Spectral Relaxations Research Articles

On inference for modularity statistics in structured networks

This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this paper contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

A hierarchy of spectral relaxations for polynomial optimization

A hierarchy of spectral relaxations for polynomial optimization

Spectral Relaxations and Branching Strategies for Global Optimization of Mixed-Integer Quadratic Programs

We consider the global optimization of nonconvex (mixed-integer) quadratic programs. We present a family of convex quadratic relaxations derived by convexifying nonconvex quadratic functions through perturbations of the quadratic matrix. We investigate the theoretical properties of these relaxations and show that they are equivalent to some particular semidefinite programs. We also introduce novel branching variable selection strategies motivated by the quadratic relaxations investigated in this paper. The proposed relaxation and branching techniques are implemented in the global optimization solver BARON and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that the proposed implementation leads to very significant reductions in BARON's computational times to solve the test problems.

An Approach to a Model Free Analysis of Excited-State Proton Transfer Kinetics in a Reverse Micelle

Time resolved area normalized emission spectra (TRANES) of a photoacid dye HPTS, confined within the water-pools of reverse micelles (RMs), are analyzed for a direct and model free observation of excited state proton transfer (ESPT) kinetics. When area normalized emission spectra of HPTS at different times are overlapped in a single window, we find population of RO–* (PTS–*) form of HPTS increases at the cost of a concomitant decreasing of the population of ROH* (HPTS*) form as time progresses. Migration of excited state population from ROH* form to RO–* form causes the emergence of an isoemissive point in TRANES, which retains throughout the entire course of ESPT process. An estimation of ESPT time scale is obtained either from the population depleting rate of ROH* form or from the population increasing rate of RO–* form; both are practically the same here. Emergence of an isoemissive point in TRANES implies that there are only two kinetically and reversibly coupled emitting species (ROH* and RO–*) of HPTS are present within the water pools of RMs. Continuous spectral relaxations of HPTS due to excited state solvation dynamics apparently have no effect on the ESPT kinetics of HPTS within the RMs; otherwise spectral shifting, caused by the excited state solvation, would have destroyed the isoemissive point of TRANES.

DS++

Correspondence problems are often modelled as quadratic optimization problems over permutations. Common scalable methods for approximating solutions of these NP-hard problems are the spectral relaxation for non-convex energies and the doubly stochastic (DS) relaxation for convex energies. Lately, it has been demonstrated that semidefinite programming relaxations can have considerably improved accuracy at the price of a much higher computational cost. We present a convex quadratic programming relaxation which is provably stronger than both DS and spectral relaxations, with the same scalability as the DS relaxation. The derivation of the relaxation also naturally suggests a projection method for achieving meaningful integer solutions which improves upon the standard closest-permutation projection. Our method can be easily extended to optimization over doubly stochastic matrices, injective matching, and problems with additional linear constraints. We employ recent advances in optimization of linear-assignment type problems to achieve an efficient algorithm for solving the convex relaxation. We present experiments indicating that our method is more accurate than local minimization or competing relaxations for non-convex problems. We successfully apply our algorithm to shape matching and to the problem of ordering images in a grid, obtaining results which compare favorably with state of the art methods. We believe our results indicate that our method should be considered the method of choice for quadratic optimization over permutations.

Graph modularity maximization as an effective method for co-clustering text data

In this paper we show how the modularity measure can serve as a useful criterion for co-clustering document-term matrices. We present and investigate the performance of CoClus, a novel, effective block-diagonal co-clustering algorithm which directly maximizes this modularity measure. The maximization is performed using an iterative alternating optimization procedure, in contrast to algorithms that use spectral relaxations of the discrete optimization problems. Extensive comparative experiments performed on various document-term datasets demonstrate that this approach is very effective, stable, and outperforms other block-diagonal co-clustering algorithms devoted to the same task. Another important advantage of using modularity in the co-clustering context is that it provides a novel, simple way of determining the appropriate number of co-clusters.Availability: an implementation of CoClus is available as part of the recently released coclust Python package which is available at: https://pypi.python.org/pypi/coclust

Global Registration of Multiple Point Clouds Using Semidefinite Programming

Consider $N$ points in ${R}^d$ and $M$ local coordinate systems that are related through unknown rigid transforms. For each point, we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the $N$ points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The least-squares formulation of this problem, although nonconvex, has a well-known closed-form solution when M=2 (based on the singular value decomposition (SVD)). However, no closed-form solution is known for $M\geq 3$. In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely, a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results from rigidity theory, we prove conditions for exact and stable recovery for the SDP relaxation. In particular, we prove that the SDP relaxation can guarantee recovery under more adversarial conditions compared to earlier proposed spectral relaxations, and we derive error bounds for the registration error incurred by the SDP relaxation. We also present results of numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the SDP (i.e., we are able to solve the original nonconvex least-squares problem) up to a certain noise threshold, and (b) the SDP performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.

Multi-way clustering and biclustering by the Ratio cut and Normalized cut in graphs

In this paper, we consider the multi-way clustering problem based on graph partitioning models by the Ratio cut and Normalized cut. We formulate the problem using new quadratic models. Spectral relaxations, new semidefinite programming relaxations and linearization techniques are used to solve these problems. It has been shown that our proposed methods can obtain improved solutions. We also adapt our proposed techniques to the bipartite graph partitioning problem for biclustering.

Quaternary Transition Pathway in Sol-Gel Encapsulated Haemoglobin Tracked by NIR and UV Spectral Relaxations

→T structural transition of haemoglobin (hb), the protein responsible for oxygen (o) transport in the red blood cells of vertebrates, is the hall mark example. This transition, which regu lates o2 uptake in the lungs and o2 release in the tissues, is a switch in the quaternary structure of the protein from a low-affinity state (T) to a high-affinity state (R), two well-characterised structures. The struc tural pathway connecting the end states of this transition remains unclear, however, although recently several experimental 1 or

Improved spectral relaxation methods for binary quadratic optimization problems

In this paper, we introduce two new methods for solving binary quadratic problems. While spectral relaxation methods have been the workhorse subroutine for a wide variety of computer vision problems—segmentation, clustering, subgraph matching to name a few—it has recently been challenged by semidefinite programming (SDP) relaxations. In fact, it can be shown that SDP relaxations produce better lower bounds than spectral relaxations on binary problems with a quadratic objective function. On the other hand, the computational complexity for SDP increases rapidly as the number of decision variables grows making them inapplicable to large scale problems. Our methods combine the merits of both spectral and SDP relaxations—better (lower) bounds than traditional spectral methods and considerably faster execution times than SDP. The first method is based on spectral subgradients and can be applied to large scale SDPs with binary decision variables and the second one is based on the trust region problem. Both algorithms have been applied to several large scale vision problems with good performance.

Valence Fluctuations of 57Fe in Disordered Li0.6FePO4

The local electronic structure around iron ions in Li0.6FePO4 was studied by 57Fe Mössbauer spectrometry at temperatures from 25 to 240 °C. The equilibrium two-phase, triphylite plus heterosite, material was compared to a disordered solid solution that was obtained by quenching from a high temperature. Substantial electronic relaxations were found in the disordered solid solution compared to the two-phase material at temperatures of 130 °C and above. Fluctuations in the electric field gradient showed an approximately Arrhenius behavior, with an activation energy of 335 ± 25meV and a prefactor of 5 × 1011 Hz, whereas Arrhenius plots for the isomer shift showed activation energies of approximately 600 meV. It is suggested that these spectral relaxations are caused by the motions of Li+ ions. A slight relaxation at 180 °C in 10% of the two-phase material can be attributed to defects in the heterosite and triphylite phases. Overall, the disordered solid solution phase shows faster electronic dynamics than the two-phase material.

Fast SDP Relaxations of Graph Cut Clustering, Transduction, and Other Combinatorial Problems

The rise of convex programming has changed the face of many research fields in recent years, machine learning being one of the ones that benefitted the most. A very recent developement, the relaxation of combinatorial problems to semi-definite programs (SDP), has gained considerable attention over the last decade (Helmberg, 2000; De Bie and Cristianini, 2004a). Although SDP problems can be solved in polynomial time, for many relaxations the exponent in the polynomial complexity bounds is too high for scaling to large problem sizes. This has hampered their uptake as a powerful new tool in machine learning. In this paper, we present a new and fast SDP relaxation of the normalized graph cut problem, and investigate its usefulness in unsupervised and semi-supervised learning. In particular, this provides a convex algorithm for transduction, as well as approaches to clustering. We further propose a whole cascade of fast relaxations that all hold the middle between older spectral relaxations and the new SDP relaxation, allowing one to trade off computational cost versus relaxation accuracy. Finally, we discuss how the methodology developed in this paper can be applied to other combinatorial problems in machine learning, and we treat the max-cut problem as an example.

Multidimensional infrared spectroscopy of water. II. Hydrogen bond switching dynamics

We use multidimensional infrared spectroscopy of the OH stretch of HOD in D2O to measure the interconversion of different hydrogen bonding environments. The OH stretching frequency distinguishes hydrogen bonded (HB) and non-hydrogen-bonded (NHB) configurations by their absorption on the low (red) and high (blue) sides of the line shape. Measured asymmetries in the two dimensional infrared OH line shapes are manifestations of the fundamentally different spectral relaxations of HB and NHB. HB oscillators exhibit coherent oscillations within the hydrogen-bonded free energy well before undergoing activated barrier crossing, resulting in the exchange of hydrogen bonded partners. Conversely, NHB oscillators rapidly return to HB frequencies within 150 fs. These results support a picture where NHB configurations are only visited transiently during large fluctuations about a hydrogen bond or during the switching of hydrogen bonding partners. The results are not consistent with the presence of entropically stabilized dangling hydrogen bonds or a conceptual picture of water as a mixture of environments with varying hydrogen bond strength separated by barriers >kT.

Femtosecond Pump−Probe Spectroscopy of the B850 Antenna Complex of Rhodobacter sphaeroides at Room Temperature

The photosynthetic bacterium Rhodobacter sphaeroides contains a light-harvesting antenna complex (LH2) with a ring of interacting bacteriochlorophyll molecules (B850). Excitation of membrane-bound LH2 complexes with low-intensity, femtosecond pulses causes changes in absorption and stimulated emission that initially depend on the excitation wavelength but relax to a quasiequilibrium with a time constant of 100 ± 20 fs. Excitation on the blue side of the B850 absorption band is followed by a shift of the signals to longer wavelengths and a decrease in amplitude, whereas the relaxations following excitation on the red side consist mainly of a decrease in amplitude. The signals have an apparent initial anisotropy of approximately 0.5 when the complex is excited with broadband pulses, and 0.35−0.4 with narrower pulses. The anisotropy decays to 0.1 with a time constant of about 30 fs. The anisotropies are similar at wavelengths on either side of the absorption band and are relatively insensitive to the excitation wavelength. Contributions of coherent pump−probe coupling and perturbed free induction decay to the measured anisotropies are considered. Pump−probe coupling could increase the initial anisotropy but cannot account for the decay kinetics. Using a density-matrix formalism, we show that the initial light-induced signals are consistent with coherent excitation of multiple exciton levels in an inhomogeneous ensemble of LH2 complexes and that the main features of the spectral relaxations and the anisotropy can be explained by electronic dephasing and thermal equilibration within the manifold of exciton levels.

Experimental time scales for dispersive kinetics in condensed media: Excess electrons in aqueous systems

Analysis of experimental results reveals that, in general, reactions of excess electrons in aqueous systems followed at the time scales electron spectral relaxations, or at shorter time scales, are dispersive while reactions followed at longer time scales than those of spectral relaxations are classical. In the latter case the internal rearrangements in the system have to be enough effective to preserve the distribution in electron reactivity during the course of reactions and because of this the classical kinetics is valid. The effects of system dynamics on kinetics of excess electrons are rationalized in terms of the renewal theory with a fractal set of renewal moments.

Dynamics of the quaternary conformational change in trout hemoglobin

The kinetics of conformational changes in trout hemoglobin I have been characterized over the temperature range 2-65 degrees C from time-resolved absorption spectra measured following photodissociation of the carbon monoxide complex. Changes in the spectra of the deoxyheme photoproduct were used to monitor changes in the protein conformation. Although the deoxyheme spectral changes are only about 8% of the total spectral change due to ligand rebinding, a combination of high-precision measurements and singular value decomposition of the data permits a detailed analysis of both their amplitudes and relaxation rates. Systematic variation of the degree of photolysis was used to alter the distribution of liganded tetramers, permitting the assignment of the spectral relaxation at 20 microseconds to the R----T quaternary conformational change of the zero-liganded and singly liganded molecules and spectral relaxations at about 50 ns and 2 microseconds to tertiary conformational changes within the R structure. Analysis of the effect of photoselection by the linearly polarized excitation pulse indicates that a major contribution to the apparent geminate rebinding in the 50-ns relaxation arises from rotational diffusion of molecules containing unphotolyzed heme-CO complexes. The activation enthalpy and activation entropy for the R0----T0 transition are +7.4 kcal/mol and -12 cal mol-1 K-1. Using the equilibrium data, delta H = +29.4 kcal/mol and delta S = +84.4 cal mol-1 K-1 [Barisas, B. G., & Gill, S. J. (1979) Biophys. Chem. 9, 235-244], the activation parameters for the T0----R0 transition are calculated to be delta H = +37 kcal/mol and delta S = +73 cal mol-1 K-1. The similarity of the equilibrium and activation parameters for the T0----R0 transition indicates that the transition state is much more R-like than T-like. This result suggests that in the path from T0 to R0 the subunits have already almost completely rearranged into the R configuration when the transition state is reached, while in the path from R0 to T0 the subunits remain in a configuration close to R in the transition state. The finding of an R-like transition state explains why the binding of ligands causes much smaller changes in the R----T rates than in the T----R rates.