Marianna Bolla

Extrema of sums of heterogeneous quadratic forms

Abstract We analyze the following problem arising in various situations in multivariate statistical analysis. We are given k symmetric, positive definite n × n matrices, A1, A2,…, Ak (k ⩽ n), and we would like to maximize the function ∑ki = 1 xTiAixi under the constraint that x1, x2,…, xk ∈ Rn form an orthonormal system. Some theoretical results as well as an algorithm are presented.

Spectra and optimal partitions of weighted graphs

Abstract The notion of the Laplacian of weighted graphs will be introduced, the eigenvectors belonging to k consecutive eigen-values of which define optimal k -dimensional Euclidean representation of the vertices. By means of these spectral techniques some combinatorial problems concerning minimal ( k +1)-cuts of weighted graphs can be handled easily with linear algebraic tools. (Here k is an arbitrary integer between 1 and the number of vertices.) The ( k +1)-variance of the optimal k -dimensional representatives is estimated from above by the k smallest positive eigenvalues and by the gap in the spectrum between the k th and ( k +1)th positive eigenvalues in increasing order.

Spectral Clustering and Biclustering

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Modularity spectra, eigen-subspaces, and structure of weighted graphs

The role of the normalized modularity matrix in finding homogeneous cuts will be presented. We also discuss the testability of the structural eigenvalues and that of the subspace spanned by the corresponding eigenvectors of this matrix. In the presence of a spectral gap between the k-1 largest absolute value eigenvalues and the remainder of the spectrum, this in turn implies the testability of the sum of the inner variances of the k clusters that are obtained by applying the k-means algorithm for the appropriately chosen vertex representatives.

Beyond the Expanders

Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the vertex subsets. Implications between spectral and regularity properties are discussed.

Singular value decomposition of large random matrices (for two-way classification of microarrays)

Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to random noise is investigated. It is proved that such an mxn random matrix almost surely has a constant number of large singular values (of order mn), while the rest of the singular values are of order m+n as m,n->~. We prove almost sure properties for the corresponding isotropic subspaces and for noisy correspondence matrices. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.

Relating multiway discrepancy and singular values of nonnegative rectangular matrices

The minimum k -way discrepancy md k ( C ) of a rectangular matrix C of nonnegative entries is the minimum of the maxima of the within- and between-cluster discrepancies that can be obtained by simultaneous k -clusterings (proper partitions) of its rows and columns. In Theorem?2, irrespective of the size of C , we give the following estimate for the k th largest nontrivial singular value of the normalized matrix: s k ? 9 md k ( C ) ( k + 2 - 9 k ln md k ( C ) ) , provided 0 < md k ( C ) < 1 and k < rank ( C ) . This statement is a certain converse of Theorem 7 of Bolla (2014), and the proof uses some lemmas and ideas of Butler (2006), where the k = 1 case is treated. The result naturally extends to the singular values of the normalized adjacency matrix of a weighted undirected or directed graph.

Submodels for the nutrient loading estimation on River Zala

Abstract An important requirement of the Balaton ecological Modelling (BEM) approach to water-body modelling is to give nutrient loading estimation for Lake Balaton. This was done for the Western Basin, that is the daily phosphorus and nitrogen loading of communal sewage and estimates were made of the loading of agricultural origin discharged by River Zala. The approaches used are described by different submodels: submodels which separate flood-waves from base-flow, regression submodels for the relationships between the discharge and different kinds of nutrient loadings, a submodel which separates the phosphorus and nitrogen of communal sewage and those of agricultural origin and an erosion submodel. On the basis of these submodels a simulation was performed for the time interval 1976–1979. The monthly and yearly averages of the simulated daily results in good agreement with the existing measurements.

Testability of minimum balanced multiway cut densities

Testability of certain balanced minimum multiway cut densities is investigated for vertex- and edge-weighted graphs with no dominant vertex-weights. We apply the results for fuzzy clustering and noisy graph sequences.

Spectra and structure of weighted graphs

Abstract This article investigates relation between spectral and structural properties of large edge-weighted graphs. In social or biological networks we frequently look for partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the vertex subsets. We estimate the constants bounding the volume regularity of the cluster pairs by means of spectral gaps and classification properties of eigenvectors. We will focus on the more than two clusters case.