Hanson-Wright inequality in Banach spaces Rafa Meller (based on - PowerPoint PPT Presentation

Hanson-Wright inequality in Banach spaces Rafał Meller (based on joint work with R. Adamczak and R. Latała) University of Warsaw Probability and Analysis, Będlewo May 2019

Notation and convention In this talk, we use the letter C to denote universal, nonnegative constant which may differ at each occurrence. So using this convention we may write 2 C ≤ C or P ( | X | ≥ Ct ) ≤ e C − 1 t 2 . We write C ( α ) if the constant may depend on some parameter α . We write a ∼ b ( a ∼ α b ) if there exists C ( C ( α )) such that a / C ≤ b ≤ aC ( a / C ( α ) ≤ b ≤ aC ( α ) ). For example 1 ∼ 2 , t 2 ∼ 2 t 2 , e x 2 ∼ e x 2 + x 8 .

Classical Hanson-Wright inequality Definition We say that a random variable X is α -subgaussian if for every � − t 2 / (2 α 2 ) � . t > 0 , P ( | X | ≥ t ) ≤ 2 exp Let us consider a sequence X 1 , X 2 , . . . of independent, mean zero and α -subgaussian random variables. The classical Hanson-Wright inequality states that for any real valued matrix A = ( a ij ) ij ≤ n  � �  � � t 2 � � t �  ≤ 2 exp � � a ij ( X i X j − E X i X j ) ≥ t − − , P C α 2 � A � op  � � C α 4 � A � 2 � � ij HS � � where � A � 2 ij a 2 HS = � ij , � A � op = sup x , y ∈ B n � ij a ij x i y j . 2

Problems with Classical Hanson-Wright inequality In many problems one need to analyze not a single quadratic form but a supremum of a collection of them i.e. expression of the form  � �  � � � a k � � P  sup ij ( X i X j − E X i X j ) ≥ t (1) � �  k ≤ n � � ij � � where A 1 = ( a 1 ij ) ij , A 2 = ( a 2 ij ) ij , . . . is a sequence of real-valued matrices. Equivalently one may need to estimate from the above the expression  � �  � � �  , � � a ij ( X i X j − E X i X j ) ≥ t (2) P  � � � � ij � � where A = ( a ij ) ij ≤ n is a matrix with values in a Banach space ( F , �·� ).

Moment estimates imply tail estimates We want to find an upper bound for  � �  � � �  = P ( S ≥ t ) , � � a ij ( X i X j − E X i X j ) ≥ t P  � � � � ij � � where A = ( a ij ) ij ≤ n is a matrix with values in a Banach space ( F , �·� ). A naive idea (which luckily is enough) is to use Chebyshev’s inequality: P ( S ≥ t ) ≤ ( � S � p / t ) p for any p ≥ 1 . So we need to estimate from the above � S � p . Standard arguments (decoupling, symmetrization and the contraction principle) yield � � � � � S � p ≤ C α 2 � � � a ij ( g i g j − δ ij ) . � � � � ij � � p

Moments of Gaussian quadratic forms Our goal is to find upper bounds (and preferably two-sided � � bounds) for moments of �� ij a ij ( g i g j − δ ij ) � (recall that ( a ij ) ij are � � from Banach space). Some results exist in the literature. Theorem (C. Borell; M. A. Arcones and E. Giné ; M. Ledoux and M. Talagrand) Let ( F , �·� ) be a Banach space and A be a symmetric, F-valued matrix. Then, for any p ≥ 1 we have � � � � � � � � � � � � � � a ij ( g i g j − δ ij ) ∼ E a ij ( g i g j − δ ij ) � � � � � � � � ij ij � � � � p � � � � + √ p E sup � � � � � � � � � � + p sup . a ij g i x j a ij x i y j � � � � x ∈ B n � � x , y ∈ B n � � ij ij 2 � � 2 � �

Problems in L q spaces The previous Theorems yields (for t > C E � � ij a ij ( g i g j − δ ij ) � ) � a ij ( g i g j − δ ij ) � ≥ C α 2 t ) P ( � ij   t 2 t ≤ 2 exp  − � 2 −   � � � � � sup x , y ∈ B n �� ij a ij x i y j E sup x ∈ B n �� � � ij a ij g i x j � � � � 2 2 Consider ( F , �·� ) = ( l q , �·� q ). Then a ij = ( a k ij ) k ≥ 1 and � q � � � � � � � � � � � � � � � q � a k � E sup a ij g i x j = E sup ij g i x j � � � � � � x ∈ B n x ∈ B n � � � � ij k ij 2 2 � � � � It is nontrivial to estimate the last expression (even in the case q = 2).

Theorem (C. Borell; M. A. Arcones and E. Giné ; M. Ledoux and M. Talagrand) Let ( F , �·� ) be a Banach space and A be a symmetric, F-valued matrix. Then, for any p ≥ 1 we have � � � � � � � � � � � � � � a ij ( g i g j − δ ij ) ∼ E a ij ( g i g j − δ ij ) � � � � � � � � ij ij � � � � p � � � � + √ p E sup � � � � � � � � � � + p sup . a ij g i x j a ij x i y j � � � � x ∈ B n � � x , y ∈ B n � � ij ij 2 � � 2 � �

Moments of Gaussian quadratic forms Theorem (R. Adamczak, R. Latała, R. Meller) Under the assumption of the previous theorem we have � � � � � � � � � � � � � � � � � � � � � a ij ( g i g j − δ ij ) � E a ij ( g i g j − δ ij ) + E a ij g ij � � � � � � � � � � � � ij ij i � = j � � � � � � p � � � � + √ p sup + √ p sup � � � � � � � � � � E a ij g i x j a ij x ij � � � � x ∈ B n � � � � x ∈ B n2 ij ij 2 � � � � 2 � � � � � � � + p sup a ij x i y j . � � x , y ∈ B n � � ij 2 � � This inequality cannot be reversed. To see this, consider p = 1 and the Banach space ( M n × n ( R ) , �·� ∗ ), where � a ij t ij . � A � ∗ = sup � T � op =1 , T ∈ M n × n

Hanson-Wright inequality in Banach spaces Theorem Let X 1 , X 2 , . . . be independent, mean-zero, α -subgaussian random variables. Then for any matrix A = ( a ij ) ij with values in ( F , �·� ) and any t ≥ C α 2 ( E � � ij a ij ( g i g j − δ ij ) � + E � � i � = j a ij g ij � ) we have  � �  � � t 2 � � t  ≤ 2 exp � � � P a ij ( X i X j − E X i X j ) ≥ t − C α 4 U 2 − , � �  C α 2 V � � ij � � � � � � � � � � � � � � � � U = sup + sup E a ij g i x j a ij x ij � � � � x ∈ B n � � � � x ∈ B n 2 ij ij 2 � � � � 2 � � � � � � � V = sup a ij x i y j . � � x , y ∈ B n � � ij 2 � �

Gaussian quadratic forms in L q spaces. Theorem In the L q spaces the following holds � � � � � � � � � � + √ p sup � � �� � ∼ q � � � � a 2 � � � a ij ( g i g j − δ ij ) a ij x ij � � � ij � � � � � � � � � x ∈ B n 2 ij ij ij � � � � � � p L q L q 2 � � � 2 � � � � �   + √ p sup � � � � � �� � � � � � � � + p sup . a ij x j a ij x i y j �  � � � x ∈ B n � � x , y ∈ B n � � i j ij 2 2 � � � � L q � � L q The reason why in L q space we have such an simplification is the following � � � � � � � � � � � � � � ≤ Cq E . E a ij g ij a ij g i g j � � � � � � � � ij ij � � � � L q L q

Hanson-Wright inequality in L q spaces Theorem Let X 1 , X 2 , . . . be independent, mean-zero, α -subgaussian random variables. Then for any matrix A = ( a ij ) ij with values in � �� � ( L q ( T ) , �·� L q ) and any t ≥ C α 2 q ij a 2 L q we have � � ij � �   � � � � t 2 � � t � � � P a ij ( X i X j − E X i X j ) ≥ t  ≤ 2 exp − C α 4 qU 2 − ,   � � C α 2 V  � � ij � � L q � � � 2 � � � � �   � � � � � � �� � � � � � U = sup + sup � a ij x ij a ij x j � � � �  � � x ∈ B n � � x ∈ B n 2 ij i j 2 � � � � 2 L q � � L q � � � � � � � V = sup a ij x i y j . � � x , y ∈ B n � � ij 2 � � L q