06 02 2025 E TS

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Course: Numerical Analysis for Machine Learning Prof. E. Miglio - February 6th 2025 Duration of the exam: 2.5 hours. IMPOR T ANT:During the exam you are allowed to use your notes, books and resources on the web but the use of ChatGPT (or other LLM) is strictly forbidden. The use of such tools will result in the invalidation of the exam. Exercise 1 The Genomics of Drug Sensitivity in Cancer (GDSC) dataset links cancer-cell lines (by “Cosmic ID”) to the natural logarithm of the drug concentration (“IC50”) required for 50% inhibition. You will work with a subset of the GDSC data in which each entry corresponds to a particular drug-cell-line pair to predict the effectivness of a drug on an untested tissue. The dataset is available on webeep and can be loaded using pandas. 1. Explore the dataset. Find out how many different drugs, tumor cells and IC50 doses are in the dataset. 2. Shuffle the dataset and split it into train and test. Build a sparse matrix X such that Xij is the IC50 for the i-th drug applied to the j-th tumor cell tissue. ( Hint: consider using the option return inverse of numpy.unique.) 3. Implement the baseline predictor as the average IC50 of a drug (ignore the zeros). 4. Implement the singular value truncation (SVT) algorithm as predictor. 5. Try to optimize by trial and error the threshold on the singular values and the number of iterations. Confront the SVT and the baseline predictor on the test dataset by introducing suitable metrics. 6. Substitute the SVD with the randomized SVD. Try to optimize the rank of the rSVD. Comment on the euristic used to tune the rank and on the relation between the threshold of the SVT and the rank of the rSVD. Exercise 2 Consider the function f(x) = x4 − 1.3x3 − 1.95x2 + 4x + 3.65. (1) 1. Compute all the minima of (1).…