Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X \in \mathbb{C}^{n \times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j a_j^*$ for some measurement vectors $a_1,...,a_m$, i.e., the measurements are given by $y_j = \mathrm{tr}(X a_j a_j^*)$. The case where the matrix $X=x x^*$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, $y_j = |\langle x,a_j\rangle|^2$ via the PhaseLift approach, which has been introduced recently. We derive bounds for the number $m$ of measurements that guarantee successful uniform recovery of Hermitian rank $r$ matrices, either for the vectors $a_j$, $j=1,...,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective $t$-design with $t=4$. In the Gaussian case, we require $m \geq C r n$ measurements, while in the case of $4$-designs we need $m \geq Cr n \log(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank $r$-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate $4$-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Comment: 24 page

Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

Quantifying nuclear wide chromatin compaction by phasor analysis of histone Förster resonance energy transfer (FRET) in frequency domain fluorescence lifetime imaging microscopy (FLIM) data.

The nanometer spacing between nucleosomes throughout global chromatin organisation modulates local DNA template access, and through continuous dynamic rearrangements, regulates genome function [1]. However, given that nucleosome packaging occurs on a spatial scale well below the diffraction limit, real time observation of chromatin structure in live cells by optical microscopy has proved technically difficult, despite recent advances in live cell super resolution imaging [2]. One alternative solution to quantify chromatin structure in a living cell at the level of nucleosome proximity is to measure and spatially map Förster resonance energy transfer (FRET) between fluorescently labelled histones - the core protein of a nucleosome [3]. In recent work we established that the phasor approach to fluorescence lifetime imaging microscopy (FLIM) is a robust method for the detection of histone FRET which can quantify nuclear wide chromatin compaction in the presence of cellular autofluorescence [4]. Here we share FLIM data recording histone FRET in live cells co-expressing H2B-eGFP and H2B-mCherry. The data was acquired in the frequency domain [5] and processed by the phasor approach to lifetime analysis [6]. The data can be valuable to researchers interested in using the histone FRET assay since it highlights the impact of cellular autofluorescence and acceptor-donor ratio on quantifying chromatin compaction. The data is related to the research article "Phasor histone FLIM-FRET microscopy quantifies spatiotemporal rearrangement of chromatin architecture during the DNA damage response" [4]