Sharp Bounds for the Concentration of the Resolvent
in Convex Concentration Settings

Initiated by Milman in the 70s [Mil71], the theory of concentration of the measure provided a wide range of concentration inequalities, mainly concerning continuous distribution (i.e. with no atoms), in particular thanks to the beautiful interpretation with the bound on the Ricci curvature [Gro79]. To give a simple fundamental example (more examples can be found in [Led05]), a random vector $Z\in\mathbb{R}^{n}$ having independent and Gaussian entries with unit variance satisfies for any $1$-Lipschitz mapping $f:\mathbb{R}^{n}\to\mathbb{R}$:

This inequality is powerful for two reasons: first, it is independent on the dimension $n$, second it concerns any $1$-Lipschitz mapping $f$. It is then interesting to formalize this behavior to introduce a class of “Lipschitz concentrated random vectors” satisfying the same concentration inequality as the Gaussian distribution (in particular, all the Lipschitz transformation of a Gaussian vector). This was done in several books and papers where this approach proved its efficiency in the study of random matrices [Tao12, Nou09, LC21b]…

We want here to extend those results to a new class of concentrated vectors discovered by Talagrand in [Tal95]. Although the concentration result looks similar, its nature is quite different as it concerns bounded distributions for which classical tools of differential geometry do not operate. In a sense, it could be seen as a combinatorial result of concentration. Given a random vector $Z\in[0,1]^{n}$ with independent entries, this result sets that for any $1$-Lipschitz and convex mapping $f:[0,1]\to\mathbb{R}$:

We can mention here the recent results of [HT21] that extend this kind of inequalities for random vectors with independent and subgaussian entries. Adopting the terminology of [VW14, MS11, Ada11], we call those vectors convexly concentrated random vector (see Definition 1.2 below). The convexity required for the observations to concentrate makes the discussion on convexly concentrated random vector far more delicate. There is no more stability towards Lipschitz transformations and given a convexly concentrated random vector $Z$, just its affine transformations are sure to be concentrated. This issue raises naturally for one of the major objects of random matrix theory, namely the resolvent $Q^{z}=(zI_{n}-X)^{-1}$ that can provide important eigen properties on $X$. In the case of convex concentration, the concentration of the resolvent $Q^{z}=(zI_{n}-X)^{-1}$ is no more a mere consequence of a bound on its differential on $X\in\mathcal{M}_{p}$. Still, as first shown by [GZ00], it is possible to obtain concentration properties on the sum of Lipschitz functionals of the eigen values. Here we pursue the study, looking at linear concentration properties of $Q^{z}$ for which similar inequalities to (1) or (2) are only satisfied by $1$-Lipschitz and linear functionals $f$. The well known identity

is true for any analytical mapping $f$ defined on the interior of a path $\gamma\in\mathbb{C}$ containing the spectrum of $X$ (or any limit of such mappings), therefore, our results on the concentration of $Q^{z}$ concern in particular the quantities studied in [GZ00].

Although it is weaker²²2Lipschitz concentrated random vectors are convexly and linearly concentrated, convexly concentrated random vectors are linearly concentrated., the class of linearly concentrated vectors behaves very well towards the dependence and the sum and allows us to obtain the concentration of the resolvent expressing it as a sum $Q^{z}=\frac{1}{z}\sum_{i=1}^{\infty}(X/z)^{i}$. The linear concentration of the powers of $X$ was justified in³³3We provide an alternative proof in the appendix. [MS11] in the case of convexly concentrated random matrix $X$. We call this weakening of the concentration property “the degeneracy of the convex concentration through multiplication”. The linear concentration of the resolvent is though sufficient for most practical applications that rely on an estimation of the Stieltjes transform $m(z)=\frac{1}{p}\operatorname{Tr}(Q^{z})$.

We present below our main contribution without the useful but non-standard formalism introduced in the rest of the article. It concerns the concentration and the estimation of⁴⁴4The concentration of $(zI_{p}-\frac{1}{\sqrt{n}}X)^{-1}$ for a positive symmetric matrix $X\in\mathcal{M}_{p}$ is immediate from our approach. The estimation of its expectation is more laborious and goes beyond the scope of the paper although it can be obtained with the same tools.

for a random matrix $X\in\mathcal{M}_{p,n}$. Following the formalism of the random matrix theory, the computable estimation of $\mathbb{E}[Q^{z}]$ will be called a “deterministic equivalent”. Its definition relies on a well known result that states that given a family of symmetric matrices $\Sigma=(\Sigma_{1},\ldots,\Sigma_{n})\in\mathcal{M}_{p}^{n}$, there exists a unique vector $\tilde{\Lambda}_{\Sigma}^{z}\in\mathbb{C}^{n}$ satisfying:

With those notations at hand, let us state:

This theorem allows us to get good inferences on the eigen values distribution through the identity (3) and the estimation of the Stieltjes transform $g(z)\equiv-\frac{1}{p}\operatorname{Tr}(Q^{z})$ satisfying the concentration inequality:

for two constants $C,c>0$ (and for $d(z,\text{Sp}(\frac{1}{n}XX^{T}))\geq O(1)$).

When the distribution of the spectrum of $\frac{1}{n}XX^{T}$ presents different bulks, this theorem also allows us to understand the eigen-spaces associated to those different bulks. Indeed, considering a path $\gamma\in\mathbb{C}$ containing a bulk of eigen-values $B\subset\text{Sp}(\frac{1}{n}XX^{T})$, if we note $E_{B}$ the associated random eigen-space and $\Pi_{B}$ the orthogonal projector on $E_{B}$, then for any deterministic matrix $A\in\mathcal{M}_{p}$:

we can estimate this projection $\Pi_{B}$ defining $E_{B}$ thanks to the concentration inequality⁵⁵5for the concentration to be valid on all the values of the path $\gamma$, one must be careful to consider a path staying at a distance $O(1)$ from the bulk, that is why we only consider here multiple bulk distributions:

for some constants $C,c>0$ and for any projector $\Pi$ defined on $\mathbb{R}^{p}$.

The approach we present here does not only allows us to set the concentration of $Q^{z}$, but also the concentration of any polynomial of finite degree taking as variable combination of $Q^{z}$, $X$ and $X^{T}$. The general idea is to develop the polynomial as an infinite series of powers of $X$ in a way that the observable diameters of the different terms of the series sum to the smallest value possible. As it is described in the proof of Proposition 4.6, the summation becomes slightly elaborate when $z$ gets close to the spectrum.

After presenting the definition and the basic properties of the convex and linear concentration (Section 1), we express the concentration of the sum of linearly concentrated random vectors (Section 2). Then we express the concentration of the entry wise product and the matricial product of convexly concentrated random vectors and matrices (Section 3). Finally we deduce the concentration of the resolvent and provide a computable deterministic equivalent (Section 4).

The concentration inequality (2) is actually also valid for quasi-convex functionals defined folowingly.

The theory of concentration of measure becomes relevant only when dimensions get big. In the cases under study in this paper, the dimension is either given by the number of entries, either by the number of columns $n$ of random matrices - the number of rows $p$ is then understood to depend on $n$, we will sometimes note $p=p_{n}$. We follow then the approach with Levy families [Lé51] whose aim is to track the concentration speed through dimensionality. Therefore, we do not talk about a static concentration of a vector but about the concentration of a sequence of random vectors as seen in the definition below. In this paper, $E_{n}$ will either be $\mathbb{R}^{n}$, $\mathbb{R}^{p_{n}}$ $\mathcal{M}_{n}$, $\mathcal{M}_{p_{n}}$ or $\mathcal{M}_{p_{n},n}$.

There will generally be three possibilities for the norms defining the Lipschitz character of the concentrated observations. Talagrand Theorem gives the concentration for the euclidean norm - i.e. the Frobenius norm for matrices - but we will see that some concentrations are expressed with the nuclear norm (the dual norm of the spectral norm). Given two integers $l,m\in\mathbb{N}$, the euclidean norm on $\mathbb{R}^{l}$ is noted $\|\cdot\|$, the spectral, Frobenius and nuclear norm are respectively defined for any $M\in\mathcal{M}_{l,m}$ with the expressions:

The Theorem of Talagrand then writes:

Convex concentration is preserved through affine transformations (as for the class of linearly concentrated vectors). Given two vector spaces, $E$ and $F$, we note $\mathcal{A}(E,F)$ the set of affine transformation from $E$ to $F$, and given $\phi\in\mathcal{A}(E,F)$, we decompose $\phi=\mathcal{L}(\phi)+\phi(0)$, where $\mathcal{L}(\phi)$ is the linear part of $\phi$ and $\phi(0)$ is the translation part. When $E=F$, $\mathcal{A}(E,F)$ is simply noted $\mathcal{A}(E)$.

We pursue our presentation with the introduction of the linear concentration. It is the “minimal” hypothesis necessary on a random vector $X$ to be able to bound quantities of the form $\mathbb{E}[\|X-\mathbb{E}[X]\|]$, as it has been explained in [LC19]. Here we will need its stability towards the sum when we will express $Q^{z}$ as an infinite series.

In the literature [BLM13], those vectors are commonly called sub-Gaussian random vectors.

The notions of linear concentration, convex concentration (and Lipschitz concentration) are equivalent for random variables and we have this important characterization with the moments:

We end with a simple lemma that allows us to state that "every deterministic vector at a distance smaller than the observable diameter to a deterministic equivalent is also a deterministic equivalent".

Independence is known to be a key elements to most of concentration inequalities. However, linear concentration behaves particularly well for the concatenation of random vectors whose dependence can not be disentangled.

The next proposition sets that the observable diameter for the $\ell^{\infty}$ norm remains unchanged through concatenation. Given a product $E\equiv\prod_{1\leq i\leq m}E_{i}$, where $(E_{1},\|\cdot\|_{\infty}),\ldots,(E_{m},\|\cdot\|_{\infty})$ are $m$ normed vector spaces we define the $\ell^{\infty}$ norm on $E$ with the following identity:

In other word, the linear observable diameter of $(Z_{1},\ldots,Z_{m})$ can not be bigger than the observable diameter of $(Z,\ldots,Z)$, where $Z$ is chosen as the worse possible random vector satisfying the hypotheses of $Z_{1},\ldots,Z_{m}$.

If we want to consider the concatenation of vectors with different observable diameter, it is more convenient to look at the concentration in a space $(\prod_{i=1}^{m}E_{i},\ell^{r})$, for any given $r>0$, where, for any $(z_{1},\ldots,z_{m})\in\prod_{i=1}^{m}E_{i}$:

Corollary 1 is very useful to set the concentration of infinite series of concentrated random variables. This is settled thanks to an elementary result of [LC19] that sets that the observable diameter of a limit of random vectors is equal to the limit of the observable vectors. Be careful that rigorously, there are two indexes, $n$ coming from Definition 1.5 that only describes the concentration of sequences of random vectors, and $m$ particular to this lemma that will tend to infinity. For clarity, we do not mention the index $n$.

The concentration of infinite series directly implies the concentration of resolvents and other related operators (like $(I_{n}-X/\sqrt{p})^{-1}X^{k}$ for instance).

In practical examples, $\|\mathcal{L}(\phi)\|$ is rarely bounded by $1-\varepsilon$ for all drawings of $\phi$ and to obtain the concentration of $\mathcal{L}(\phi)^{k}$ with an observable diameter of order $\sigma(1-\varepsilon)^{k}$, one needs to place oneself on an event $\mathcal{A}_{\phi}$ satisfying $\mathcal{A}_{\phi}\subset\{\left\|\mathcal{L}(\phi)\right\|\leq 1-\varepsilon\}$. Then, thanks to a simple adaptation of Lemma 4.2 below to the case of linear concentration, we have the concentration $(Y\ |\ \mathcal{A}_{\phi})\in\mathcal{E}_{2}(\sigma)$. When $\mathbb{E}[\|\mathcal{L}(\phi)\|]\leq 1-2\varepsilon$ for $\varepsilon\geq O(1)$ and $\phi$ is sufficiently concentrated, it is generally possible to chose an event $\mathcal{A}_{\phi}$ of overwhelming probability.

As it will be seen in Subsection 4, this corollary finds its relevancy under convex concentration hypotheses, where the linear concentration seems to be the best concentration property to obtain on the resolvent $Q^{z}=(zI_{p}-\frac{1}{n}XX^{T})^{-1}$.

In order to satisfy the hypothesis of Corollary 3, but also for independent interest, we are now going to express the concentration of the product of convexly concentrated random matrices.

Given two convexly concentrated random vectors $X,Y\in E$ satisfying $X,Y\propto_{c}\mathcal{E}_{2}(\sigma)$, the convex concentration of the couple $(X,Y)\propto_{c}\mathcal{E}_{2}(\sigma)$ is ensured if:

1. 1.

   $X$ and $Y$ are independent

2. 2.

   $(X,Y)=u(Z)$ with $u$ affine and $Z\propto_{c}\mathcal{E}_{2}(\sigma)$.

We can then in particular state the concentration of $X+Y$ as it is a linear transformation of $(X,Y)$. For the product it is not as simple as for the Lipschitz concentration, let us first consider the particular case of the entry-wise product in $E=\mathbb{R}^{p}$. Since this result is not important for the rest of the paper, we left its proof in A.

The result of concentration of a product of matrices convexly concentrated was already proven in [MS11] but since their formulation is slightly different, we reprove in A the following result with the formulation required for the study of the resolvent.