Everything you wanted to know
(and more!)
about PyTorch tensors

Author

Marie-Hélène Burle

Content from the webinar slides for easier browsing.

Acknowledgements

Many drawings in this webinar come from the book:

The section on storage is also highly inspired by it

Using tensors locally

You need to have Python and PyTorch installed

Additionally, you might want to use an IDE such as elpy if you are an Emacs user, JupyterLab, etc.

Note that PyTorch does not yet support Python 3.10 except in some Linux distributions or on systems where a wheel has been built For the time being, you might have to use it with Python 3.9

Using tensors on CC clusters

List available wheels and compatible Python versions (in the terminal):

avail_wheels "torch*"

List available Python versions:

module avail python

Get setup:

module load python/3.9.6             # Load a sensible Python version
virtualenv --no-download env         # Create a virtual env
source env/bin/activate              # Activate the virtual env
pip install --no-index --upgrade pip # Update pip
pip install --no-index torch         # Install PyTorch

You can then launch jobs with sbatch or salloc
Leave the virtual env with the command: deactivate

Outline

What is a PyTorch tensor?
Memory storage
Data type (dtype)
Basic operations
Working with NumPy
Linear algebra
Harvesting the power of GPUs
Distributed operations

ANN do not process information directly

Modified from Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

It needs to be converted to numbers

Modified from Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

These numbers must be stored in a data structure

PyTorch tensors are Python objects holding multidimensional arrays

Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

Why a new object when NumPy already exists?

Can run on accelerators (GPUs, TPUs…)
Keep track of computation graphs, allowing automatic differentiation
Future plan for sharded tensors to run distributed computations

What is a PyTorch tensor?

PyTorch is foremost a deep learning library

In deep learning, the information contained in objects of interest (e.g. images, texts, sounds) is converted to floating-point numbers (e.g. pixel values, token values, frequencies)

As this information is complex, multiple dimensions are required (e.g. two dimensions for the width and height of an image, plus one dimension for the RGB colour channels)

Additionally, items are grouped into batches to be processed together, adding yet another dimension

Multidimensional arrays are thus particularly well suited for deep learning

Artificial neurons perform basic computations on these tensors

Their number however is huge and computing efficiency is paramount

GPUs/TPUs are particularly well suited to perform many simple operations in parallel

The very popular NumPy library has, at its core, a mature multidimensional array object well integrated into the scientific Python ecosystem

But the PyTorch tensor has additional efficiency characteristics ideal for machine learning and it can be converted to/from NumPy’s ndarray if needed

Efficient memory storage

In Python, collections (lists, tuples) are groupings of boxed Python objects

PyTorch tensors and NumPy ndarrays are made of unboxed C numeric types

Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

They are usually contiguous memory blocks, but the main difference is that they are unboxed: floats will thus take 4 (32-bit) or 8 (64-bit) bytes each

Boxed values take up more memory (memory for the pointer + memory for the primitive)

Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

Implementation

Under the hood, the values of a PyTorch tensor are stored as a torch.Storage instance which is a one-dimensional array

import torch
t = torch.arange(10.).view(2, 5); print(t) # Functions explained later

tensor([[ 0.,  1.,  2., 3.,  4.],
        [ 5.,  6.,  7.,  8.,  9.]])

storage = t.storage(); print(storage)

 0.0
 1.0
 2.0
 3.0
 4.0
 5.0
 6.0
 7.0
 8.0
 9.0
[torch.FloatStorage of size 10]

The storage can be indexed

storage[3]

3.0

storage[3] = 10.0; print(storage)

 0.0
 1.0
 2.0
 10.0
 4.0
 5.0
 6.0
 7.0
 8.0
 9.0
[torch.FloatStorage of size 10]

To view a multidimensional array from storage, we need metadata:

the size (shape in NumPy) sets the number of elements in each dimension
the offset indicates where the first element of the tensor is in the storage
the stride establishes the increment between each element

Storage metadata

Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

t.size()
t.storage_offset()
t.stride()

torch.Size([2, 5])
0
(5, 1)

Transposing in 2 dimensions

t = torch.tensor([[3, 1, 2], [4, 1, 7]]); print(t)
t.size()
t.t()
t.t().size()

tensor([[3, 1, 2],
        [4, 1, 7]])
torch.Size([2, 3])
tensor([[3, 4],
        [1, 1],
        [2, 7]])
torch.Size([3, 2])

This is the same as flipping the stride elements around

Stevens, E., Antiga, L., & Viehmann, T. (2020). Deep learning with PyTorch. Manning Publications

Transposing in higher dimensions

torch.t() is a shorthand for torch.transpose(0, 1):

torch.equal(t.t(), t.transpose(0, 1))

True

While torch.t() only works for 2D tensors, torch.transpose() can be used to transpose 2 dimensions in tensors of any number of dimensions

t = torch.zeros(1, 2, 3); print(t)

t.size()
t.stride()

tensor([[[0., 0., 0.],
         [0., 0., 0.]]])

torch.Size([1, 2, 3])
(6, 3, 1)

t.transpose(0, 1)

t.transpose(0, 1).size()
t.transpose(0, 1).stride()

tensor([[[0., 0., 0.]],
        [[0., 0., 0.]]])

torch.Size([2, 1, 3])
(3, 6, 1)  # Notice how transposing flipped 2 elements of the stride

t.transpose(0, 2)

t.transpose(0, 2).size()
t.transpose(0, 2).stride()

tensor([[[0.],
         [0.]],
        [[0.],
         [0.]],
        [[0.],
         [0.]]])

torch.Size([3, 2, 1])
(1, 3, 6)

t.transpose(1, 2)

t.transpose(1, 2).size()
t.transpose(1, 2).stride()

tensor([[[0., 0.],
         [0., 0.],
         [0., 0.]]])

torch.Size([1, 3, 2])
(6, 1, 3)

Default dtype

Since PyTorch tensors were built with utmost efficiency in mind for neural networks, the default data type is 32-bit floating points

This is sufficient for accuracy and much faster than 64-bit floating points

Note that, by contrast, NumPy ndarrays use 64-bit as their default

List of PyTorch tensor dtypes

torch.float16 / torch.half		16-bit / half-precision floating-point
torch.float32 / torch.float		32-bit / single-precision floating-point
torch.float64 / torch.double		64-bit / double-precision floating-point

torch.uint8		unsigned 8-bit integers
torch.int8		signed 8-bit integers
torch.int16 / torch.short		signed 16-bit integers
torch.int32 / torch.int		signed 32-bit integers
torch.int64 / torch.long		signed 64-bit integers

torch.bool		boolean

Checking and changing dtype

t = torch.rand(2, 3)
print(t)

# Remember that the default dtype for PyTorch tensors is float32
t.dtype

# If dtype ≠ default, it is printed
t2 = t.type(torch.float64)
print(t2)

t2.dtype

tensor([[0.8130, 0.3757, 0.7682],
        [0.3482, 0.0516, 0.3772]])

torch.float32

tensor([[0.8130, 0.3757, 0.7682],
        [0.3482, 0.0516, 0.3772]], dtype=torch.float64)

torch.float64

Creating tensors

torch.tensor: Input individual values
torch.arange: Similar to range but creates a 1D tensor
torch.linspace: 1D linear scale tensor
torch.logspace: 1D log scale tensor
torch.rand: Random numbers from a uniform distribution on [0, 1)
torch.randn: Numbers from the standard normal distribution
torch.randperm: Random permutation of integers
torch.empty: Uninitialized tensor
torch.zeros: Tensor filled with 0
torch.ones: Tensor filled with 1
torch.eye: Identity matrix

torch.manual_seed(0)  # If you want to reproduce the result
torch.rand(1)

torch.manual_seed(0)  # Run before each operation to get the same result
torch.rand(1).item()  # Extract the value from a tensor

tensor([0.4963])

0.49625658988952637

torch.rand(1)
torch.rand(1, 1)
torch.rand(1, 1, 1)
torch.rand(1, 1, 1, 1)

tensor([0.6984])
tensor([[0.5675]])
tensor([[[0.8352]]])
tensor([[[[0.2056]]]])

torch.rand(2)
torch.rand(2, 2, 2, 2)

tensor([0.5932, 0.1123])
tensor([[[[0.1147, 0.3168],
          [0.6965, 0.9143]],
         [[0.9351, 0.9412],
          [0.5995, 0.0652]]],
        [[[0.5460, 0.1872],
          [0.0340, 0.9442]],
         [[0.8802, 0.0012],
          [0.5936, 0.4158]]]])

torch.rand(2)
torch.rand(3)
torch.rand(1, 1)
torch.rand(1, 1, 1)
torch.rand(2, 6)

tensor([0.7682, 0.0885])
tensor([0.1320, 0.3074, 0.6341])
tensor([[0.4901]])
tensor([[[0.8964]]])
tensor([[0.4556, 0.6323, 0.3489, 0.4017, 0.0223, 0.1689],
        [0.2939, 0.5185, 0.6977, 0.8000, 0.1610, 0.2823]])

torch.rand(2, 4, dtype=torch.float64)  # You can set dtype
torch.ones(2, 1, 4, 5)

tensor([[0.6650, 0.7849, 0.2104, 0.6767],
        [0.1097, 0.5238, 0.2260, 0.5582]], dtype=torch.float64)
tensor([[[[1., 1., 1., 1., 1.],
          [1., 1., 1., 1., 1.],
          [1., 1., 1., 1., 1.],
          [1., 1., 1., 1., 1.]]],
        [[[1., 1., 1., 1., 1.],
          [1., 1., 1., 1., 1.],
          [1., 1., 1., 1., 1.],
          [1., 1., 1., 1., 1.]]]])

t = torch.rand(2, 3); print(t)
torch.zeros_like(t)             # Matches the size of t
torch.ones_like(t)
torch.randn_like(t)

tensor([[0.4051, 0.6394, 0.0871],
        [0.4509, 0.5255, 0.5057]])
tensor([[0., 0., 0.],
        [0., 0., 0.]])
tensor([[1., 1., 1.],
        [1., 1., 1.]])
tensor([[-0.3088, -0.0104,  1.0461],
        [ 0.9233,  0.0236, -2.1217]])

torch.arange(2, 10, 4)    # From 2 to 10 in increments of 4
torch.linspace(2, 10, 4)  # 4 elements from 2 to 10 on the linear scale
torch.logspace(2, 10, 4)  # Same on the log scale
torch.randperm(4)
torch.eye(3)

tensor([2, 6])
tensor([2.0000,  4.6667,  7.3333, 10.0000])
tensor([1.0000e+02, 4.6416e+04, 2.1544e+07, 1.0000e+10])
tensor([1, 3, 2, 0])
tensor([[1., 0., 0.],
        [0., 1., 0.],
        [0., 0., 1.]])

Tensor information

t = torch.rand(2, 3); print(t)
t.size()
t.dim()
t.numel()

tensor([[0.5885, 0.7005, 0.1048],
        [0.1115, 0.7526, 0.0658]])
torch.Size([2, 3])
2
6

Tensor indexing

x = torch.rand(3, 4)
x[:]                 # With a range, the comma is implicit: same as x[:, ]
x[:, 2]
x[1, :]
x[2, 3]

tensor([[0.6575, 0.4017, 0.7391, 0.6268],
        [0.2835, 0.0993, 0.7707, 0.1996],
        [0.4447, 0.5684, 0.2090, 0.7724]])
tensor([0.7391, 0.7707, 0.2090])
tensor([0.2835, 0.0993, 0.7707, 0.1996])
tensor(0.7724)

x[-1:]        # Last element (implicit comma, so all columns)

# No range, no implicit comma
# Indexing from a list of tensors, so the result is a one dimensional tensor
# (Each dimension is a list of tensors of the previous dimension)
x[-1]

x[-1].size()  # Same number of dimensions than x (2 dimensions)

x[-1:].size() # We dropped one dimension

tensor([[0.8168, 0.0879, 0.2642, 0.3777]])

tensor([0.8168, 0.0879, 0.2642, 0.3777])

torch.Size([4])

torch.Size([1, 4])

x[0:1]     # Python ranges are inclusive to the left, not the right
x[:-1]     # From start to one before last (and implicit comma)
x[0:3:2]   # From 0th (included) to 3rd (excluded) in increment of 2

tensor([[0.5873, 0.0225, 0.7234, 0.4538]])
tensor([[0.5873, 0.0225, 0.7234, 0.4538],
        [0.9525, 0.0111, 0.6421, 0.4647]])
tensor([[0.5873, 0.0225, 0.7234, 0.4538],
        [0.8168, 0.0879, 0.2642, 0.3777]])

x[None]          # Adds a dimension of size one as the 1st dimension
x.size()
x[None].size()

tensor([[[0.5873, 0.0225, 0.7234, 0.4538],
         [0.9525, 0.0111, 0.6421, 0.4647],
         [0.8168, 0.0879, 0.2642, 0.3777]]])
torch.Size([3, 4])
torch.Size([1, 3, 4])

A word of caution about indexing

While indexing elements of a tensor to extract some of the data as a final step of some computation is fine, you should not use indexing to run operations on tensor elements in a loop as this would be extremely inefficient

Instead, you want to use vectorized operations

Vectorized operations

Since PyTorch tensors are homogeneous (i.e. made of a single data type), as with NumPy’s ndarrays, operations are vectorized and thus staggeringly fast

NumPy is mostly written in C and PyTorch in C++. With either library, when you run vectorized operations on arrays/tensors, you don’t use raw Python (slow) but compiled C/C++ code (much faster)

Here is an excellent post explaining Python vectorization and why it makes such a big difference

Vectorized operations: comparison

Raw Python method

# Create tensor. We use float64 here to avoid truncation errors
t = torch.rand(10**6, dtype=torch.float64)

# Initialize sum
su# Run loop
for i in range(len(t)): sum += t[i]

# Print result
print(sum)

Vectorized function

t.sum()

Both methods give the same result

This is why we used float64:
While the accuracy remains excellent with float32 if we use the PyTorch function torch.sum(), the raw Python loop gives a fairly inaccurate result

tensor(500023.0789, dtype=torch.float64)
tensor(500023.0789, dtype=torch.float64)

Vectorized operations: timing

Let’s compare the timing with PyTorch built-in benchmark utility

# Load utility
import torch.utils.benchmark as benchmark

# Create a function for our loop
def sum_loop(t, sum):
    for i in range(len(t)): sum += t[i]

Now we can create the timers

t0 = benchmark.Timer(
    stmt='sum_loop(t, sum)',
    setup='from __main__ import sum_loop',
    globals={'t': t, 'sum': sum})

t1 = benchmark.Timer(
    stmt='t.sum()',
    globals={'t': t})

Let’s time 100 runs to have a reliable benchmark

print(t0.timeit(100))
print(t1.timeit(100))

I ran the code on my laptop with a dedicated GPU and 32GB RAM

Timing of raw Python loop

sum_loop(t, sum)
setup: from __main__ import sum_loop
  1.37 s
  1 measurement, 100 runs , 1 thread

Timing of vectorized function

t.sum()
  191.26 us
  1 measurement, 100 runs , 1 thread

Speedup:

1.37/(191.26 * 10**-6) = 7163

The vectorized function runs more than 7,000 times faster!!!

Even more important on GPUs

We will talk about GPUs in detail later

Timing of raw Python loop on GPU (actually slower on GPU!)

sum_loop(t, sum)
setup: from __main__ import sum_loop
  4.54 s
  1 measurement, 100 runs , 1 thread

Timing of vectorized function on GPU (here we do get a speedup)

t.sum()
  50.62 us
  1 measurement, 100 runs , 1 thread

Speedup:

4.54/(50.62 * 10**-6) = 89688

On GPUs, it is even more important not to index repeatedly from a tensor

On GPUs, the vectorized function runs almost 90,000 times faster!!!

Simple mathematical operations

t1 = torch.arange(1, 5).view(2, 2); print(t1)
t2 = torch.tensor([[1, 1], [0, 0]]); print(t2)

t1 + t2 # Operation performed between elements at corresponding locations
t1 + 1  # Operation applied to each element of the tensor

tensor([[1, 2],
        [3, 4]])
tensor([[1, 1],
        [0, 0]])

tensor([[2, 3],
        [3, 4]])
tensor([[2, 3],
        [4, 5]])

Reduction

t = torch.ones(2, 3, 4); print(t)
t.sum()   # Reduction over all entries

tensor([[[1., 1., 1., 1.],
         [1., 1., 1., 1.],
         [1., 1., 1., 1.]],
        [[1., 1., 1., 1.],
         [1., 1., 1., 1.],
         [1., 1., 1., 1.]]])
tensor(24.)

Other reduction functions (e.g. mean) behave the same way

# Reduction over a specific dimension
t.sum(0)
t.sum(1)
t.sum(2)

tensor([[2., 2., 2., 2.],
        [2., 2., 2., 2.],
        [2., 2., 2., 2.]])
tensor([[3., 3., 3., 3.],
        [3., 3., 3., 3.]])
tensor([[4., 4., 4.],
        [4., 4., 4.]])

# Reduction over multiple dimensions
t.sum((0, 1))
t.sum((0, 2))
t.sum((1, 2))

tensor([6., 6., 6., 6.])
tensor([8., 8., 8.])
tensor([12., 12.])

In-place operations

With operators post-fixed with _:

t1 = torch.tensor([1, 2]); print(t1)
t2 = torch.tensor([1, 1]); print(t2)
t1.add_(t2); print(t1)
t1.zero_(); print(t1)

tensor([1, 2])
tensor([1, 1])
tensor([2, 3])
tensor([0, 0])

In-place operations vs reassignments

t1 = torch.ones(1); t1, hex(id(t1))
t1.add_(1); t1, hex(id(t1))        # In-place operation: same address
t1 = t1.add(1); t1, hex(id(t1))    # Reassignment: new address in memory
t1 = t1 + 1; t1, hex(id(t1))       # Reassignment: new address in memory

(tensor([1.]), '0x7fc61accc3b0')
(tensor([2.]), '0x7fc61accc3b0')
(tensor([3.]), '0x7fc61accc5e0')
(tensor([4.]), '0x7fc61accc6d0')

Tensor views

t = torch.tensor([[1, 2, 3], [4, 5, 6]]); print(t)
t.size()
t.view(6)
t.view(3, 2)
t.view(3, -1) # Same: with -1, the size is inferred from other dimensions

tensor([[1, 2, 3],
        [4, 5, 6]])
torch.Size([2, 3])
tensor([1, 2, 3, 4, 5, 6])
tensor([[1, 2],
        [3, 4],
        [5, 6]])

Note the difference

t1 = torch.tensor([[1, 2, 3], [4, 5, 6]]); print(t1)
t2 = t1.t(); print(t2)
t3 = t1.view(3, 2); print(t3)

tensor([[1, 2, 3],
        [4, 5, 6]])
tensor([[1, 4],
        [2, 5],
        [3, 6]])
tensor([[1, 2],
        [3, 4],
        [5, 6]])

Logical operations

t1 = torch.randperm(5); print(t1)
t2 = torch.randperm(5); print(t2)
t1 > 3                            # Test each element
t1 < t2                           # Test corresponding pairs of elements

tensor([4, 1, 0, 2, 3])
tensor([0, 4, 2, 1, 3])
tensor([ True, False, False, False, False])
tensor([False,  True,  True, False, False])

Conversion without copy

PyTorch tensors can be converted to NumPy ndarrays and vice-versa in a very efficient manner as both objects share the same memory

t = torch.rand(2, 3); print(t)     # PyTorch Tensor
t_np = t.numpy(); print(t_np)      # NumPy ndarray

tensor([[0.8434, 0.0876, 0.7507],
        [0.1457, 0.3638, 0.0563]])   

[[0.84344184 0.08764815 0.7506627 ]
 [0.14567494 0.36384273 0.05629885]]

Mind the different defaults

t_np.dtype

dtype('float32')

Remember that PyTorch tensors use 32-bit floating points by default
(because this is what you want in neural networks)

But NumPy defaults to 64-bit
Depending on your workflow, you might have to change dtype

From NumPy to PyTorch

import numpy as np
a = np.random.rand(2, 3); print(a)
a_pt = torch.from_numpy(a); print(a_pt)    # From ndarray to tensor

[[0.55892276 0.06026952 0.72496545]
 [0.65659463 0.27697739 0.29141587]]

tensor([[0.5589, 0.0603, 0.7250],
        [0.6566, 0.2770, 0.2914]], dtype=torch.float64)

Here again, you might have to change dtype

Notes about conversion without copy

t and t_np are objects of different Python types, so, as far as Python is concerned,
they have different addresses

id(t) == id(t_np)

False

However—that’s quite confusing—they share an underlying C array in memory and modifying one in-place also modifies the other

t.zero_()
print(t_np)

tensor([[0., 0., 0.],
        [0., 0., 0.]])

[[0. 0. 0.]
 [0. 0. 0.]]

Lastly, as NumPy only works on CPU, to convert a PyTorch tensor allocated to the GPU, the content will have to be copied to the CPU first

torch.linalg module

All functions from numpy.linalg implemented (with accelerator and automatic differentiation support) + additional functions

Requires torch >= 1.9
Linear algebra support was less developed before the introduction of this module

System of linear equations solver

Let’s have a look at an extremely basic example:

2x + 3y - z = 5
x - 2y + 8z = 21
6x + y - 3z = -1

We are looking for the values of x, y, and z that would satisfy this system

We create a 2D tensor A of size (3, 3) with the coefficients of the equations
and a 1D tensor b of size 3 with the right hand sides values of the equations

A = torch.tensor([[2., 3., -1.], [1., -2., 8.], [6., 1., -3.]]); print(A)
b = torch.tensor([5., 21., -1.]); print(b)

tensor([[ 2.,  3., -1.],
        [ 1., -2.,  8.],
        [ 6.,  1., -3.]])
tensor([ 5., 21., -1.])

Solving this system is as simple as running the torch.linalg.solve function:

x = torch.linalg.solve(A, b); print(x)

tensor([1., 2., 3.])

Our solution is:

x = 1
y = 2
z = 3

Verify our result

torch.allclose(A @ x, b)

True

System of linear equations solver

Here is another simple example:

# Create a square normal random matrix
A = torch.randn(4, 4); print(A)
# Create a tensor of right hand side values
b = torch.randn(4); print(b)

# Solve the system
x = torch.linalg.solve(A, b); print(x)

# Verify
torch.allclose(A @ x, b)

(Results)

A (coefficients):

tensor([[ 1.5091,  2.0820,  1.7067,  2.3804],
        [-1.1256, -0.3170, -1.0925, -0.0852],
        [ 0.3276, -0.7607, -1.5991,  0.0185],
        [-0.7504,  0.1854,  0.6211,  0.6382]])

b (right hand side values):

tensor([-1.0886, -0.2666,  0.1894, -0.2190])

x (our solution):

tensor([ 0.1992, -0.7011,  0.2541, -0.1526])

Verification:

True

With 2 multidimensional tensors

A = torch.randn(2, 3, 3)              # Must be batches of square matrices
B = torch.randn(2, 3, 5)              # Dimensions must be compatible
X = torch.linalg.solve(A, B); print(X)
torch.allclose(A @ X, B)

tensor([[[-0.0545, -0.1012,  0.7863, -0.0806, -0.0191],
         [-0.9846, -0.0137, -1.7521, -0.4579, -0.8178],
         [-1.9142, -0.6225, -1.9239, -0.6972,  0.7011]],
        [[ 3.2094,  0.3432, -1.6604, -0.7885,  0.0088],
         [ 7.9852,  1.4605, -1.7037, -0.7713,  2.7319],
         [-4.1979,  0.0849,  1.0864,  0.3098, -1.0347]]])
True

Matrix inversions

It is faster and more numerically stable to solve a system of linear equations directly than to compute the inverse matrix first

Limit matrix inversions to situations where it is truly necessary

A = torch.rand(2, 3, 3)      # Batch of square matrices
A_inv = torch.linalg.inv(A)  # Batch of inverse matrices
A @ A_inv                    # Batch of identity matrices

tensor([[[ 1.0000e+00, -6.0486e-07,  1.3859e-06],
         [ 5.5627e-08,  1.0000e+00,  1.0795e-06],
         [-1.4133e-07,  7.9992e-08,  1.0000e+00]],
        [[ 1.0000e+00,  4.3329e-08, -3.6741e-09],
         [-7.4627e-08,  1.0000e+00,  1.4579e-07],
         [-6.3580e-08,  8.2354e-08,  1.0000e+00]]])

Other linear algebra functions

torch.linalg contains many more functions:

torch.tensordot which generalizes matrix products
torch.linalg.tensorsolve which computes the solution X to the system torch.tensordot(A, X) = B
torch.linalg.eigvals which computes the eigenvalues of a square matrix
…

Device attribute

Tensor data can be placed in the memory of various processor types:

the RAM of CPU
the RAM of a GPU with CUDA support
the RAM of a GPU with AMD’s ROCm support
the RAM of an XLA device (e.g. Cloud TPU) with the torch_xla package

The values for the device attributes are:

CPU: 'cpu'
GPU (CUDA and AMD’s ROCm): 'cuda'
XLA: xm.xla_device()

This last option requires to load the torch_xla package first:

import torch_xla
import torch_xla.core.xla_model as xm

Creating a tensor on a specific device

By default, tensors are created on the CPU

t1 = torch.rand(2); print(t1)

tensor([0.1606, 0.9771])  # Implicit: device='cpu'

Printed tensors only display attributes with values ≠ default values

You can create a tensor on an accelerator by specifying the device attribute

t2_gpu = torch.rand(2, device='cuda'); print(t2_gpu)

tensor([0.0664, 0.7829], device='cuda:0')  # :0 means the 1st GPU

Copying a tensor to a specific device

You can also make copies of a tensor on other devices

# Make a copy of t1 on the GPU
t1_gpu = t1.to(device='cuda'); print(t1_gpu)
t1_gpu = t1.cuda()  # Same as above written differently

# Make a copy of t2_gpu on the CPU
t2 = t2_gpu.to(device='cpu'); print(t2)
t2 = t2_gpu.cpu()   # For the altenative form

tensor([0.1606, 0.9771], device='cuda:0')
tensor([0.0664, 0.7829]) # Implicit: device='cpu'

Multiple GPUs

If you have multiple GPUs, you can optionally specify which one a tensor should be created on or copied to

t3_gpu = torch.rand(2, device='cuda:0')  # Create a tensor on 1st GPU
t4_gpu = t1.to(device='cuda:0')          # Make a copy of t1 on 1st GPU
t5_gpu = t1.to(device='cuda:1')          # Make a copy of t1 on 2nd GPU

Or the equivalent short forms for the last two:

t4_gpu = t1.cuda(0)
t5_gpu = t1.cuda(1)

Timing

Let’s compare the timing of some matrix multiplications on CPU and GPU with PyTorch built-in benchmark utility

# Load utility
import torch.utils.benchmark as benchmark
# Define tensors on the CPU
A = torch.randn(500, 500)
B = torch.randn(500, 500)
# Define tensors on the GPU
A_gpu = torch.randn(500, 500, device='cuda')
B_gpu = torch.randn(500, 500, device='cuda')

I ran the code on my laptop with a dedicated GPU and 32GB RAM

Let’s time 100 runs to have a reliable benchmark

t0 = benchmark.Timer(
    stmt='A @ B',
    globals={'A': A, 'B': B})

t1 = benchmark.Timer(
    stmt='A_gpu @ B_gpu',
    globals={'A_gpu': A_gpu, 'B_gpu': B_gpu})

print(t0.timeit(100))
print(t1.timeit(100))

A @ B
  2.29 ms
  1 measurement, 100 runs , 1 thread

A_gpu @ B_gpu
  108.02 us
  1 measurement, 100 runs , 1 thread

Speedup:

(2.29 * 10**-3)/(108.02 * 10**-6) = 21

This computation was 21 times faster on my GPU than on CPU

By replacing 500 with 5000, we get:

A @ B
  2.21 s
  1 measurement, 100 runs , 1 thread

A_gpu @ B_gpu
  57.88 ms
  1 measurement, 100 runs , 1 thread

Speedup:

2.21/(57.88 * 10**-3) = 38

The larger the computation, the greater the benefit: now 38 times faster

Parallel tensor operations

PyTorch already allows for distributed training of ML models

The implementation of distributed tensor operations—for instance for linear algebra—is in the work through the use of a ShardedTensor primitive that can be sharded across nodes

See also this issue for more comments about upcoming developments on (among other things) tensor sharding

Acknowledgements

Using tensors locally

Using tensors on CC clusters

Outline

What is a PyTorch tensor?

Memory storage

Data type (dtype)

Basic operations

Working with NumPy

Linear algebra

Harvesting the power of GPUs

Distributed operations

ANN do not process information directly

It needs to be converted to numbers

These numbers must be stored in a data structure

Why a new object when NumPy already exists?

What is a PyTorch tensor?

Efficient memory storage

Implementation

Storage metadata

Sharing storage

Transposing in 2 dimensions

Transposing in higher dimensions

Default dtype

List of PyTorch tensor dtypes

Checking and changing dtype

Creating tensors

Tensor information

Tensor indexing

A word of caution about indexing

Vectorized operations

Vectorized operations: comparison

Vectorized operations: timing

Even more important on GPUs

Simple mathematical operations

Reduction

In-place operations

In-place operations vs reassignments

Tensor views

Note the difference

Logical operations

Conversion without copy

Mind the different defaults

From NumPy to PyTorch

Notes about conversion without copy

torch.linalg module

System of linear equations solver

Verify our result

System of linear equations solver

With 2 multidimensional tensors

Matrix inversions

Other linear algebra functions

Device attribute

Creating a tensor on a specific device

Copying a tensor to a specific device

Multiple GPUs

Timing

Parallel tensor operations