arXiv subjects

Krishnan Mody

Publications and source records attributed to Krishnan Mody.

6 recordsLinked to original sources

arXiv2023

Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs

This work addresses the block-diagonal semidefinite program (SDP) relaxations for the clique number of the Paley graphs. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; a Paley graph is a deterministic graph where two vertices are connected if their difference is a quadratic residue (square) in a finite field with the number of elements given by certain primes and prime powers. Improving the upper bound for the Paley graph clique number for prime powers that are non-squares is an open problem in combinatorics. Moreover, since quadratic residues exhibit pseudorandom properties, Paley graphs are related to the construction of deterministic restricted isometries, an open problem in compressed sensing. Recent work provides numerical evidence that the current upper bounds can be improved by the sum-of-squares (SOS) relaxations. In particular, the bounds given by the SOS relaxations of degree 4 (SOS-4) have been empirically observed to be growing at an order smaller than square root of the prime. However, computations of SOS-4 appear to be intractable with respect to large graphs. Gvozdenovic et al. introduced a more computationally efficient block-diagonal hierarchy of SDPs and computed the values of these SDPs of degrees 2 (L2) for the Paley graph clique numbers associated with primes p less or equal to 809, which bound from above the corresponding SOS-4 relaxations. We compute the values of the L2 relaxations for p's between 821 and 997. Our results provide some numerical evidence that these relaxations, and therefore also the SOS-4 relaxations, may be scaling at an order smaller than the square root of p. However, due to the size of the SDPs, we have not been able to compute L2 relaxations for p's greater than 997. Therefore, our scaling estimate is not conclusive and presents an interesting open problem for further study.

cs.DS↗

arXiv2023

The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated

We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum.

math.PR↗

arXiv2022

Optimal local law and central limit theorem for $β$-ensembles

In the setting of generic $β$-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order $(\log N)/N$ in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale $\sqrt{\log N}/N$. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian $β$-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.

math.PR↗

arXiv2019

Gaussian fluctuations of the determinant of Wigner Matrices

We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised by Tao and Vu.

math.PR↗

arXiv2018

Topological Data Analysis of Decision Boundaries with Application to Model Selection

We propose the labeled Čech complex, the plain labeled Vietoris-Rips complex, and the locally scaled labeled Vietoris-Rips complex to perform persistent homology inference of decision boundaries in classification tasks. We provide theoretical conditions and analysis for recovering the homology of a decision boundary from samples. Our main objective is quantification of deep neural network complexity to enable matching of datasets to pre-trained models; we report results for experiments using MNIST, FashionMNIST, and CIFAR10.

stat.ML↗

arXiv2012

One Thousand and One Clusters: Measuring the Bulk Flow with the Planck ESZ and X-Ray Selected Galaxy Cluster Catalogs

We present our measurement of the "bulk flow" using the kinetic Sunyaev-Zel'dovich (kSZ) effect in the WMAP 7-year data. As the tracer of peculiar velocities, we use Planck Early Sunyaev-Zel'dovich Detected Cluster Catalog and a compilation of X-ray detected galaxy cluster catalogs based on ROSAT All-Sky Survey. We build a full-sky kSZ template and fit it to the WMAP data in W-band. Using a Wiener filter we maximize the signal to noise ratio of the kSZ cluster signal in the data. We find no significant detection of the bulk flow, and our results are consistent with the LCDM prediction.

astro-ph.CO↗