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<!DOCTYPE html>

<head>

<title>ot.utils — POT Python Optimal Transport 0.9.7 documentation</title>

</head>

POT Python Optimal Transport

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<li class="breadcrumb-item"><a href="../index.html">Module code</a></li>

<li class="breadcrumb-item active">ot.utils</li>

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<h1>Source code for ot.utils</h1><div class="highlight"><pre>

# -*- coding: utf-8 -*-

"""

Various useful functions

"""

# Author: Remi Flamary <remi.flamary@unice.fr>

#

# License: MIT License

from functools import reduce

import time

import numpy as np

from scipy.spatial.distance import cdist

import sys

import warnings

from inspect import signature

from .backend import get_backend, Backend, NumpyBackend, JaxBackend

__time_tic_toc = time.perf_counter()

def tic():

r"""Python implementation of Matlab tic() function"""

global __time_tic_toc

__time_tic_toc = time.perf_counter()</div>

def toc(message="Elapsed time : {} s"):

r"""Python implementation of Matlab toc() function"""

t = time.perf_counter()

print(message.format(t - __time_tic_toc))

return t - __time_tic_toc</div>

def toq():

r"""Python implementation of Julia toc() function"""

t = time.perf_counter()

return t - __time_tic_toc</div>

def kernel(x1, x2, method="gaussian", sigma=1, **kwargs):

r"""Compute kernel matrix"""

nx = get_backend(x1, x2)

if method.lower() in ["gaussian", "gauss", "rbf"]:

K = nx.exp(-dist(x1, x2) / (2 * sigma**2))

return K</div>

def laplacian(x):

r"""Compute Laplacian matrix"""

nx = get_backend(x)

L = nx.diag(nx.sum(x, axis=0)) - x

return L</div>

def list_to_array(*lst, nx=None):

r"""Convert a list if in numpy format"""

# Filter to non-empty, non-list items (handle sparse matrices gracefully)

lst_not_empty = []

for a in lst:

if isinstance(a, list):

continue

try:

if len(a) > 0:

lst_not_empty.append(a)

except TypeError:

# Sparse matrices don't support len(), but they're not empty

lst_not_empty.append(a)

if nx is None: # find backend

if len(lst_not_empty) == 0:

type_as = np.zeros(0)

nx = get_backend(type_as)

else:

nx = get_backend(*lst_not_empty)

type_as = lst_not_empty[0]

else:

type_as = None

else:

type_as = lst_not_empty[0]

if len(lst) > 1:

return [

nx.from_numpy(np.array(a), type_as=type_as) if isinstance(a, list) else a

for a in lst

]

else:

if isinstance(lst[0], list):

return nx.from_numpy(np.array(lst[0]), type_as=type_as)

else:

return lst[0]</div>

def proj_simplex(v, z=1):

r"""Compute the closest point (orthogonal projection) on the

generalized `(n-1)`-simplex of a vector :math:`\mathbf{v}` wrt. to the Euclidean

distance, thus solving:

.. math::

\mathcal{P}(w) \in \mathop{\arg \min}_\gamma \| \gamma - \mathbf{v} \|_2

s.t. \ \gamma^T \mathbf{1} = z

\gamma \geq 0

If :math:`\mathbf{v}` is a 2d array, compute all the projections wrt. axis 0

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends.

Parameters

----------

v : {array-like}, shape (n, d)

z : int, optional

'size' of the simplex (each vectors sum to z, 1 by default)

Returns

-------

h : ndarray, shape (`n`, `d`)

Array of projections on the simplex

"""

nx = get_backend(v)

n = v.shape[0]

if v.ndim == 1:

d1 = 1

v = v[:, None]

else:

d1 = 0

d = v.shape[1]

# sort u in ascending order

u = nx.sort(v, axis=0)

# take the descending order

u = nx.flip(u, 0)

cssv = nx.cumsum(u, axis=0) - z

ind = nx.arange(n, type_as=v)[:, None] + 1

cond = u - cssv / ind > 0

rho = nx.sum(cond, 0)

theta = cssv[rho - 1, nx.arange(d)] / rho

w = nx.maximum(v - theta[None, :], nx.zeros(v.shape, type_as=v))

if d1:

return w[:, 0]

else:

return w</div>

def projection_sparse_simplex(V, max_nz, z=1, axis=None, nx=None):

r"""Projection of :math:`\mathbf{V}` onto the simplex with cardinality constraint (maximum number of non-zero elements) and then scaled by `z`.

.. math::

P\left(\mathbf{V}, \text{max_nz}, z\right) = \mathop{\arg \min}_{\substack{\mathbf{y} >= 0 \\ \sum_i \mathbf{y}_i = z} \\ ||p||_0 \le \text{max_nz}} \quad \|\mathbf{y} - \mathbf{V}\|^2

Parameters

----------

V: 1-dim or 2-dim ndarray

max_nz: int

Maximum number of non-zero elements in the projection.

If `max_nz` is larger than the number of elements in `V`, then

the projection is equivalent to `proj_simplex(V, z)`.

z: float or array

If array, len(z) must be compatible with :math:`\mathbf{V}`

axis: None or int

- axis=None: project :math:`\mathbf{V}` by :math:`P(\mathbf{V}.\mathrm{ravel}(), \text{max_nz}, z)`

- axis=1: project each :math:`\mathbf{V}_i` by :math:`P(\mathbf{V}_i, \text{max_nz}, z_i)`

- axis=0: project each :math:`\mathbf{V}_{:, j}` by :math:`P(\mathbf{V}_{:, j}, \text{max_nz}, z_j)`

Returns

-------

projection: ndarray, shape :math:`\mathbf{V}`.shape

References

----------

.. [1] Sparse projections onto the simplex

Anastasios Kyrillidis, Stephen Becker, Volkan Cevher and, Christoph Koch

ICML 2013

https://arxiv.org/abs/1206.1529

"""

if nx is None:

nx = get_backend(V)

if V.ndim == 1:

return projection_sparse_simplex(

# V[nx.newaxis, :], max_nz, z, axis=1).ravel()

V[None, :],

max_nz,

z,

axis=1,

).ravel()

if V.ndim > 2:

raise ValueError("V.ndim must be <= 2")

if axis == 1:

# For each row of V, find top max_nz values; arrange the

# corresponding column indices such that their values are

# in a descending order.

max_nz_indices = nx.argsort(V, axis=1)[:, -max_nz:]

max_nz_indices = nx.flip(max_nz_indices, axis=1)

row_indices = nx.arange(V.shape[0])

row_indices = row_indices.reshape(-1, 1)

print(row_indices.shape)

# Extract the top max_nz values for each row

# and then project to simplex.

U = V[row_indices, max_nz_indices]

z = nx.ones(len(U)) * z

cssv = nx.cumsum(U, axis=1) - z[:, None]

cond = U - cssv / ind > 0

# rho = nx.count_nonzero(cond, axis=1)

theta = cssv[nx.arange(len(U)), rho - 1] / rho

nz_projection = nx.maximum(U - theta[:, None], 0)

# Put the projection of max_nz_values to their original column indices

# while keeping other values zero.

sparse_projection = nx.zeros(V.shape, type_as=nz_projection)

if isinstance(nx, JaxBackend):

# in Jax, we need to use the `at` property of `jax.numpy.ndarray`

# to do in-place array modifications.

sparse_projection = sparse_projection.at[row_indices, max_nz_indices].set(

nz_projection

)

else:

sparse_projection[row_indices, max_nz_indices] = nz_projection

return sparse_projection

elif axis == 0:

return projection_sparse_simplex(V.T, max_nz, z, axis=1).T

else:

V = V.ravel().reshape(1, -1)

return projection_sparse_simplex(V, max_nz, z, axis=1).ravel()</div>

def unif(n, type_as=None):

r"""

Return a uniform histogram of length `n` (simplex).

Parameters

----------

n : int

number of bins in the histogram

type_as : array-like

array of the same type of the expected output (numpy/pytorch/jax)

Returns

-------

h : array-like, shape (n,)

histogram of length `n` such that :math:`\forall i, \mathbf{h}_i = \frac{1}{n}`

"""

if type_as is None:

return np.ones((n,)) / n

else:

nx = get_backend(type_as)

return nx.ones((n,), type_as=type_as) / n</div>

def clean_zeros(a, b, M):

r"""Remove all components with zeros weights in :math:`\mathbf{a}` and :math:`\mathbf{b}`"""

M2 = M[a > 0, :][:, b > 0].copy() # copy force c style matrix (froemd)

a2 = a[a > 0]

b2 = b[b > 0]

return a2, b2, M2</div>

def euclidean_distances(X, Y, squared=False, nx=None):

r"""

Considering the rows of :math:`\mathbf{X}` (and :math:`\mathbf{Y} = \mathbf{X}`) as vectors, compute the

distance matrix between each pair of vectors.

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends.

Parameters

----------

X : array-like, shape (n_samples_1, n_features)

Y : array-like, shape (n_samples_2, n_features)

squared : boolean, optional

Return squared Euclidean distances.

Returns

-------

distances : array-like, shape (`n_samples_1`, `n_samples_2`)

"""

if nx is None:

nx = get_backend(X, Y)

a2 = nx.einsum("ij,ij->i", X, X)

b2 = nx.einsum("ij,ij->i", Y, Y)

c = -2 * nx.dot(X, nx.transpose(Y))

c += a2[:, None]

c += b2[None, :]

c = nx.maximum(c, 0)

if not squared:

c = nx.sqrt(c)

if X is Y:

c = c * (1 - nx.eye(X.shape[0], type_as=c))

return c</div>

def sparse_ot_dist(

x1,

x2,

i,

j,

w=None,

metric="sqeuclidean",

p=2,

batch_size=None,

):

r"""Compute ot distance between samples in :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`

with sparse weights given by `w` for the pairs of samples with indices `i` and `j`.

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends for the following metrics:

'sqeuclidean', 'euclidean', 'cityblock', 'minkowski'.

Parameters

----------

x1 : array-like, shape (n1,d)

matrix with `n1` samples of size `d`

x2 : array-like, shape (n2,d), optional

matrix with `n2` samples of size `d` (if None then :math:`\mathbf{x_2} = \mathbf{x_1}`)

i : array-like, shape (k,)

indices of samples in `x1` to compute distance from

j : array-like, shape (k,)

indices of samples in `x2` to compute distance to

w : array-like, shape (k,), optional

weights for each pair of samples to compute distance between.

If None, all pairs are weighted equally (=1/k).

metric : str | callable, optional

'sqeuclidean', 'euclidean', 'cityblock' or 'minkowski'.

p : float, optional

p-norm for the Minkowski metric. Default value is 2.

batch_size : int, optional

If specified, compute the distance in batches of size `batch_size` to avoid memory issues for large datasets. Default is None (no batching).

Returns

-------

dist : float

sum of the distance between :math:`\mathbf{x_1}_i` and :math:`\mathbf{x_2}_j` computed with given metric and weighted by `w`

"""

assert x1.ndim == 2, f"x1 must be a 2d array, got {x1.ndim}d array instead"

assert x2.ndim == 2, f"x2 must be a 2d array, got {x2.ndim}d array instead"

assert len(i) == len(

j

), f"i and j must have the same length, got {len(i)} and {len(j)}"

if w is not None:

assert len(w) == len(

i

), f"w must have the same length as i and j, got {len(w)} and {len(i)}"

assert metric in ("minkowski", "euclidean", "cityblock", "sqeuclidean"), (

"sparse_dist work only with metrics from the following list: "

+ "`['sqeuclidean', 'minkowski', 'cityblock', 'euclidean']`"

)

assert (

x1.shape[1] == x2.shape[1]

), f"x1 ({x1.shape}) and x2 ({x2.shape}) must have the same number of columns"

if metric == "euclidean":

p = 2

elif metric == "cityblock":

p = 1

def dist_idxs(idx_x1, idx_x2):

if metric == "sqeuclidean":

return nx.sum((x1[idx_x1] - x2[idx_x2]) ** 2, axis=1)

else:

return nx.sum(nx.abs(x1[idx_x1] - x2[idx_x2]) ** p, axis=1) ** (1 / p)

if w is None:

w = nx.ones(len(i), type_as=x1) / len(i)

d = 0

if batch_size is None:

batch_size = len(i)

for b in range(0, len(i), batch_size):

d += nx.sum(

dist_idxs(i[b : b + batch_size], j[b : b + batch_size])

* w[i[b : b + batch_size]]

)

return d</div>

def dist(

x1,

x2=None,

metric="sqeuclidean",

p=2,

w=None,

backend="auto",

nx=None,

use_tensor=False,

):

r"""Compute distance between samples in :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends for the following metrics:

'sqeuclidean', 'euclidean', 'cityblock', 'minkowski', 'cosine', 'correlation'.

Parameters

----------

x1 : array-like, shape (n1,d)

matrix with `n1` samples of size `d`

x2 : array-like, shape (n2,d), optional

matrix with `n2` samples of size `d` (if None then :math:`\mathbf{x_2} = \mathbf{x_1}`)

metric : str | callable, optional

'sqeuclidean' or 'euclidean' on all backends. On numpy the function also

accepts from the scipy.spatial.distance.cdist function : 'braycurtis',

'canberra', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice',

'euclidean', 'hamming', 'jaccard', 'kulczynski1', 'mahalanobis',

'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean',

'sokalmichener', 'sokalsneath', 'sqeuclidean', 'wminkowski', 'yule'.

p : float, optional

p-norm for the Minkowski and the Weighted Minkowski metrics. Default value is 2.

w : array-like, rank 1

Weights for the weighted metrics.

backend : str, optional

Backend to use for the computation. If 'auto', the backend is

automatically selected based on the input data. if 'scipy',

the ``scipy.spatial.distance.cdist`` function is used (and gradients are

detached).

use_tensor : bool, optional

If true use tensorized computation for the distance matrix which can

cause memory issues for large datasets. Default is False and the

parameter is used only for the 'cityblock' and 'minkowski' metrics.

nx : Backend, optional

Backend to perform computations on. If omitted, the backend defaults to that of `x1`.

Returns

-------

M : array-like, shape (`n1`, `n2`)

distance matrix computed with given metric

"""

if nx is None:

if x2 is None:

x2 = x1

if backend == "scipy": # force scipy backend with cdist function

x1 = nx.to_numpy(x1)

x2 = nx.to_numpy(x2)

if isinstance(metric, str) and metric.endswith("minkowski"):

return nx.from_numpy(cdist(x1, x2, metric=metric, p=p, w=w))

if w is not None:

return nx.from_numpy(cdist(x1, x2, metric=metric, w=w))

elif metric == "sqeuclidean":

return euclidean_distances(x1, x2, squared=True, nx=nx)

elif metric == "euclidean":

return euclidean_distances(x1, x2, squared=False, nx=nx)

elif metric == "cityblock":

if use_tensor:

return nx.sum(nx.abs(x1[:, None, :] - x2[None, :, :]), axis=2)

else:

M = 0.0

for i in range(x1.shape[1]):

M += nx.abs(x1[:, i][:, None] - x2[:, i][None, :])

return M

elif metric == "minkowski":

if w is None:

if use_tensor:

return nx.power(

nx.sum(

nx.power(nx.abs(x1[:, None, :] - x2[None, :, :]), p), axis=2

),

1 / p,

)

else:

M = 0.0

return M ** (1 / p)

else:

if use_tensor:

return nx.power(

nx.sum(

w[None, None, :]

* nx.power(nx.abs(x1[:, None, :] - x2[None, :, :]), p),

axis=2,

),

1 / p,

)

else:

M = 0.0

M += w[i] * nx.abs(x1[:, i][:, None] - x2[:, i][None, :]) ** p

elif metric == "cosine":

nx1 = nx.sqrt(nx.einsum("ij,ij->i", x1, x1))

nx2 = nx.sqrt(nx.einsum("ij,ij->i", x2, x2))

return 1.0 - (nx.dot(x1, nx.transpose(x2)) / nx1[:, None] / nx2[None, :])

elif metric == "correlation":

x1 = x1 - nx.mean(x1, axis=1)[:, None]

x2 = x2 - nx.mean(x2, axis=1)[:, None]

else:

if not get_backend(x1, x2).__name__ == "numpy":

raise NotImplementedError()

else:

return cdist(x1, x2, metric=metric, p=p, w=w)

# Only pass w parameter for metrics that support it

# According to SciPy docs, only 'minkowski' and 'wminkowski' support w

if w is not None and metric in ["minkowski", "wminkowski"]:

return cdist(x1, x2, metric=metric, w=w)

def dist0(n, method="lin_square"):

r"""Compute standard cost matrices of size (`n`, `n`) for OT problems

Parameters

----------

n : int

Size of the cost matrix.

method : str, optional

Type of loss matrix chosen from:

* 'lin_square' : linear sampling between 0 and `n-1`, quadratic loss

Returns

-------

M : ndarray, shape (`n1`, `n2`)

Distance matrix computed with given metric.

"""

res = 0

if method == "lin_square":

x = np.arange(n, dtype=np.float64).reshape((n, 1))

res = dist(x, x)

return res</div>

def cost_normalization(C, norm=None, return_value=False, value=None):

r"""Apply normalization to the loss matrix

Parameters

----------

C : ndarray, shape (n1, n2)

The cost matrix to normalize.

norm : str

Type of normalization from 'median', 'max', 'log', 'loglog'. Any

other value do not normalize.

Returns

-------

C : ndarray, shape (`n1`, `n2`)

The input cost matrix normalized according to given norm.

"""

nx = get_backend(C)

if norm is None:

pass

elif norm == "median":

if value is None:

value = nx.median(C)

C /= value

elif norm == "max":

if value is None:

value = nx.max(C)

C /= float(value)

elif norm == "log":

elif norm == "loglog":

C = nx.log(1 + nx.log(1 + C))

else:

raise ValueError(

"Norm %s is not a valid option.\n"

"Valid options are:\n"

"median, max, log, loglog" % norm

)

if return_value:

return C, value

else:

return C</div>

def dots(*args):

r"""dots function for multiple matrix multiply"""

nx = get_backend(*args)

return reduce(nx.dot, args)</div>

def is_all_finite(*args):

r"""Tests element-wise for finiteness in all arguments."""

nx = get_backend(*args)

return all(not nx.any(~nx.isfinite(arg)) for arg in args)</div>

def label_normalization(y, start=0, nx=None):

r"""Transform labels to start at a given value

Parameters

----------

y : array-like, shape (n, )

The vector of labels to be normalized.

start : int

Desired value for the smallest label in :math:`\mathbf{y}` (default=0)

nx : Backend, optional

Backend to perform computations on. If omitted, the backend defaults to that of `y`.

Returns

-------

y : array-like, shape (`n1`, )

The input vector of labels normalized according to given start value.

"""

if nx is None:

nx = get_backend(y)

diff = nx.min(y) - start

return y if diff == 0 else (y - diff)</div>

def labels_to_masks(y, type_as=None, nx=None):

r"""Transforms (n_samples,) vector of labels into a (n_samples, n_labels) matrix of masks.

Parameters

----------

y : array-like, shape (n_samples, )

The vector of labels.

type_as : array_like

Array of the same type of the expected output.

nx : Backend, optional

Backend to perform computations on. If omitted, the backend defaults to that of `y`.

Returns

-------

masks : array-like, shape (n_samples, n_labels)

The (n_samples, n_labels) matrix of label masks.

"""

if nx is None:

nx = get_backend(y)

if type_as is None:

type_as = y

labels_u, labels_idx = nx.unique(y, return_inverse=True)

n_labels = labels_u.shape[0]

masks = nx.eye(n_labels, type_as=type_as)[labels_idx]

return masks</div>

def parmap(f, X, nprocs="default"):

r"""parallel map for multiprocessing.

The function has been deprecated and only performs a regular map.

"""

return list(map(f, X))</div>

def check_params(**kwargs):

r"""check_params: check whether some parameters are missing"""

missing_params = []

check = True

for param in kwargs:

if kwargs[param] is None:

missing_params.append(param)

if len(missing_params) > 0:

print("POT - Warning: following necessary parameters are missing")

for p in missing_params:

print("\n", p)

check = False

return check</div>

def check_random_state(seed):

r"""Turn `seed` into a np.random.RandomState instance

Parameters

----------

seed : None | int | instance of RandomState

If `seed` is None, return the RandomState singleton used by np.random.

If `seed` is an int, return a new RandomState instance seeded with `seed`.

If `seed` is already a RandomState instance, return it.

Otherwise raise ValueError.

"""

if seed is None or seed is np.random:

return np.random.mtrand._rand

if isinstance(seed, (int, np.integer)):

return np.random.RandomState(seed)

if isinstance(seed, np.random.RandomState):

return seed

raise ValueError(

"{} cannot be used to seed a numpy.random.RandomState instance".format(seed)

)</div>

def get_coordinate_circle(x):

r"""For :math:`x\in S^1 \subset \mathbb{R}^2`, returns the coordinates in

turn (in [0,1[).

.. math::

u = \frac{\pi + \mathrm{atan2}(-x_2,-x_1)}{2\pi}

Parameters

----------

x: ndarray, shape (n, 2)

Samples on the circle with ambient coordinates

Returns

-------

x_t: ndarray, shape (n,)

Coordinates on [0,1[

Examples

--------

>>> u = np.array([[0.2,0.5,0.8]]) * (2 * np.pi)

>>> x1, y1 = np.cos(u), np.sin(u)

>>> x = np.concatenate([x1, y1]).T

>>> get_coordinate_circle(x)

array([0.2, 0.5, 0.8])

"""

nx = get_backend(x)

x_t = (nx.atan2(-x[:, 1], -x[:, 0]) + np.pi) / (2 * np.pi)

return x_t</div>

def reduce_lazytensor(a, func, axis=None, nx=None, batch_size=100):

"""Reduce a LazyTensor along an axis with function fun using batches.

When axis=None, reduce the LazyTensor to a scalar as a sum of fun over

batches taken along dim.

.. warning::

This function works for tensor of any order but the reduction can be done

only along the first two axis (or global). Also, in order to work, it requires that the slice of size `batch_size` along the axis to reduce (or axis 0 if `axis=None`) is can be computed and fits in memory.

Parameters

----------

a : LazyTensor

LazyTensor to reduce

func : callable

Function to apply to the LazyTensor

axis : int, optional

Axis along which to reduce the LazyTensor. If None, reduce the

LazyTensor to a scalar as a sum of fun over batches taken along axis 0.

If 0 or 1 reduce the LazyTensor to a vector/matrix as a sum of fun over

batches taken along axis.

nx : Backend, optional

Backend to use for the reduction

batch_size : int, optional

Size of the batches to use for the reduction (default=100)

Returns

-------

res : array-like

Result of the reduction

"""

if nx is None:

nx = get_backend(a[0:1])

if axis is None:

res = 0.0

for i in range(0, a.shape[0], batch_size):

res += func(a[i : i + batch_size])

return res

elif axis == 0:

res = nx.zeros(a.shape[1:], type_as=a[0])

if nx.__name__ in ["jax", "tf"]:

lst = []

for j in range(0, a.shape[1], batch_size):

lst.append(func(a[:, j : j + batch_size], 0))

return nx.concatenate(lst, axis=0)

else:

res[j : j + batch_size] = func(a[:, j : j + batch_size], axis=0)

return res

elif axis == 1:

if len(a.shape) == 2:

shape = a.shape[0]

else:

shape = (a.shape[0], *a.shape[2:])

res = nx.zeros(shape, type_as=a[0])

lst = []

lst.append(func(a[i : i + batch_size], 1))

else:

res[i : i + batch_size] = func(a[i : i + batch_size], axis=1)

return res

else:

raise (NotImplementedError("Only axis=None, 0 or 1 is implemented for now."))</div>

def get_lowrank_lazytensor(Q, R, d=None, nx=None):

"""Get a low rank LazyTensor T=Q@R^T or T=Q@diag(d)@R^T

Parameters

----------

Q : ndarray, shape (n, r)

First factor of the lowrank tensor

R : ndarray, shape (m, r)

Second factor of the lowrank tensor

d : ndarray, shape (r,), optional

Diagonal of the lowrank tensor

nx : Backend, optional

Backend to use for the reduction

Returns

-------

T : LazyTensor

Lowrank tensor T=Q@R^T or T=Q@diag(d)@R^T

"""

if nx is None:

nx = get_backend(Q, R, d)

shape = (Q.shape[0], R.shape[0])

if d is None:

def func(i, j, Q, R):

return nx.dot(Q[i], R[j].T)

T = LazyTensor(shape, func, Q=Q, R=R)

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