * (corr_res + rhos_c[-1] * D1_t)
        return x_t, model_t

    def sample(
        self,
        x,
        timesteps,
        t_start=None,
        t_end=None,
        order=3,
        skip_type="time_uniform",
        method="singlestep",
        lower_order_final=True,
        denoise_to_zero=False,
        solver_type="dpm_solver",
        atol=0.0078,
        rtol=0.05,
        corrector=False,
        callback=None,
        disable_pbar=False,
    ):
        steps = len(timesteps) - 1
        if method == "multistep":
            assert steps >= order
            assert timesteps.shape[0] - 1 == steps
            for step_index in trange(steps, disable=disable_pbar):
                if step_index == 0:
                    vec_t = timesteps[0].expand((x.shape[0]))
                    model_prev_list = [self.model_fn(x, vec_t)]
                    t_prev_list = [vec_t]
                elif step_index < order:
                    init_order = step_index
                    vec_t = timesteps[init_order].expand(x.shape[0])
                    x, model_x = self.multistep_uni_pc_update(x, model_prev_list, t_prev_list, vec_t, init_order, use_corrector=True)
                    if model_x is None:
                        model_x = self.model_fn(x, vec_t)
                    model_prev_list.append(model_x)
                    t_prev_list.append(vec_t)
                else:
                    extra_final_step = 0
                    if step_index == (steps - 1):
                        extra_final_step = 1
                    for step in range(step_index, step_index + 1 + extra_final_step):
                        vec_t = timesteps[step].expand(x.shape[0])
                        if lower_order_final:
                            step_order = min(order, steps + 1 - step)
                        else:
                            step_order = order
                        if step == steps:
                            use_corrector = False
                        else:
                            use_corrector = True
                        x, model_x = self.multistep_uni_pc_update(x, model_prev_list, t_prev_list, vec_t, step_order, use_corrector=use_corrector)
                        for i in range(order - 1):
                            t_prev_list[i] = t_prev_list[i + 1]
                            model_prev_list[i] = model_prev_list[i + 1]
                        t_prev_list[-1] = vec_t
                        if step < steps:
                            if model_x is None:
                                model_x = self.model_fn(x, vec_t)
                            model_prev_list[-1] = model_x
                if callback is not None:
                    callback({"x": x, "i": step_index, "denoised": model_prev_list[-1]})
        else:
            raise NotImplementedError()
        return x


#############################################################
# other utility functions
#############################################################


def interpolate_fn(x, xp, yp):
    N, K = x.shape[0], xp.shape[1]
    all_x = torch.cat([x.unsqueeze(2), xp.unsqueeze(0).repeat((N, 1, 1))], dim=2)
    sorted_all_x, x_indices = torch.sort(all_x, dim=2)
    x_idx = torch.argmin(x_indices, dim=2)
    cand_start_idx = x_idx - 1
    start_idx = torch.where(
        torch.eq(x_idx, 0),
        torch.tensor(1, device=x.device),
        torch.where(
            torch.eq(x_idx, K),
            torch.tensor(K - 2, device=x.device),
            cand_start_idx,
        ),
    )
    end_idx = torch.where(torch.eq(start_idx, cand_start_idx), start_idx + 2, start_idx + 1)
    start_x = torch.gather(sorted_all_x, dim=2, index=start_idx.unsqueeze(2)).squeeze(2)
    end_x = torch.gather(sorted_all_x, dim=2, index=end_idx.unsqueeze(2)).squeeze(2)
    start_idx2 = torch.where(
        torch.eq(x_idx, 0),
        torch.tensor(0, device=x.device),
        torch.where(
            torch.eq(x_idx, K),
            torch.tensor(K - 2, device=x.device),
            cand_start_idx,
        ),
    )
    y_positions_expanded = yp.unsqueeze(0).expand(N, -1, -1)
    start_y = torch.gather(y_positions_expanded, dim=2, index=start_idx2.unsqueeze(2)).squeeze(2)
    end_y = torch.gather(y_positions_expanded, dim=2, index=(start_idx2 + 1).unsqueeze(2)).squeeze(2)
    cand = start_y + (x - start_x) * (end_y - start_y) / (end_x - start_x)
    return cand


def expand_dims(v, dims):
    return v[(...,) + (None,) * (dims - 1)]