After successful completion of the module students will be able to understand and apply the basic notions, concepts, and methods of computational linear algebra, convex optimization and differential geometry used for data analysis. In particular, they will master the use of singular value decomposition method as well as random matrices for low dimensional data representations, including fundamentals of sparse recovery problems, as e.g., compressed sensing, low rank matrix recovery, and dictionary learning algorithms. The students will be also able to manage the representation of data as clusters around manifolds in high dimensions and in random graphs, acquiring methods to construct local charts and clusters for the data. In complementary laboratory sessions they will get acquainted with suitable programming tools and environment in order to analyse relevant case studies.
*Introduction to signal processing continuous and discrete
- Fourier Transform, Discrete Fourier Transform, Discrete Time Fourier Transform.
- Fast Fourier Transform algorithm.
- Application to signal and image analysis: denoising, compression.
* Singular Value Decomposition:
- Best k-rank approximation, Randomize SVD
- Principal Component Analysis, Pseudo-Inverse.
* Compressed Sensing
- Basis pursuit problem: l1-minimization and sparse recovery
- Application to signals and images reconstruction.
* Data Analysis
- Dimensionality reduction techniques: (Local Linear Embedding, ISOMAP, diffusion map).
- Supervised learning for classification: Support Vector Machine
- Unsupervised learning for clustering: K-means.
- Artificial Neural Networks and applications.
|Stephane Mallat||A Wavelet Tour of Signal Processing (Edizione 2)||Academic Press||1999||9780124666061|
|Avrim Blum, John Hopcroft, Ravi Kannan,||Foundations of Data Science||Cambridge University Press||2020|
|John A. Lee, Michel Verleysen||Nonlinear Dimensionality Reduction||Springer||2006|
The exam consists of an oral examination with written questions and discussion. The development of a project is encouraged (but not mandatory) as an integration of the oral examination.
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