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#### Mathematical Foundations of Deep Learning: a Theoretical Framework for Generalization

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor) ; Hadji Mirsadeghi, Mir Omid (Co-Supervisor)
Abstract

Deep Neural Networks, are predictive models in Machine Learning, that during the last decade they've had a great success. However being in an over-parametrized and highly non-convex regime, the analytical examinations of these models is quite a challenging task to do. The empirical developments of Neural Networks, and their distinguishing performance in prediction problems, has motivated researchers, to formalize a theoretical foundations for these models and provide us with a framework, in which one can explain and justify their behavior and properties. this framework is of great importance because it would help us to come to a better understanding of how these models work and also enables...

#### Irregularities of Some Random Point Processes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this dissertation we study the spherical ensemble and its local repelling property by...

#### Limiting spectral distribution of the sample covariance matrix of the windowed array data

, Article Eurasip Journal on Advances in Signal Processing ; Volume 2013, Issue 1 , 2013 ; 16876172 (ISSN) ; Gazor, S ; Bastani, M. H ; Sharif University of Technology
2013

Abstract

In this article, we investigate the limiting spectral distribution of the sample covariance matrix (SCM) of weighted/windowed complex data. We use recent advances in random matrix theory and describe the distribution of eigenvalues of the doubly correlated Wishart matrices. We obtain an approximation for the spectral distribution of the SCM obtained from windowed data. We also determine a condition on the coefficients of the window, under which the fragmentation of the support of noise eigenvalues can be avoided, in the noise-only data case. For the commonly used exponential window, we derive an explicit expression for the l.s.d of the noise-only data. In addition, we present a method to...

#### Spectral distribution of the exponentially windowed sample covariance matrix

, Article ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, 25 March 2012 through 30 March 2012, Kyoto ; 2012 , Pages 3529-3532 ; 15206149 (ISSN) ; 9781467300469 (ISBN) ; Bastani, M. H ; Gazor, S ; Sharif University of Technology
IEEE
2012

Abstract

In this paper, we investigate the effect of applying an exponential window on the limiting spectral distribution (l.s.d.) of the exponentially windowed sample covariance matrix (SCM) of complex array data. We use recent advances in random matrix theory which describe the distribution of eigenvalues of the doubly correlated Wishart matrices. We derive an explicit expression for the l.s.d. of the noise-only data. Simulations are performed to support our theoretical claims

#### Determinantal Processes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

Determinantal processes are a special family of stochastic processes that arise in physics (fermions), random matrices (eigenvalues), and in combinatorics (random spanning trees and non-intersecting paths). These processes have repelling property (points close to each other are chosen with low probability). Because of this repelling property, determinantal processes are approporiat for modeling some physical quantities (e.g. the position of electrons). Their probabilistic structure is described by operators on complex vector spaces and their eigenvalues. Determinantal processes have interesting properties, e.g. number of points in a region is a sum of independent Bernoulli random variables....

#### Source Enumeration and Identification in Array Processing Systems

, Ph.D. Dissertation Sharif University of Technology ; Bastani, Mohammad Hasan (Supervisor)
Abstract

Employing array of antennas in amny signal processing application has received considerable attention in recent years due to major advances in design and implementation of large dimentional antennas. In many applications we deal with such large dimentional antennas which challenge the traditional signal processing algorithms. Since most of traditional signal processing algorithms assume that the number of samples is much more than the number of array elements while it is not possible to collect so many samples due to hardware and time constraints.

In this thesis we exploit new results in random matrix theory to charachterize and describe the properties of Sample Covariance Matrices...

In this thesis we exploit new results in random matrix theory to charachterize and describe the properties of Sample Covariance Matrices...

#### Review of Thermalization in Closed Quantum Systems

, M.Sc. Thesis Sharif University of Technology ; Rezakhani, Ali (Supervisor)
Abstract

For a physical system, equilibrium is defined as a state in which the values of macroscopic quantities describing the system do not change in time. We observe systems around us reaching equilibrium every day. Describing such a seemingly simple phenomena has remained as one of the important challenges in theoretical physics. So far, various explanations for thermalization (approach to equilibrium) have been offered within classical and quantum thermodynamics. In classical statistical mechanics, the microcanonical ensemble provides a suitable prediction of the thermal behavior of a closed system. In this ensemble, using ergodic hypothesis and chaos theory one can assume that all microstates...

#### Source enumeration in large arrays based on moments of eigenvalues in sample starved conditions

, Article IEEE Workshop on Signal Processing Systems, SiPS: Design and Implementation, 17 October 2012 through 19 October 2012, Quebec ; October , 2012 , Pages 79-84 ; 15206130 (ISSN) ; 9780769548562 (ISBN) ; Bastani, M. H ; Gazor, S ; Sharif University of Technology
2012

Abstract

This paper presents a scheme to enumerate the incident waves impinging on a high dimensional uniform linear array using relatively few samples. The approach is based on Minimum Description Length (MDL) criteria and statistical properties of eigenvalues of the Sample Covariance Matrix (SCM). We assume that several models, with each model representing a certain number of sources, will compete and MDL criterion will select the best model with the minimum model complexity and maximum model decision. Statistics of noise eigenvalue of SCM can be approximated by the distributional properties of the eigenvalues given by Marcenko-Pastur distribution in the signal-free SCM. In this paper we use random...

#### Eigenvalue estimation of the exponentially windowed sample covariance matrices

, Article IEEE Transactions on Information Theory ; Volume 62, Issue 7 , 2016 , Pages 4300-4311 ; 00189448 (ISSN) ; Gazor, S ; Bastani, M. H ; Sharifitabar, M ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2016

Abstract

In this paper, we consider an exponentially windowed sample covariance matrix (EWSCM) and propose an improved estimator for its eigenvalues. We use new advances in random matrix theory, which describe the limiting spectral distribution of the large dimensional doubly correlated Wishart matrices to find the support and distribution of the eigenvalues of the EWSCM. We then employ the complex integration and residue theorem to design an estimator for the eigenvalues, which satisfies the cluster separability condition, assuming that the eigenvalue multiplicities are known. We show that the proposed estimator is consistent in the asymptotic regime and has good performance in finite sample size...