A rectangular additive convolution for polynomials

We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava. We then prove a sliding bound on the largest root of this convolution. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges. The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest

Crystallization of random matrix orbits

Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated to general values of $\beta>0$ through associated special functions. We show that $\beta\to\infty$ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta$ self-adjoint matrix with fixed eigenvalues is known as $\beta$-corners process. We show that as $\beta\to\infty$ these eigenvalues crystallize on the irregular lattice of all the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which provides a new explanation of why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.Comment: 25 pages. v2: misprints corrected, to appear in IMR

Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced

Exploring the Potential of Feature Selection Methods in the Classification of Urban Trees Using Field Spectroscopy Data

Mapping of vegetation at the species level using hyperspectral satellite data can be effective and accurate because of its high spectral and spatial resolutions that can detect detailed information of a target object. Its wide application, however, not only is restricted by its high cost and large data storage requirements, but its processing is also complicated by challenges of what is known as the Hughes effect. The Hughes effect is where classification accuracy decreases once the number of features or wavelengths passes a certain limit. This study aimed to explore the potential of feature selection methods in the classification of urban trees using field hyperspectral data. We identified the best feature selection method of key wavelengths that respond to the target urban tree species for effective and accurate classification. The study compared the effectiveness of Principal Component Analysis Discriminant Analysis (PCA-DA), Partial Least Squares Discriminant Analysis (PLS-DA) and Guided Regularized Random Forest (GRRF) in feature selection of the key wavelengths for classification of urban trees. The classification performance of Random Forest (RF) and Support Vector Machines (SVM) algorithms were also compared to determine the importance of the key wavelengths selected for the detection of the target urban trees. The feature selection methods managed to reduce the high dimensionality of the hyperspectral data. Both the PCA-DA and PLS-DA selected 10 wavelengths and the GRRF algorithm selected 13 wavelengths from the entire dataset (n = 1523). Most of the key wavelengths were from the short-wave infrared region (1300-2500 nm). SVM outperformed RF in classifying the key wavelengths selected by the feature selection methods. The SVM classifier produced overall accuracy values of 95.3%, 93.3% and 86% using the GRRF, PLS-DA and PCA-DA techniques, respectively, whereas those for the RF classifier were 88.7%, 72% and 56.8%, respectively

Bidirectional relationship between functional connectivity and amyloid-β deposition in mouse brain

Brain region-specific deposition of extracellular amyloid plaques principally composed of aggregated amyloid-β (Aβ) peptide is a pathological signature of Alzheimer’s disease (AD). Recent human neuroimaging data suggest that resting-state functional connectivity strength is reduced in patients with AD, cognitively normal elderly harboring elevated amyloid burden, and in advanced aging. Interestingly, there exists a striking spatial correlation between functional connectivity strength in cognitively normal adults and the location of Aβ plaque deposition in AD. However, technical limitations have heretofore precluded examination of the relationship between functional connectivity, Aβ deposition, and normal aging in mouse models. Using a novel functional connectivity optical intrinsic signal (fcOIS) imaging technique, we demonstrate that Aβ deposition is associated with significantly reduced bilateral functional connectivity in multiple brain regions of older APP/PS1 transgenic mice. The amount of Aβ deposition in each brain region was associated with the degree of local, age-related bilateral functional connectivity decline. Normal aging was associated with reduced bilateral functional connectivity specifically in retrosplenial cortex. Furthermore, we found that the magnitude of regional bilateral functional correlation in young APP/PS1 mice prior to Aβ plaque formation was proportional to the amount of region-specific plaque deposition seen later in older APP/PS1 mice. Together, these findings suggest that Aβ deposition and normal aging are associated with region-specific disruption of functional connectivity and that the magnitude of local bilateral functional connectivity predicts regional vulnerability to subsequent Aβ deposition in mouse brain

Analysis of Climate Policy Targets under Uncertainty

Abstract and PDF report are also available on the MIT Joint Program on the Science and Policy of Global Change website (http://globalchange.mit.edu/).Although policymaking in response to the climate change is essentially a challenge of risk management, most studies of the relation of emissions targets to desired climate outcomes are either deterministic or subject to a limited representation of the underlying uncertainties. Monte Carlo simulation, applied to the MIT Integrated Global System Model (an integrated economic and earth system model of intermediate complexity), is used to analyze the uncertain outcomes that flow from a set of century-scale emissions targets developed originally for a study by the U.S. Climate Change Science Program. Results are shown for atmospheric concentrations, radiative forcing, sea ice cover and temperature change, along with estimates of the odds of achieving particular target levels, and for the global costs of the associated mitigation policy. Comparison with other studies of climate targets are presented as evidence of the value, in understanding the climate challenge, of more complete analysis of uncertainties in human emissions and climate system response.This study received support from the MIT Joint Program on the Science and Policy of Global Change, which is funded by a consortium of government, industry and foundation sponsors