Random Projections and the Johnson-Lindenstrauss Lemma Quiz Quiz

Challenge your understanding of random projections and the Johnson-Lindenstrauss lemma, key concepts in dimensionality reduction and data science. This quiz covers fundamental ideas, mathematical properties, and applications of random projection techniques in high-dimensional spaces.

1. Purpose of Random Projection

   What is the main purpose of using random projection in high-dimensional data analysis?

   1. To reduce dimensionality while approximately preserving distances
   2. To exactly reconstruct original data
   3. To increase the number of features
   4. To perform classification directly

   Explanation: The main purpose of random projection is to reduce the dimensionality of data while approximately preserving the pairwise distances between points, which helps retain the data’s structure. Increasing the number of features is not achieved by random projection; in fact, it reduces features. Exact reconstruction of the original data is not possible with this method due to the loss of information. Random projection is a preprocessing step and not a classification method by itself.

2. Johnson-Lindenstrauss Lemma Guarantee

   According to the Johnson-Lindenstrauss lemma, what property is almost preserved when projecting points into lower dimensions?

   1. Total variance of data
   2. Individual feature values
   3. Angle between axes only
   4. Pairwise Euclidean distances

   Explanation: The Johnson-Lindenstrauss lemma guarantees that pairwise Euclidean distances between points are almost preserved under certain random projections, even after dimensionality is greatly reduced. The lemma does not claim preservation of individual feature values or angles between axes alone. While some total variance may be conserved, the main focus of the lemma is on pairwise distances, not total variance.

3. Required Dimensions and Distortion

   If you want to embed n points into a lower-dimensional space with only a small distortion ε, which factor does the minimal required dimension k mainly depend upon?

   1. It is independent of ε
   2. It must always be greater than n
   3. It grows logarithmically with n and inversely with ε squared
   4. It equals the original dimension m

   Explanation: The minimal required dimension k for the embedding is proportional to the logarithm of n (number of points) divided by ε squared, where ε is the allowed distortion. It does not equal the original dimension m, since the goal is dimensionality reduction. The required dimension does depend on ε, contrary to the statement that it is independent. It does not always have to be greater than n; in fact, k is usually much smaller.

4. Random Projection Matrix Construction

   Which of the following is a commonly used method for constructing a random projection matrix?

   1. Use a diagonal matrix of all ones
   2. Fill entries with independent standard normal values
   3. Arrange all zeros except for ones on the border
   4. Copy the original identity matrix

   Explanation: A random projection matrix is often constructed by filling its entries with independent standard normal (Gaussian) random values, which ensures the randomness and independence needed for the lemma’s guarantee. Using a diagonal matrix of ones or the identity matrix does not project the data to a lower dimension. Matrices filled mostly with zeros and ones on the border are not standard and do not provide the necessary randomness.

5. Practical Application

   In which scenario is random projection especially useful for machine learning tasks?

   1. When dealing with very high-dimensional, sparse data such as text documents
   2. When visualizing data in two dimensions
   3. When fitting a small dataset with few features
   4. When exact reconstruction is required after transformation

   Explanation: Random projection is particularly beneficial for reducing the dimensionality of very high-dimensional, sparse data (like text documents in bag-of-words format), making downstream computations more efficient. Visualization in two dimensions may use other techniques like PCA or t-SNE. For small datasets, dimensionality reduction is less critical. Exact reconstruction is not feasible with random projection, as it is a lossy transformation.