Portrait of Raghav Somani

Raghav Somani

Quantitative Researcher

Blog Notes

2021

Note on the Kadison-Singer Problem and its Solution

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The Kadison-Singer problem arose from the work on quantum mechanics done by Paul Dirac in the 1930s. The problem is equivalent to fundamental problems in areas like Operator theory, Hilbert and Banach space theory, Frame theory, Harmonic Analysis, Discrepancy theory, Graph theory, Signal Processing and theoretical Computer Science. The Kadison-Singer problem had been long standing and defied the efforts of most Mathematicians until it was recently solved by Adam Wade Marcus, Daniel Alan Spielman and Nikhil Srivastava in 2013. Read more

2020

A note on Conformal Symplectic and Relativistic Optimization

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This note on a spotlight paper at NeurIPS 2020, has been made while I had been reading the literature on the principle connections between continuous and discrete optimization. The motivation is to understand and create accelerated discrete large scale optimization algorithms from first principles via considering the geometry of phase spaces and numerical integration, specifically symplectic integration. Recent works successfully have been able to throw sufficient light on the two and therefore has attracted my attention. Read more

Geometry of Relativistic Spacetime Physics

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This article introduces the mathematical structures needed to understand relativistic spacetime physics. The self-referential and self-contained nature of mathematics provides a rigorous language for formulating the components of Einstein’s general theory of relativity—spacetime, matter, and gravity—along with their behavior and interactions. These notes begin with smooth manifolds and then add the necessary and sufficient differential-geometric structures. Read more

Dual spaces and the Fenchel conjugate

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Dual spaces lie at the core of linear algebra and allow us to reason formally about duality in mathematics. Duality appears naturally in measure theory, functional analysis, and mathematical optimization. In this post, I explore dual spaces and their interpretation in linear algebra, motivated by the so-called convex conjugate, or Fenchel conjugate, in mathematical optimization. Read more

A survey on Strongly Rayleigh measures and their mixing time analysis

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Strongly Rayleigh measures are natural generalizations of measures that satisfy the notion of negative dependence. The class of Strongly Rayleigh measures provides the most useful characterization of Negative Dependence by grounding it in the theory of multivariate stable polynomials. This post attempts to throw some light on the origin of Strongly Rayleigh measures and Determinantal Point Processes and highlights the fast mixing time analysis of the natural MCMC chain in the support of a Strongly Rayleigh measure as shown by Anari, Gharan and Rezaei 2016. Read more

2019

Analysis of Newton’s Method

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In optimization, Newton’s method is used to find roots of the derivative of a twice-differentiable function given oracle access to its gradient and Hessian. By using super-linear memory in the dimension of the ambient space, Newton’s method can take advantage of second-order curvature and optimize the objective function at a quadratically convergent rate. Here I consider the case in which the objective function is smooth and strongly convex. Read more

Deriving the Fokker-Planck equation

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In the theory of dynamical systems, the Fokker–Planck equation describes the time evolution of a probability density function. It describes how the density of a stochastic process changes over time under the influence of a potential field. Common applications include Brownian motion, the Ornstein–Uhlenbeck process, and statistical physics. My motivation for understanding the derivation is to study Lévy flight processes. Read more

2018

Nesterov’s Acceleration

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This post contains an error vector analysis of the Nesterov’s accelerated gradient descent method and some insightful implications that can be derived from it. Read more

2017

A survey on Large Scale Optimization

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This post contains a summary and survey of the theoretical understandings of Large Scale Optimization by referring some talks, papers, and lectures that I have come across in the recent. Read more