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Diffstat (limited to 'doc/source/user/tutorial-svd.rst')
| -rw-r--r-- | doc/source/user/tutorial-svd.rst | 33 |
1 files changed, 22 insertions, 11 deletions
diff --git a/doc/source/user/tutorial-svd.rst b/doc/source/user/tutorial-svd.rst index 086e0a6de..fd9e366e0 100644 --- a/doc/source/user/tutorial-svd.rst +++ b/doc/source/user/tutorial-svd.rst @@ -9,7 +9,8 @@ Tutorial: Linear algebra on n-dimensional arrays import numpy as np np.random.seed(1) -**Prerequisites** +Prerequisites +------------- Before reading this tutorial, you should know a bit of Python. If you would like to refresh your memory, take a look at the @@ -19,7 +20,8 @@ If you want to be able to run the examples in this tutorial, you should also have `matplotlib <https://matplotlib.org/>`_ and `SciPy <https://scipy.org>`_ installed on your computer. -**Learner profile** +Learner profile +--------------- This tutorial is for people who have a basic understanding of linear algebra and arrays in NumPy and want to understand how n-dimensional @@ -28,7 +30,8 @@ you don't know how to apply common functions to n-dimensional arrays (without using for-loops), or if you want to understand axis and shape properties for n-dimensional arrays, this tutorial might be of help. -**Learning Objectives** +Learning Objectives +------------------- After this tutorial, you should be able to: @@ -38,7 +41,8 @@ After this tutorial, you should be able to: arrays without using for-loops; - Understand axis and shape properties for n-dimensional arrays. -**Content** +Content +------- In this tutorial, we will use a `matrix decomposition <https://en.wikipedia.org/wiki/Matrix_decomposition>`_ from linear algebra, the @@ -78,7 +82,8 @@ We can see the image using the `matplotlib.pyplot.imshow` function:: If you are executing the commands above in the IPython shell, it might be necessary to use the command ``plt.show()`` to show the image window. -**Shape, axis and array properties** +Shape, axis and array properties +-------------------------------- Note that, in linear algebra, the dimension of a vector refers to the number of entries in an array. In NumPy, it instead defines the number of axes. For @@ -162,7 +167,8 @@ syntax:: >>> green_array = img_array[:, :, 1] >>> blue_array = img_array[:, :, 2] -**Operations on an axis** +Operations on an axis +--------------------- It is possible to use methods from linear algebra to approximate an existing set of data. Here, we will use the `SVD (Singular Value Decomposition) @@ -290,7 +296,8 @@ diagonal and with the appropriate dimensions for multiplying: in our case, Now, we want to check if the reconstructed ``U @ Sigma @ Vt`` is close to the original ``img_gray`` matrix. -**Approximation** +Approximation +------------- The `linalg` module includes a ``norm`` function, which computes the norm of a vector or matrix represented in a NumPy array. For @@ -360,7 +367,8 @@ Now, you can go ahead and repeat this experiment with other values of `k`, and each of your experiments should give you a slightly better (or worse) image depending on the value you choose. -**Applying to all colors** +Applying to all colors +---------------------- Now we want to do the same kind of operation, but to all three colors. Our first instinct might be to repeat the same operation we did above to each color @@ -411,7 +419,8 @@ matrices into the approximation. Now, note that To build the final approximation matrix, we must understand how multiplication across different axes works. -**Products with n-dimensional arrays** +Products with n-dimensional arrays +---------------------------------- If you have worked before with only one- or two-dimensional arrays in NumPy, you might use `numpy.dot` and `numpy.matmul` (or the ``@`` operator) @@ -495,7 +504,8 @@ Even though the image is not as sharp, using a small number of ``k`` singular values (compared to the original set of 768 values), we can recover many of the distinguishing features from this image. -**Final words** +Final words +----------- Of course, this is not the best method to *approximate* an image. However, there is, in fact, a result in linear algebra that says that the @@ -504,7 +514,8 @@ terms of the norm of the difference. For more information, see *G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985*. -**Further reading** +Further reading +--------------- - :doc:`Python tutorial <python:tutorial/index>` - :ref:`reference` |