HW0: PyTorch Primer

HW0: PyTorch Primer#

We’ll use Python and PyTorch for the assignments in this course. This lab is to help you get up to speed. It will introduce:

Tensors: PyTorch’s equivalent of NumPy arrays, but with more bells and whistles for running on GPUs and supporting automatic differentiation.
Broadcasting and Fancy Indexing: If you’re coming from Matlab or NumPy, you probably know that you can avoid costly for-loops by broadcasting computation over dimensions of an array (here, tensor) and using fancy indexing tricks.
Distributions: PyTorch has an excellent library of distributions for sampling, evaluating log probabilities, and much more.

import torch
import torch.distributions as dist
import matplotlib.pyplot as plt

1. Constructing Tensors#

Tensors are PyTorch’s equivalent of NumPy arrays. The PyTorch documentation already has a great tutorial on tensors. Rather than recreate the wheel, please start by reading that.

Once you’ve read through that, try using torch functions like arange, reshape, etc. to construct the following tensors.

Problem 1.1#

Construct the following tensor:

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

Note: For this problems and the ones below, don’t literally construct the tensor from the specified list. Use torch functions.

# YOUR CODE HERE

Problem 1.2#

Construct the following tensor:

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

# YOUR CODE HERE

Problem 1.3#

Construct the following tensor:

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

Note: Here the sequence is repeated 3 times. Does your code support arbitrary numbers of repeats?

# YOUR CODE HERE

Problem 1.4#

Construct the following tensor:

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

# YOUR CODE HERE

Problem 1.5#

Construct the following tensor:

tensor([[ 1., -2.,  0.,  0.],
        [-2.,  1., -2.,  0.],
        [ 0., -2.,  1., -2.],
        [ 0.,  0., -2.,  1.]])

# YOUR CODE HERE

Problem 1.6#

Construct the following tensor:

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

# YOUR CODE HERE

2. Broadcasting and Fancy Indexing#

Your life will be much easier and your code will be much faster once you get the hang of broadcasting and indexing. Start by reading the PyTorch documentation.

Problem 2.1#

Construct a tensor X where X[i,j] = i + j by broadcasting a sum of two 1-dimensional tensors.

For example, broadcast a sum to construct the following tensor,

tensor([[0, 1, 2],
        [1, 2, 3],
        [2, 3, 4],
        [3, 4, 5]])

# YOUR CODE HERE

Problem 2.2#

Compute a distance matrix D where D[i,j] is the Euclidean distance between X[i] and X[j], with

X = torch.arange(10, dtype=float).reshape(5, 2)

Your answer should be,

tensor([[ 0.0000,  2.8284,  5.6569,  8.4853, 11.3137],
        [ 2.8284,  0.0000,  2.8284,  5.6569,  8.4853],
        [ 5.6569,  2.8284,  0.0000,  2.8284,  5.6569],
        [ 8.4853,  5.6569,  2.8284,  0.0000,  2.8284],
        [11.3137,  8.4853,  5.6569,  2.8284,  0.0000]])

X = torch.arange(10, dtype=float).reshape(5, 2)
# YOUR CODE HERE

Problem 2.3#

Extract the submatrix of rows [2,3] and columns [0,1,4] of the tensor,

A = torch.arange(25).reshape(5, 5)

Your answer should be,

tensor([[10, 11, 14],
        [15, 16, 19]])

A = torch.arange(25).reshape(5, 5)
# YOUR CODE HERE

Problem 2.4#

Create a binary mask matrix M of the same shape as A where M[i,j] is True if and only if A[i,j] is divisible by 7. Let

A = torch.arange(25).reshape(5, 5)

Your answer should be

tensor([[ True, False, False, False, False],
        [False, False,  True, False, False],
        [False, False, False, False,  True],
        [False, False, False, False, False],
        [False,  True, False, False, False]])

A = torch.arange(25).reshape(5, 5)
# YOUR CODE HERE

Problem 2.5#

Add one to the entries in A that are divisible by 7. After updating in place, A should be,

tensor([[ 1,  1,  2,  3,  4],
        [ 5,  6,  8,  8,  9],
        [10, 11, 12, 13, 15],
        [15, 16, 17, 18, 19],
        [20, 22, 22, 23, 24]])

# YOUR CODE HERE

3. Distributions#

PyTorch has an excellent library of distributions in torch.distributions. Read the docs here.

We will use these distribution objects to construct and fit a Poisson mixture model.

Problem 3.1#

Draw 50 samples from a Poisson distribution with rate 10.

# YOUR CODE HERE

Problem 3.2#

One of the awesome thing about PyTorch distributions is that they support broadcasting too.

Construct a matrix P where P[i,j] equals \(\mathrm{Pois}(x=j; \lambda=i)\) for \(i=0,\ldots,4\) and \(j=0,\ldots,4\).

Your answer should be,

tensor([[1.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.3679, 0.3679, 0.1839, 0.0613, 0.0153],
        [0.1353, 0.2707, 0.2707, 0.1804, 0.0902],
        [0.0498, 0.1494, 0.2240, 0.2240, 0.1680],
        [0.0183, 0.0733, 0.1465, 0.1954, 0.1954]])

# YOUR CODE HERE

Problem 3.3#

Evaluate the log probability of the points [1.5, 3., 4.2] under a gamma distribution with shape (aka concentration) 2.0 and inverse scale (aka rate) 1.5.

Your answer should be,

tensor([-1.0336, -2.5905, -4.0540])

# YOUR CODE HERE

Problem 3.4#

Draw 1000 samples from a Poisson mixture model,

\[ \begin{align*} p(x) &= \frac{1}{2} \mathrm{Pois}(10.0) + \frac{1}{2} \mathrm{Pois}(2.0) \end{align*} \]

Use matplotlib.pyplot.hist to plot a normalized histogram of the samples.

# YOUR CODE HERE
# data = ...

4. PyTorch Distributions#

Problem 4.1#

Use dist.Normal to draw a batch of shape (100, 4) independent standard normal random variables.

# YOUR CODE HERE

Problem 4.2#

Use dist.Normal to draw a batch independent normal random variables with shape (5,5), variance 1.0, and means

tensor([[ 0,  1,  2,  3,  4],
        [ 5,  6,  8,  8,  9],
        [10, 11, 12, 13, 15],
        [15, 16, 17, 18, 19],
        [20, 22, 22, 23, 24]])

# YOUR CODE HERE