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To give an example of real world sparse matrices, power grids can have thousands and thousands of nodes, but most of those nodes only connect to a few other nod

To give an example of real world sparse matrices, power grids can have thousands and thousands of nodes, but most of those nodes only connect to a few other nodes at most that are local neighbors. Systems are highly sparse as a result

westurner 4y ago
Integer factor graphs are sparse. https://en.wikipedia.org/wiki/Factor_graph#Message_passing_on_factor_graphs https://en.wikipedia.org/wiki/Factor_graph#Message_passing_o... Compared to the Powerset graph that includes all possible operators and parameter values and parentheses in infix but not Reverse Polish Notation, a correlation graph is sparse: most conditional probabilities should be expected to tend toward the Central Limit Theorem, so if you subtract (or substitute) a constant noise scalar, a factor graph should be extra-sparse. https://en.wikipedia.org/wiki/Central_limit_theorem_for_directional_statistics https://en.wikipedia.org/wiki/Central_limit_theorem_for_dire... What do you call a factor graph with probability distribution functions (PDFs) instead of float64s? Are Path graphs and Path graphs with cycles extra sparse? An adjacency matrix for all possible paths through a graph is also mostly zeroes. https://en.wikipedia.org/wiki/Path_graph https://en.wikipedia.org/wiki/Path_graph Methods of feature reduction use and affect the sparsity of a sparse matrix (that does not have elements for confounding variables). For example, from "Exploratory factor analysis (EFA) versus principal components analysis (PCA)" https://en.wikipedia.org/wiki/Factor_analysis https://en.wikipedia.org/wiki/Factor_analysis : > For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.

westurner 4y ago

Integer factor graphs are sparse. https://en.wikipedia.org/wiki/Factor_graph#Message_passing_on_factor_graphs https://en.wikipedia.org/wiki/Factor_graph#Message_passing_o... Compared to the Powerset graph that includes all possible operators and parameter values and parentheses in infix but not Reverse Polish Notation, a correlation graph is sparse: most conditional probabilities should be expected to tend toward the Central Limit Theorem, so if you subtract (or substitute) a constant noise scalar, a factor graph should be extra-sparse. https://en.wikipedia.org/wiki/Central_limit_theorem_for_directional_statistics https://en.wikipedia.org/wiki/Central_limit_theorem_for_dire... What do you call a factor graph with probability distribution functions (PDFs) instead of float64s? Are Path graphs and Path graphs with cycles extra sparse? An adjacency matrix for all possible paths through a graph is also mostly zeroes. https://en.wikipedia.org/wiki/Path_graph https://en.wikipedia.org/wiki/Path_graph Methods of feature reduction use and affect the sparsity of a sparse matrix (that does not have elements for confounding variables). For example, from "Exploratory factor analysis (EFA) versus principal components analysis (PCA)" https://en.wikipedia.org/wiki/Factor_analysis https://en.wikipedia.org/wiki/Factor_analysis : > For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.