Chapter 8 Dimensionality Reducti

1. What are the main motivations for reducing a dataset's dimensionality? What are the main darwbacks?

The main motivations for dimensionality reductions are:

• To spped up a subsequent training algorithm (in some cases it ay even remove noise and redundant features, making the training algorithm perform better)
  • To visualize the data and gain insights on the moset import features .
  • To save space (compression)

The main drawbacks are:

• Some information is lost, possibly degrading the performance of subsequent training algorithms.
• It can be computationally intensive
• It adds some complexity to your Machine Learning pipelines.
• Transformed features are often hard to interpret.

2. What is the curse of dimensionality ?

The cures of dimensionality refers to the fact that many problems that do not exist in low-dimensional space arise in high-dimensional space.In Machine Learning, one common manifestation is the fact that randomly sampled high-dimensional vectors are generally far from one another, increasing the risk of overfitting and making is very difficult to identify patterns without having plenty of plenty of training data.

3. Once a dataset's dimensionality has been reduced, is it possible to revers the operation ? If so, how ? If not , why ?

Once a dataset's dimensionality has been reduced using one of the algorithms we dicussed, it is almost always impossible to perfectly reverse the operation, because some information gets lost during dimensionality reductions. Moreover, while some algorithms (such as PCA) have a simple reverse transformation procedure that can reconstruct a dataset relatively similar to the original, other algorithms (such as t-SNE) do not.

4. Can PCA be used to reduce the dimensionality of a hightly nonlinear dataset?

PCA can be used to significantly reduce the dimensionality of most datasets, even if they are highly nonlinear, because it can at least get rid of useless dimensions. However, if there are no useless dimensions-as in the Swiss roll dataset-then reducing dimensionality with PCA will lose too much information . You want to unroll the Swiss roll, not squash it.

5. Suppose you perform PCA on a 1,000-dimensional dataset, setting the explained variance ratio to 95%. How manh dimensions will the resulting dataset have?

That's a trick question: it depends on the dataset. Let's look at two extrem examples. First, suppose the dataset is composed of points that are almost perfectly aligned. In this case, PCA can reduce the dataset down to jjust one dimension will still preserving 95% of variance. Now imagine that the dataset is composed of perfectly random points, scattered all around dimensions. In this case roughly 950 dimensions are required to preserve 95% of variance. So the answer is , it depends on the dataset, and it could be any number between 1 and 950. Plotting the explained variance as a function of the number of dimensions is one way to get a rough idea of the dataset's intrinsic dimensionality

6. In what cases would you use regular PCA, incremental PCA, randomized PCA, or random projection?

Regular PCA is the default, but it works only if the dataset fits in memory. Incremental PCA is useful for large datasets that don't fit in memory, but it is slower that regular PCA, so it the dataset fit in memory you should prefer regular PCA, Incremental PCA is also useful for online task, when you need to apply PCA on the fly, every time a new instance arrives. Randomized PCA is useful when you want to considerably reduce dimensionality and the dataset fit in memory; in this case, it is much faster than regular PCA. Finally, Random Projection is great for very high-dimensional datasets.

7. How can you evaluate the performance of a dimensionality reduction algorithm on your dataset ?

Intuitively, a dimensionality reduction algorithm performs well if it eliminates a lot of dimensions from the dataset without losing too much information. One way to measure this is to apply the reverse transofrmation and measure the reconstruction error. However, not all dimensionality reduction algorithms provide a reverse transformation. Alternatively, if you are using dimensionality reduction as a preprocessing step before another Machine Leaning algorithm (e.g., a Random Forset classifier), then you can simply measure the performance of that second algorithm; if dimensionality reduction did not lose too much information, then the algorithm should perform just as well as when using the original dataset.

8. Does it make any sense to chain two different dimensionality reduction algorithms ?

I can absolutely make sense to chain two different dimensionality reduction algorithms. A common example is using PCA or Random Projection to quickly get rid of a large number of usless dimensions, then applying another much slower dimensionality reduction algorithm, such LLE. This two-step approach will likely yield roughly the same performance as using LLE only, but in a fraction of the time.