MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel

This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but standard assumptions motivated by statistical physics, we characterize the minimum mean squared error (MMSE) achievable information theoretically and with the AMP algorithm. Unlike in related problems of linear estimation, in the present setting the MMSE depends on the output channel only trough a single parameter - its Fisher information. We illustrate this striking finding by analysis of submatrix localization, and of detection of communities hidden in a dense stochastic block model. For this example we locate the computational and statistical boundaries that are not equal for rank larger than four.Comment: 10 pages, Allerton Conference on Communication, Control, and Computing 201

Phase Transitions in Sparse PCA

We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state evolution to analyze what is the information theoretically minimal mean-squared error and the one achieved by AMP in the limit of large sizes. For a special case of rank one and large enough density of non-zeros Deshpande and Montanari [1] proved that AMP is asymptotically optimal. We show that both for low density and for large rank the problem undergoes a series of phase transitions suggesting existence of a region of parameters where estimation is information theoretically possible, but AMP (and presumably every other polynomial algorithm) fails. The analysis of the large rank limit is particularly instructive.Comment: 6 pages, 3 figure

Discretization error dominance over subgrid terms in large eddy simulation of compressible shear layers in 2D

Second- and fourth-order-accurate spatial discretization methods give rise to discretization errors which are large than the corresponding subgrid terms in large eddy simulation of compressible shear layers in 2D, if the ratio between the filter width and the grid spacing is close to one. Even if an exact representation for the subgrid-scale contributions is assumed, large eddy simulation is accurate only if this ratio is sufficiently larger than one. In that regime fourth-order methods are more accurate than second-order methods. An analysis of the data obtained from two-dimensional direct numerical simulations of compressible shear layers substantiates these assertions

A cooperative Pd-Cu system for direct C-H bond arylation

The authors are grateful to the Royal Society (University Research Fellowship to CSJC) for financial support.A novel and efficient method for C-H arylation using well-defined Pd- and Cu-NHC systems has been developed. This process promotes the challenging construction of C-C bonds from arenes or heteroarenes using aryl bromides and chlorides. Mechanistic studies show that [Cu(OH)(NHC)] plays a key role in the C-H activation and is involved in the transmetallation with the Pd-NHC co-catalyst.Publisher PDFPeer reviewe

Measured quantum groupoids

In this article, we give a definition for measured quantum groupoids. We want to get objects with duality extending both quantum groups and groupoids. We base ourselves on J. Kustermans and S. Vaes' works about locally compact quantum groups that we generalize thanks to formalism introduced by M. Enock and J.M. Vallin in the case of inclusion of von Neumann algebras. From a structure of Hopf-bimodule with left and right invariant operator-valued weights, we define a fundamental pseudo-multiplicative unitary. To get a satisfying duality in the general case, we assume the existence of an antipode given by its polar decomposition. This theory is illustrated with many examples among others inclusion of von Neumann algebras (M. Enock) and a sub family of measured quantum groupoids with easier axiomatic.Comment: 139 pages. Retenu pour publication aux M{\'e}moires de la SM