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qchu.wordpress.com | ||
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peterbloem.nl
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| | | | | [AI summary] The pseudo-inverse is a powerful tool for solving matrix equations, especially when the inverse does not exist. It provides exact solutions when they exist and least squares solutions otherwise. If multiple solutions exist, it selects the one with the smallest norm. The pseudo-inverse can be computed using the singular value decomposition (SVD), which is numerically stable and handles cases where the matrix does not have full column rank. The SVD approach involves computing the SVD of the matrix, inverting the non-zero singular values, and then reconstructing the pseudo-inverse using the modified SVD components. This method is preferred due to its stability and ability to handle noisy data effectively. | |
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www.jeremykun.com
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| | | | | The singular value decomposition (SVD) of a matrix is a fundamental tool in computer science, data analysis, and statistics. It's used for all kinds of applications from regression to prediction, to finding approximate solutions to optimization problems. In this series of two posts we'll motivate, define, compute, and use the singular value decomposition to analyze some data. (Jump to the second post) I want to spend the first post entirely on motivation and background. | |
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nhigham.com
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| | | | | In many applications a matrix $latex A\in\mathbb{R}^{m\times n}$ has less than full rank, that is, $latex r = \mathrm{rank}(A) < \min(m,n)$. Sometimes, $latex r$ is known, and a full-rank factorization $LATEX A = GH$ with $latex G\in\mathbb{R}^{m \times r}$ and $latex H\in\mathbb{R}^{r \times n}$, both of rank $latex r$, is given-especially when $latex r =... | |
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francisbach.com
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| | | [AI summary] The blog post discusses non-convex quadratic optimization problems and their solutions, including the use of strong duality, semidefinite programming (SDP) relaxations, and efficient algorithms. It highlights the importance of these problems in machine learning and optimization, particularly for non-convex problems where strong duality holds. The post also mentions the equivalence between certain non-convex problems and their convex relaxations, such as SDP, and provides examples of when these relaxations are tight or not. Key concepts include the role of eigenvalues in quadratic optimization, the use of Lagrange multipliers, and the application of methods like Newton-Raphson for solving these problems. The author also acknowledges contributions... | ||
