34 lines
1.5 KiB
TeX
34 lines
1.5 KiB
TeX
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\chapter{Introduction}\label{ch: introduction}
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(P) is the linear least squares problem
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\[\min_{w}\ \left\lVert \hat{X}w-\hat{y} \right\rVert\]
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where
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\[
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\hat{X} =
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\begin{bmatrix}
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X^T \\
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\lambda I_m
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\end{bmatrix},
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\ \
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\hat{y} =
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\begin{bmatrix}
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y \\
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0
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\end{bmatrix},
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\]
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with $X$ the (tall thin) matrix from the ML-cup dataset by prof. Micheli, $\lambda > 0$ and $y$ is a random vector.
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\begin{itemize}
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\item[--] (A1) is an algorithm of the class of limited-memory quasi-Newton methods.
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\item[--] (A2) is a cothin QR factorization with Householder reflectors, in the variant where one does not form the matrix $Q$, but stores the Householder vectors $u_k$ and uses them to perform (implicitly) products with $Q$ and $Q^T$.
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\end{itemize}
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No off-the-shelf solvers allowed. In particular you must implement yourself the thin QR factorization, and the computational cost of your implementation should be at most quadratic in $m$.
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\subsection*{Outline}
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This report is organized as follows:
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\begin{description}
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\item[\autoref{ch: problem definition},] in which the problem is reformulated under the mathematical aspect;
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\item[\autoref{ch: algorithms},] where we will include the implemented algorithms, with the analysis of convergence and complexity;
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\item[\autoref{ch: experiments},] to evaluate and compare (A1) with (A2) for this task and provide different tests in order to examine deeper the algorithms;
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\item[\autoref{ch: conclusion},] in which conclusions are drawn, offering a critical analysis of the results obtained.
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\end{description}
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