Quadratic forms, lattice points and interference alignment Faustin - PowerPoint PPT Presentation

Introduction The algebraic approach Some words about the geometric approach Workshop on Interactions between Number Theory and Wireless Communication, University of York ; July 05, 2016. Quadratic forms, lattice points and interference alignment Faustin ADICEAM (joint with Evgeniy ZORIN) University of York Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach Plan 1 Introduction 2 The algebraic approach Some words about the geometric approach 3 Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach Motivation Let m , n ≥ 1 be integers and let γ, c > 0 be real numbers. Define H ∈ R n × m : det γ I m + H T · H � � � � H m , n ( γ, c ) := = c . Initial Problem Assume that the set H m , n ( γ, c ) is equipped with a “uniform” probability measure. Let s ≥ 0 . What is the probability that the quantity a ∈ Z m \{ 0 } a T · � γ I m + H T · H � min · a (1) should be less than s ? In other words, what is the cumulative distribution function of (1) seen as a random variable ? Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach Motivation Let m , n ≥ 1 be integers and let γ, c > 0 be real numbers. Define H ∈ R n × m : det γ I m + H T · H � � � � H m , n ( γ, c ) := = c . Initial Problem Assume that the set H m , n ( γ, c ) is equipped with a “uniform” probability measure. Let s ≥ 0 . What is the probability that the quantity a ∈ Z m \{ 0 } a T · � γ I m + H T · H � min · a (1) should be less than s ? In other words, what is the cumulative distribution function of (1) seen as a random variable ? Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach Rephrasing the Initial Problem in a more general context Recall that H ∈ R n × m : det γ I m + H T · H � � � � H m , n ( γ, c ) := = c . Given H ∈ H m , n ( γ, c ) , let Σ H := c − 1 / m · γ I m + H T · H � � ∈ Σ ++ m , where Σ ++ is the set of positive definite matrices with determinant one . m Given Σ ∈ Σ ++ d , set a ∈ Z d \{ 0 } a T · Σ · a M d (Σ) := min (which is easily seen to be well-defined). Main Problem Assume that the set Σ ∈ Σ ++ is equipped with a probability measure. What is d the cumulative distribution function of the random variable M d (Σ) ? Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach Rephrasing the Initial Problem in a more general context Recall that H ∈ R n × m : det γ I m + H T · H � � � � H m , n ( γ, c ) := = c . Given H ∈ H m , n ( γ, c ) , let Σ H := c − 1 / m · γ I m + H T · H � � ∈ Σ ++ m , where Σ ++ is the set of positive definite matrices with determinant one . m Given Σ ∈ Σ ++ d , set a ∈ Z d \{ 0 } a T · Σ · a M d (Σ) := min (which is easily seen to be well-defined). Main Problem Assume that the set Σ ∈ Σ ++ is equipped with a probability measure. What is d the cumulative distribution function of the random variable M d (Σ) ? Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach Rephrasing the Initial Problem in a more general context Recall that H ∈ R n × m : det γ I m + H T · H � � � � H m , n ( γ, c ) := = c . Given H ∈ H m , n ( γ, c ) , let Σ H := c − 1 / m · γ I m + H T · H � � ∈ Σ ++ m , where Σ ++ is the set of positive definite matrices with determinant one . m Given Σ ∈ Σ ++ d , set a ∈ Z d \{ 0 } a T · Σ · a M d (Σ) := min (which is easily seen to be well-defined). Main Problem Assume that the set Σ ∈ Σ ++ is equipped with a probability measure. What is d the cumulative distribution function of the random variable M d (Σ) ? Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach The non–probabilistic case : the Hermite constant Recall that for Σ ∈ Σ ++ d , a ∈ Z d \{ 0 } a T · Σ · a M d (Σ) := min Hermite proved that there exists a constant γ d > 0 such that, for any Σ ∈ Σ ++ d , M d (Σ) ≤ γ d . d (dimension) 1 2 3 4 5 6 7 8 24 γ d 4 24 1 4/3 2 4 8 64/3 64 256 d T ABLE : Known values of the Hermite constant γ d One can also prove for instance that V − 2 / d ≤ γ d ≤ 4 · V − 2 / d , d d where V d is the volume of the Euclidean unit ball in dimension d ≥ 1. Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach The non–probabilistic case : the Hermite constant Recall that for Σ ∈ Σ ++ d , a ∈ Z d \{ 0 } a T · Σ · a M d (Σ) := min Hermite proved that there exists a constant γ d > 0 such that, for any Σ ∈ Σ ++ d , M d (Σ) ≤ γ d . d (dimension) 1 2 3 4 5 6 7 8 24 γ d 4 24 1 4/3 2 4 8 64/3 64 256 d T ABLE : Known values of the Hermite constant γ d One can also prove for instance that V − 2 / d ≤ γ d ≤ 4 · V − 2 / d , d d where V d is the volume of the Euclidean unit ball in dimension d ≥ 1. Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach The non–probabilistic case : the Hermite constant Recall that for Σ ∈ Σ ++ d , a ∈ Z d \{ 0 } a T · Σ · a M d (Σ) := min Hermite proved that there exists a constant γ d > 0 such that, for any Σ ∈ Σ ++ d , M d (Σ) ≤ γ d . d (dimension) 1 2 3 4 5 6 7 8 24 γ d 4 24 1 4/3 2 4 8 64/3 64 256 d T ABLE : Known values of the Hermite constant γ d One can also prove for instance that V − 2 / d ≤ γ d ≤ 4 · V − 2 / d , d d where V d is the volume of the Euclidean unit ball in dimension d ≥ 1. Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach How to tackle the Main Problem ? Two possible approaches : a purely algebraic one : ◮ essentially based on the Cholesky decomposition of an element in Σ ++ , d ◮ will provide an answer to the Initial Problem ; a purely geometric one : ◮ based on the spectral decomposition of an element in Σ ++ , d ◮ much more theoretical (only a few words in this talk). Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The algebraic approach Some words about the geometric approach How to tackle the Main Problem ? Two possible approaches : a purely algebraic one : ◮ essentially based on the Cholesky decomposition of an element in Σ ++ , d ◮ will provide an answer to the Initial Problem ; a purely geometric one : ◮ based on the spectral decomposition of an element in Σ ++ , d ◮ much more theoretical (only a few words in this talk). Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The General Theory The algebraic approach Application to Signal Processing Some words about the geometric approach Plan 1 Introduction 2 The algebraic approach The General Theory Application to Signal Processing 3 Some words about the geometric approach Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The General Theory The algebraic approach Application to Signal Processing Some words about the geometric approach Plan Introduction 1 The algebraic approach 2 The General Theory Application to Signal Processing Some numerical values Some words about the geometric approach 3 Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment

Introduction The General Theory The algebraic approach Application to Signal Processing Some words about the geometric approach Two preliminary remarks The problem is SL d ( Z ) –invariant in the sense that for any Σ ∈ Σ ++ and d any A ∈ SL d ( Z ) , A T · Σ · A � � M d = M d (Σ) Any Σ ∈ Σ ++ can be decomposed as d Σ = L T · L , where L belongs to the set of upper triangular matrices. This decomposition is furthermore unique if one requires that L should have strictly positive diagonal entries (it is then known as the Cholesky decomposition of a positive definite matrix). Faustin ADICEAM (joint with Evgeniy ZORIN) Quadratic forms, lattice points an interference alignment