Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model

Abstract

We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation.

In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker η that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker η* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix.

Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.

1. Introduction

Suppose we observe p-dimensional Gaussian vectors $\begin{array}{cc}{X}_i\stackrel{i.i.d}{~}\mathcal{N}\left(0,{\mathrm{\Sigma }}_p\right),& i=1,\dots ,n,\end{array}$ with Ʃ = Ʃ_p the underlying p-by-p population covariance matrix. To estimate Ʃ, we form the empirical (sample) covariance matrix $S=S_{n,p}=n^{-1}{\sum }_{i=1}^n{X}_i{X}_i^{\prime };$ this is the maximum likelihood estimator. Stein [1, 2] observed that the maximum likelihood estimator S ought to be improvable by eigenvalue shrinkage.

Write $S=V\mathrm{\Lambda }{V}^{\prime }$ for the eigendecomposition of S, where V is orthogonal and the diagonal matrix $\mathrm{\Lambda }=\text{diag}\left({\mathrm{\lambda }}_1,\dots ,{\mathrm{\lambda }}_p\right)$ contains the empirical eigenvalues. Stein [2] proposed to shrink the eigenvalues by applying a specific nonlinear mapping φ producing the estimate ${\widehat{\mathrm{\Sigma }}}_{\phi }=V\phi \left(\mathrm{\Lambda }\right)V^{\prime },$ where φ maps the space of positive diagonal matrices onto itself. In the ensuing half century, research on eigenvalue shrinkers has flourished, producing an extensive literature. We can point here only to a fraction, with pointers organized into early decades [3, 4, 5, 6, 7, 8], the middle decades [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], and the last decade [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Such papers typically choose some loss function $L_p:S_p^{+}\times S_p^{+}\to \left[0,\infty \right),$ where $S_p^{+}$ is the space of positive semidefinite p-by-p matrices, and develop a shrinker η with “favorable” risk $\mathbb{E}{L}_p\left(\mathrm{\Sigma },{\widehat{\mathrm{\Sigma }}}_{\eta }\left(S\right)\right).$

In high dimensional problems, p and n are often of comparable magnitude. There, the maximum likelihood estimator is no longer a reasonable choice for covariance estimation and the need to shrink becomes acute.

In this paper, we consider a popular large n, large p setting with p comparable to n, and a set of assumptions about Ʃ known as the Spiked Covariance Model [30]. We study a variety of loss functions derived from or inspired by the literature, and show that to each “reasonable” nonlinearity η there corresponds a well-defined asymptotic loss.

In the sibling problem of matrix denoising under a similar setting, it has been shown that there exists a unique asymptotically admissible shrinker [31, 32]. The same phenomenon is shown to exist here: for many different loss functions, we show that there exists a unique optimal nonlinearity η*, which we explicitly provide. Perhaps surprisingly, η* is the only asymptotically admissible nonlinearity, namely, it offers equal or better asymptotic loss than that of any other choice of η, across all possible Spiked Covariance models.

1.1. Estimation in the Spiked Covariance Model

Consider a sequence of covariance estimation problems, satisfying two basic assumptions.

[ASY(γ)] The number of observations n and the number of variables p_n in the n-th problem follows the proportional-growth limit p_n/n → γ, as n → ∞, for a certain 0 < γ ≤ 1.

Denote the population and sample covariances in the n-th problem by Σ = Σ_pn and S = S_n,pn and assume that the eigenvalues ℓ_i of Σp_n satisfy:

[Spike(ℓ₁, …, ℓ_r)] The r “spikes” ℓ₁ > … > ℓ_r ≥ 1 are fixed independently of n and p_n, and ℓ_r+1 = … = ℓ_pn = 1.

The spiked model exhibits three important phenomena, not seen in classical fixed-p asymptotics, that play an essential role in the construction of optimal estimators. Drawing on results from [33, 34, 35, 36, 37, 38], we highlight:

• a.
  Eigenvalue spreading. Consider model [Asy(γ)] in the null case ℓ₁ = … = ℓ_r = 1. The empirical distribution of the sample eigenvalues λ_1n, …, λ_pn converges as n → ∞ to a non-degenerate absolutely continuous distribution, the Marcenko-Pastur or ‘quarter-circle’ law [33]. The distribution, or ‘bulk’, is supported on a single interval, whose limiting ‘bulk edges’ are given by

  $${\mathrm{\lambda }}_{\pm }\left(\gamma \right)={\left(1\pm \sqrt{\gamma }\right)}^2.$$ (1.1)
• b.
  Top eigenvalue bias. Consider models [Asy(γ)] and [Spike(ℓ₁, …, ℓ_r)]. For i = 1, …, r, the leading sample eigenvalues satisfy

  $${\mathrm{\lambda }}_{in}\stackrel{a.s.}{\to }\mathrm{\lambda }\left({\ell }_i\right),$$ (1.2)

  where the ‘biasing’ function

  $$\begin{array}{cc}\mathrm{\lambda }\left(\ell \right)=\ell +\gamma \ell /\left(\ell -1\right),& \ell \ge {\ell }_{+}\left(\gamma \right),\end{array}$$ (1.3)

  $\mathrm{\lambda }\left(\ell \right)\equiv {\left(1+\sqrt{\gamma }\right)}^2={\mathrm{\lambda }}_{+}\left(\gamma \right)$ for ℓ ≤ ℓ₊(γ), the Baik-Ben Arous-Peché transition point

  $${\ell }_{+}\left(\gamma \right)=1+\sqrt{\gamma }.$$ (1.4)

  Thus the empirical eigenvalues λ_i are shifted upwards from their theoretical counterparts ℓ_i by an asymptotically predictable amount, of a size that exceeds γ even for very large signal strengths ℓ_i.

• c.
  Top eigenvector inconsistency. Again consider models [Asy(γ)] and [Spike(ℓ₁, …, ℓ_r)], noting that ℓ₁ > … > ℓ_r are distinct. The angles between the sample eigenvectors v_1n, …, v_pn, and the corresponding “true” population eigenvectors u_1n, …, u_pn, have non-zero limits:

  $$\begin{array}{cc}\left|\langle u_{in},{\upsilon }_{jn}\rangle \right|\stackrel{a.s.}{\to }{\delta }_{i,j}\cdot c\left({\ell }_i\right)& 1\le i,j\le r\end{array},$$ (1.5)

  where the cosine function is given by

  $$c\left(\ell \right)=\begin{array}{cc}\sqrt{\frac{1-\gamma /{\left(\ell -1\right)}^2}{1+\gamma /\left(\ell -1\right)}}& \ell \ge {\ell }_{+}\left(\gamma \right)\end{array},$$ (1.6)

  and c(ℓ) = 0 for ℓ ≤ ℓ₊(γ).

Loss functions and optimal estimation. Now consider a class of estimators for the population covariance , based on individual shrinkage of the sample eigenvalues. Specifically,

$$\widehat{\mathrm{\Sigma }}={\widehat{\mathrm{\Sigma }}}_{\eta }=\eta \left({\mathrm{\lambda }}_1\right)v_1{v}_1^{\prime }+\dots +\eta \left({\mathrm{\lambda }}_p\right)v_p{v}_p^{\prime },$$ (1.7)

where v_i is the sample eigenvector with sample eigenvalue λ_i and η(λ) is a scalar nonlinearity, η : ℝ⁺ → [1, ∞), so that the same function acts on each sample eigenvalue. While this appears to be a significant restriction from Stein’s use of vector functions φ [2], the discussion in Section 8 shows that nothing is lost in our setting by the restriction to scalar shrinkers.

Consider a family of loss functions $L={\left\{L_p\right\}}_{p=1}^{\infty }$ and a fixed nonlinearity η : [0, ∞) → ℝ. Define the asymptotic loss relative to L of the shrinkage estimator ${\widehat{\mathrm{\Sigma }}}_{\eta }$ in model [Spike(ℓ₁, …, ℓ_r)] by

$$L_{\infty }\left({\ell }_1,\dots ,{\ell }_r|\eta \right)=\underset{n\to \infty }{\lim }{L}_{p_n}\left({\mathrm{\Sigma }}_{p_n},{\widehat{\mathrm{\Sigma }}}_{\eta }\left(S_{n,p_n}\right)\right),$$ (1.8)

assuming such limit exists. If a nonlinearity η* satisfies

$$L_{\infty }\left({\ell }_1,\dots ,{\ell }_r|{\eta }^{*}\right)\le L_{\infty }\left({\ell }_1,\dots ,{\ell }_r|\eta \right)$$ (1.9)

for any other nonlinearity η, any r and any spikes ℓ₁, … , ℓ_r, and if for any η the inequality is strict at some choice of ℓ₁, … , ℓ_r, then we say that η* is the unique asymptotically admissible nonlinearity (nicknamed “optimal”) for the loss sequence L.

In constructing estimators, it is natural to expect that the effect of the biasing function λ(ℓ) in (1.3) might be undone simply by applying its inverse function ℓ(λ) given by

$$\begin{array}{cc}\ell \left(\mathrm{\lambda }\right)=\frac{\left(\mathrm{\lambda }+1-\gamma \right)+\sqrt{{\left(\mathrm{\lambda }+1-\gamma \right)}^2-4\mathrm{\lambda }}}{2}& \mathrm{\lambda }>{\mathrm{\lambda }}_{+}\left(\gamma \right).\end{array}$$ (1.10)

However, eigenvector inconsistency makes the situation more complicated (and interesting!), as we illustrate using Figure 1. Focus on the plane spanned by u₁, the top population eigenvector, and by v₁, its sample counterpart. We represent ${\ell }_1{u}_1{u}_1^{\prime },$ the top rank one component of Ʃ, by the vector ${\ell }_1{u}_1.$ The corresponding top rank one component of S is ${\mathrm{\lambda }}_1{v}_1{v}_1^{\prime },$ represented by λ₁v₁. If we apply the inverse function (1.10) to λ₁, we obtain $\ell \left({\mathrm{\lambda }}_1\right)v_1{v}_1^{\prime }.$ Since v₁ is not collinear with u₁, there is a non-vanishing error $\ell \left({\mathrm{\lambda }}_1\right)v_1{v}_1^{\prime }-{\ell }_1{u}_1{u}_1^{\prime }$ that remains, even though $\ell \left({\mathrm{\lambda }}_1\right)-{\ell }_1=O_p\left(n^{-1/2}\right).$ As the picture suggests, it is quite possible that a different amount of shrinkage, $\eta \left({\mathrm{\lambda }}_1\right)v_1{v}_1^{\prime }$ will lead to smaller error. However, we will see that the optimal choice of η depends greatly on the particular error measure $L_p\left(\mathrm{\Sigma },\widehat{\mathrm{\Sigma }}\right)$ that is chosen.

Figure 1:

Shrinking empirical eigenvalue λ₁ to a value η(λ₁) that is smaller than the inverse function ℓ(λ₁) may reduce the error of estimation.

To give the flavor of results to be developed systematically later, we now look at four error measures in common use. The first three, based on the operator, Frobenius and nuclear norms, use the singular values σ_j of $\widehat{\Sigma }-\Sigma$:

$$\begin{array}{l}{L}^O\left(\Sigma ,\widehat{\Sigma }\right)=\Vert \widehat{\Sigma }-\Sigma {\Vert }_{\infty }=\underset{i}{\max }{\sigma }_i,\text{}\\ L^F\left(\Sigma ,\widehat{\Sigma }\right)=\Vert \widehat{\Sigma }-\Sigma {\Vert }_2={\left(\sum _i{\sigma }_i^2\right)}^{1/2},\text{}\\ L^N\left(\Sigma ,\widehat{\Sigma }\right)=\Vert \widehat{\Sigma }-\Sigma {\Vert }_1=\sum _i{\sigma }_i,\text{}\\ L^{\text{St}}\left(\Sigma ,\widehat{\Sigma }\right)=\text{tr}\left({\Sigma }^{-1}\widehat{\Sigma }-I\right)-\text{log\hspace{0.17em}det}\left({\Sigma }^{-1}\widehat{\Sigma }\right).\end{array}$$ (1.11)

The fourth is Stein’s loss, widely studied in covariance estimation [1, 9, 39].

For convenience, we begin with the single spike model Spike(ℓ), so that $\mathrm{\Sigma }={\mathrm{\Sigma }}_{\ell }=I+\left(\ell -1\right)u_1{u}_1^{\prime }.$ When η is continuous, the losses have a deterministic asymptotic limit $L_{\infty }\left(\ell |\eta \right)$ defined in−(1.8).

For many losses, including (1.11), this deterministic limiting loss has a simple form, and we can evaluate, often analytically, the optimal shrinkage function, namely the shrinkage function satisfying (1.9). For example, writing ${\eta }^{*}\left(\mathrm{\lambda }\right)={\eta }_{*}\left(\ell \left(\mathrm{\lambda }\right)\right),$ for the four popular loss functions (1.11) we find that on $\ell >1+\sqrt{\gamma }$ the corresponding four optimal shrinkers are

$$\begin{array}{ll}{\eta }_{*}^O\left(\ell \right)=\ell \hfill & {\eta }_{*}^F\left(\ell \right)=\ell c^2+s^2\hfill \\ \text{}{\eta }_{*}^N\left(\ell \right)=\text{max}\left(1+\left(\ell -1\right)\left(1-2s^2\right),1\right)\hfill & {\eta }_{*}^{\text{St}}\left(\ell \right)=\ell /\left(c^2+\ell s^2\right),\hfill \end{array}$$ (1.12)

where s² = 1 − c². Figure 2 shows these four optimal shrinkers as a function of the sample eigen value λ. These are just four examples; The full list of optimal shrinkers we discover in this paper appears in Table 2 below. In all cases, ${\eta }_{*}^{}\left(\ell \right)\equiv 1$ for $\ell \le 1+\sqrt{\gamma }.$ Figure 3 in Section 6 below shows all the full list of optimal shrinkers when γ = 1.

Figure 2:

Vertical axis: optimal shrinkers η_* from (1.12), shown as functions η_*(ℓ(λ)) of the empirical eigenvalue λ, horizontal axis. Here γ = lim p_n/n = 1, so λ₊(γ) = 4. (Color online.)

Table 2:

Optimal shrinkers η^*(λ) for 18 of the loss families L discussed. Values shown are shrinkers for λ > λ₊(γ). All shrinkers obey η^*(λ) = 1 for λ≤λ₊(γ). Here, ℓ, c and s depend on λ (and implicitly on γ) according to (1.10), (1.6) and $s=\sqrt{1-c^2}$. In cases marked “N/A” the optimal shrinker does not seem to admit a simple closed form, but can be easily calculated numerically.

Pivot	MatrixNorm
	Frobenius	Operator	Nuclear
A − B	ℓc2 + s2	ℓ	max(l + (ℓ − l)(l − 2s2), l)
A−1 − B−1	$\frac{\ell }{c^2+\ell s^2}$	ℓ	$\text{max\hspace{0.17em}}\left(\frac{\ell }{c^2+(2\ell -1)s^2},1\right)$
A−1B − I	$\frac{\ell c^2+{\ell }^2{s}^2}{c^2+{\ell }^2{s}^2}$	N/A	$\text{max\hspace{0.17em}}\left(\frac{\ell }{c^2+{\ell }^2{s}^2},1\right)$
B−1A − I	$\frac{{\ell }^2{c}^2+s^2}{\ell c^2+s^2}$	N/A	$\text{max\hspace{0.17em}}\left(\frac{{\ell }^2{c}^2+s^2}{\ell },1\right)$
A−1/2 BA−1/2 − I	$1+\frac{\left(\ell -1\right)c^2}{{\left(c^2+\ell s^2\right)}^2}$	$1+\frac{\ell -1}{c^2+\ell s^2}$	$\text{max\hspace{0.17em}}\left(\frac{\ell -{\left(\ell -1\right)}^2{c}^2{s}^2}{{\left(c^2+\ell s^2\right)}^2},1\right)$
	Statistical Measures
	St	Ent	Div
Stein	$\frac{\ell }{c^2+\ell s^2}$	ℓc2 + s2	$\sqrt{\frac{{\ell }^2{c}^2+\ell s^2}{c^2+\ell s^2}}$
Fréchet	${\left(\sqrt{\ell }{c}^2+s^2\right)}^2$
Affine	$\frac{\left(1+c^2\right)\ell +s^2}{1+c^2+\ell s^2}$

Figure 3:

Optimal Shrinkers for 26 Component Loss Functions for γ = 1 and 4 ≤ λ ≤ 10. Upper Left:Frobenius-norm-based losses; Lower Left: Nuclear-Norm based losses; Upper Right: Operator-norm-based losses; Lower Right: Statistical Discrepancies. (Color online; curves jittered in vertical axis to avoid overlap.) The supplemental article [40] contains an larger version of these plots. Reproducibility advisory: The code supplement [41] includes a script that reproduces any one of these individual curves.

The main conclusion is that the optimal shrinkage function depends strongly on the loss function chosen. The operator norm shrinker ${\eta }_{*}^O$ simply inverts the biasing function λ(ℓ), while the other functions shrink by much larger, and very different, amounts, with ${\eta }_{*}^{\text{St}}$ typically shrinking most. There are also important qualitative differences in the optimal shrinkers: ${\eta }_{*}^O$ is discontinuous at the bulk edge $\mathrm{\lambda }={\mathrm{\lambda }}_{+}\left(\gamma \right).$ The others are continuous, but ${\eta }_{*}^N$ has the additional feature that it shrinks a neighborhood of the bulk to 1.

Remark. The optimal shrinker also depends on γ, so we might write η*(λ, γ). In model [Asy(γ)], one can use the same γ for each problem size n. Alternatively, in the n-th problem, one might use γ_n = p_n/n. The former choice is simpler, as η* can be regarded as a univariate function of λ, and so we make it in Sections 1–6. The latter choice is preferable technically, and perhaps also in practice, when one has p and n, but not γ. It does, however, require us to treat η(λ, c) as a bivariate function – see Section 7.

1.2. Some key observations

The sections to follow construct a framework for evaluating and optimizing the asymptotic loss(1.8). We highlight here some observations that will play an important role. Beforehand, let us introduce a useful modification of (1.7) to a rank-aware shrinkage rule:

$${\widehat{\mathrm{\Sigma }}}_{\eta ,r}={\displaystyle \sum }_{i=1}^r\eta \left({\mathrm{\lambda }}_i\right)v_i{v}_i^{\prime }+{\displaystyle \sum }_{i=r+1}^p{v}_i{v}_i^{\prime },$$ (1.13)

where the dimension r of the spiked model is taken as known. While our main results concern estimators ${\widehat{\mathrm{\Sigma }}}_{\eta }$ that naturally do not require r to be known in advance, it will be easier conceptually and technically to analyze rank-aware shrinkage rules as a preliminary step.

[Obs. 1] Simultaneous block diagonalization. (Lemmas 1 and 5). There exists a (random) basis W such that

$$\begin{array}{l}{W}^{\prime }\text{Σ}W=\left({\oplus }_i{A}_i\right)\oplus I_{p-2r}\\ W^{\prime }{\widehat{\Sigma }}_{\eta ,r}W=\left({\oplus }_i{B}_i\right)\oplus I_{p-2r},\end{array}$$

where A_i and B_i are square blocks of equal size d_i, and Σd_i = 2r. (Here and below, A ⊕ B denotes a block-diagonal matrix with blocks A and B).

[Obs. 2] Decomposable loss functions. The loss functions (1.11) and many others studied below satisfy

$$L_p\left(\mathrm{\Sigma },{\widehat{\mathrm{\Sigma }}}_{\eta ,r}\right)={\displaystyle \sum }_i{L}_{di}\left(A_i,B_i\right)$$

or the corresponding equality with sum replaced by max.

[Obs. 3] Asymptotic deterministic loss. (Lemmas 3 and 7). For rank-aware estimators, when η and L are suitably continuous, almost surely

$$L_{\infty }\left({\ell }_1,\dots ,{\ell }_r|\eta \right)=\underset{p\to \infty }{\lim }{L}_p\left(\mathrm{\Sigma },{\widehat{\mathrm{\Sigma }}}_{\eta ,r}\right).$$

[Obs. 4] Asymptotic equivalence of losses.(Proposition 2). Conclusions derived for rank-aware estimators (1.13) carry over to the original estimators (1.7) because, under suitable condition

$$L_p\left(\mathrm{\Sigma },{\widehat{\mathrm{\Sigma }}}_{\eta }\right)-L_p\left(\mathrm{\Sigma },{\widehat{\mathrm{\Sigma }}}_{\eta ,r}\right){\to }_{P}0.$$

This relies on the fact that in the [Spike(ℓ₁, …, ℓ_r)] model, the sample noise eigenvalues λ_in, i ≥ r+1 “stick to the bulk” in an appropriate sense.

1.3. Organization of the paper

For simplicity of exposition, we assume a single spike, r = 1, in the first half of the paper. [Obs. 1], [Obs. 2] and [Obs. 3] are developed respectively in Sections 2, 3 and 4, arriving at an explicit formula for the asymptotic loss of a shrinker. Section 5 illustrates the assumptions with our list of 26 decomposable matrix loss functions. In Section 6 we use the formula to characterize the asymptotically unique admissible nonlinearity for any decomposable loss, provide an algorithm for computing the optimal nonlinearity, and provide analytical formulas for many of the 26 losses. Section 7 extends the results to the general case where r > 1 spikes are present. We develop [Obs. 4] , remove the rank-aware assumption and explore some new phenomena that arise in cases where the optimal shrinker turns out to be discontinuous. In Section 8 we show, at least for Frobenius and Stein losses, that our optimal univariate shrinkage estimator, which applies the same scalar function to each sample eigenvalue, in fact asymptotically matches the performance of the best orthogonally-equivariant covariance estimator under assumption [Spike(ℓ₁, …, ℓ_r)]. Section 9 extends to the more general spiked model with ${\mathrm{\Sigma }}_p=diag\left({\ell }_1,\dots ,{\ell }_r,{\sigma }^2,\dots ,{\sigma }^2\right)$ for σ > 0 known or unknown. Section 10 discusses our results in light of the high-dimensional covariance estimation work of El Karoui [24] and Ledoit and Wolf [26]. Some proofs and calculations are deferred to the supplementary article [40], where we also evaluate and document the strong signal (large-ℓ) asymptotics of the optimal shrinkage estimators, and the asymptotic percent improvement over naive hard thresholding of the sample covariance eigenvalues. Additional technical details and software are provided in the Code Supplement available online as a permanent URL from the Stanford Digital Repository [41].

2. Simultaneous Block-Diagonalization

We first develop [Obs. 1] in the simplest case, r = 1, assumping a rank-aware shrinker. In general, the estimator ${\widehat{\mathrm{\Sigma }}}_{\eta }$ and estimand Ʃ are not simultaneously diagonalizable. However, in the particular case that both are rank-one perturbations of the identity, we will see that simultaneous block diagonalization is possible.

Some notation is needed. We denote the eigenvalues and eigenvectors of the spectral decompostion $S_{n,p_n}=V\mathrm{\Lambda }{V}^{\prime }$ by

$$spec\left(S_{n,p_n}\right)=\left[\left({\mathrm{\lambda }}_{1n},\dots ,{\mathrm{\lambda }}_{pn}\right),\left(v_{1n},\dots ,v_{pn}\right)\right].$$

Whenever possible, we supress the index n and write e.g. S, λ_i and v_i instead. Similarly, we often write Ʃ_p or even Ʃ for ${\mathrm{\Sigma }}_{p_n}.$

Lemma 1. Let Ʃ and $\widehat{\mathrm{\Sigma }}$ be (fixed, nonrandom) p-by-p symmetric positive definite matrices with

$$spec\left(\mathrm{\Sigma }\right)=\left[\left(\ell ,1,\dots ,1\right),\left(u_1,\dots ,u_p\right)\right]$$ (2.1)

$$spec\left(\widehat{\mathrm{\Sigma }}\right)=\left[\left(\eta ,1,\dots ,1\right),\left(v_1,\dots ,v_p\right)\right].$$ (2.2)

Let $c=\langle u_1,v_1\rangle$ and $s=\sqrt{1-c^2}.$ Then there exists an orthogonal matrix W, which depends on Ʃ and $\widehat{\mathrm{\Sigma }},$ such that

$$W^{\prime }\mathrm{\Sigma }W=A\left(\ell \right)\oplus I_{p-2},$$ (2.3)

$$W^{\prime }\widehat{\Sigma }W=B\left(\eta ,c\right)\oplus I_{p-2},$$ (2.4)

where the fundamental 2 × 2 matrices A and B are given by

$$\begin{array}{cc}A\left(\ell \right)=\left[\begin{array}{cc}\ell & 0\\ 0& 1\end{array}\right],& B\left(\eta ,c\right)=I_2+\left(\eta -1\right)\left[\begin{array}{c}c\\ s\end{array}\right]\begin{array}{c}\left[\begin{array}{cc}c& s\end{array}\right]\\ \hspace{0.17em}\end{array}.\end{array}$$ (2.5)

Proof. Let $\mathrm{\Delta }=\text{diag}\left(\eta ,1,\dots ,1\right)=I+\left(\eta -1\right)e_1{e}_1^{\prime },$ where e₁ denotes the unit vector in the first co-ordinate direction. It is evident that

$$\begin{array}{cc}\mathrm{\Sigma }=I+\left(\ell -1\right)u_1{u}_1^{\prime },& \widehat{\mathrm{\Sigma }}=I+\left(\eta -1\right)v_1{v}_1^{\prime }.\end{array}$$ (2.6)

It is natural, then, to work in the “common” basis of u₁ and v₁. We apply one step of Gram-Schmidt if we can, setting

$$z=\{\begin{array}{ll}\left(v_1-c{u}_1\right)/s\hfill & \text{if }s\ne 0\hfill \\ u_p\hfill & \text{if }s=0.\hfill \end{array}$$

In the second–exceptional–case, v₁ = ±u₁, so we pick a convenient vector orthogonal to u₁. In either case, the columns of the p × 2 matrix W₂ = [u₁ z] are orthonormal and their span contains both u₁ and v₁. Now fill out W₂ to an orthogonal matrix $W=\left[\begin{array}{cc}{W}_2& W_2^{\perp }\end{array}\right].$ Observe now that if y lies in the column span of W₂ and α is a scalar, then necessarily

$$\begin{array}{ll}{W}^{\prime }\left(I_p+\alpha y{y}^{\prime }\right)W=\left(I_2+\alpha \check{y}\check{y}\right)|\oplus I_{p-2},\hfill & \check{y}=W_2^{\prime }y.\hfill \end{array}$$

The expressions (2.3) – (2.5) now follow from the rank one perturbation forms (2.6) along with

$$W_2^{\prime }{u}_1=\left[\begin{array}{c}{u}_1^{\prime }{u}_1\\ z^{\prime }{u}_1\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right],\text{ and }{W^{\prime }}_2{\upsilon }_1=\left[\begin{array}{c}{u}_1^{\prime }{\upsilon }_1\\ z^{\prime }{\upsilon }_1\end{array}\right]=\left[\begin{array}{c}c\\ s\end{array}\right].$$

3. Decomposable Loss Functions

Here and below, by loss function L_p we mean a function of two p-by-p positive semidefinite matrix arguments obeying L_p ≥ 0, with L_p(A, B) = 0 if and only if A = B. A loss family is a sequence $L={\left\{L_p\right\}}_{p=1}^{\infty },$ one for each matrix size p. We often write loss function and refer to the entire family. [Obs. 2] calls out a large class of loss functions which naturally exploit the simultaneously block-diagonalizability property of Lemma 1; we now develop this observation.

Definition 1. Orthogonal Invariance. We say the loss function L_p(A, B) is orthogonally invariant if for each orthogonal p-by-p matrix O,

$$L_p\left(A,B\right)=L_p\left(OA{O}^{\prime },OB{O}^{\prime }\right).$$

For given p and a given sequence of block {d_i} sizes ∑_i d_i=p, consider block-diagonal matrix decompositions of p by p matrices A and B into blocks Aⁱ and Bⁱ of size d_i:

$$A={\oplus }_i{A}^i B={\oplus }_i{B}^i.$$ (3.1)

Definition 2. Sum-Decomposability and Max-Decomposability. We say the loss function L_p(A, B) is sum-decomposable if for all decompositions (3.1),

$$L_p\left(A,B\right)={\displaystyle \sum }_i{L}_{d_i}\left(A^i,B^i\right).$$

We say that it is max-decomposable if if for all decompositions (3.1),

$$L_p\left(A,B\right)=\underset{i}{\max }{L}_{d_i}\left(A^i,B^i\right).$$

Clearly, such loss functions can exploit the simultaneous block diagonalization of Lemma 1. Indeed,

Lemma 2. Reduction to Two-Dimensional Problem. Consider an orthogonally invariant loss function, L_p, which is sum- or max-decomposable. Suppose that Ʃ and $\widehat{\Sigma }$ satisfy (2.1) and (2.2) respectively. Then

$$L_p\left(\Sigma ,\widehat{\Sigma }\right)=L_2\left(A\left(\ell \right),B\left(\eta ,c\right)\right).$$

Proof. Lemma 1 provides a change of basis W yielding decompositions (2.3) and (2.4). From the invariance and decomposability hypotheses,

$$\begin{array}{ll}{L}_p\left(\Sigma ,\widehat{\Sigma }\right)\hfill & =L_p\left(W^{\prime }\Sigma W,W^{\prime }\widehat{\Sigma }W\right)\hfill \\ \hfill & \text{}=L_p\left(A\left(\ell \right)\oplus I_{p-2},B\left(\eta \right),c\right)\oplus I_{p-2})\hfill \\ \hfill & \text{}=L_2\left(A\left(\ell \right),B\left(\eta ,c\right)\right).\hfill \end{array}$$

4. Asymptotic Loss in the Spiked Covariance Model

Consider the spiked model with a single spike, r = 1, namely, make assumptions [Asy(γ)] and [Spike(ℓ)]. The principal 2× 2 block estimator occurring in Lemmas 1 and 2 is B(η(λ_1n),c_1n) where λ_1n is the largest eigenvalue of S_n and $c_{1n}=\langle u_{1n},{\upsilon }_{1n}\rangle .$If η is continuous, then the convergence results (1.2) and (1.5) imply that the principal block converges as n → ∞. Specifically,

$$B\left(\eta \left({\mathrm{\lambda }}_{\text{1}n}\right),c_{1n}\right)\stackrel{a.s.}{\to }B\left(\eta \left(\mathrm{\lambda }\left(\ell \right)\right),c\left(\ell \right)\right)=:B\left(\ell ,\eta \right),$$ (4.1)

say, with the convergence occurring in all norms on 2 × 2 matrices.

In accord with [Obs. 3], we now show that the asymptotic loss (1.8) is a deterministic, explicit function of the population spike ℓ. For now, we will continue to assume that the shrinker η is rank-aware. Alternatively, we can make a different simplifying assumption on η, which will be useful in what follows:

Definition 3. We say that a scalar function η : [0, ∞) → [1, ∞) is a bulk shrinker if η(λ) = 1 when λ ≤ λ₊(γ), and a neighborhood bulk shrinker if for some ∊ > 0, η(λ) = 1 whenever λ ≤ λ₊(γ) + ∊.

The neighborhood bulk shrinker condition on η is rather strong, but does hold for ${\eta }_{*}^N$ in (1.12), for example. (Note that our definitions ignore the lower bulk edge $\mathrm{\lambda }\_\left(\text{γ}\right),$ which is of less interest in the spiked model.)

Lemma 3. A Formula for the Asymptotic Loss. Adopt models [Asy(γ)] and [Spike(ℓ)] with ℓ₁, …, ℓ_r > ℓ₊(γ). Suppose (a) that the family L = {L_p} of loss functions is orthogonally invariant and sum- or max-decomposable, and that $B\mapsto L_2\left(A,B\right)$ is continuous. Let ${\widehat{\Sigma }}_{\eta }={\widehat{\Sigma }}_{\eta }\left(S_{n,p_n}\right)$ be given by (1.7), and let ${\widehat{\Sigma }}_{\eta ,1}$ be the corresponding rank-aware shrinkage rule (1.13) for r = 1. Suppose the scalar nonlinearity η is continuous on (λ₊(γ), ∞) . Then

$$L_{p_n}\left({\Sigma }_{p_n},{\widehat{\Sigma }}_{\eta ,1}\right)\stackrel{a.s.}{\to }{L}_2\left(A\left(\ell \right),B\left(\ell ,\eta \right)\right),$$ (4.2)

Furthermore, if (b) η is a neighborhood bulk shrinker, then $L_{p_n}\left({\Sigma }_{p_n}{\widehat{\Sigma }}_{\eta }\right)$ also has this limit a.s.

Each of the 26 losses considered in this paper satisfies conditions (a).

Proof. In the rank-aware case ${\widehat{\Sigma }}_{\eta }={\widehat{\Sigma }}_{\eta ,1}$ satisfies

$$\text{spec}\left({\widehat{\Sigma }}_{\eta }\right)=\left[\left(\eta \left({\mathrm{\lambda }}_{\text{1}n}\right),1\dots ,1\right),\left({\upsilon }_{1n},\dots ,{\upsilon }_{pn}\right)\right],$$

Lemma 2 implies that

$$L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta }\right)=L_2\left(A\left(\ell \right),B\left(\eta \left({\mathrm{\lambda }}_{\text{1}n}\right),c_{1n}\right)\right)\stackrel{a.s.}{\to }{L}_2\left(A\left(\ell \right),B\left(\ell ,\eta \right)\right),$$

where the limit on the right hand side follows from convergence (4.1) and the assumed continuity of L₂.

Now assume that η is a neighborhood bulk shrinker. From (1.2) we know that ${\mathrm{\lambda }}_{1n}\stackrel{a.s.}{\to }\mathrm{\lambda }\left(\ell \right)$ From eigenvalue interlacing (see (7.11) below) we have ${\mathrm{\lambda }}_{2n}\le {\mu }_{1n}$, where μ_1n is the largest eigenvalue of a white Wishart matrix W_pn-1(n,I), and satisfies ${\mu }_{1n}\stackrel{a.s.}{\to }{\mathrm{\lambda }}_{+},$ from [42]. Let ∊ > 0 be small enough that ${\mathrm{\lambda }}_{+}+\in <\mathrm{\lambda }\left(\ell \right)$ and also lies in the neighborhood shrunk to 1 by η. Hence, there exists a random variable $\widehat{n}$ such that almost surely, ${\mathrm{\lambda }}_{2n}<{\lambda }_{+}+ϵ<{\mathrm{\lambda }}_{1n}$ for all $n>\widehat{n}.$ For such n, the first display above of this proof applies and we then obtain the second display as before.

5. Examples of Decomposable Loss Functions

Many of the loss functions that appear in the literature are Pivot-Losses. They can be obtained via the following common recipe:

Definition 4. Pivots. A matrix pivot is a matrix-valued function Δ(A, B) of two real positive definitee matrices A, B such that: (i) Δ(A, B) = 0 if and only if A = B, (ii) Δ is orthogonally equivariant and (iii) Δ respects block structure in the sense that

$$\Delta \left(OA{O}^{\prime },OB{O}^{\prime }\right)=O\Delta \left(A,B\right)O^{\prime },$$ (5.1)

$$\Delta \left(\oplus A^i,\oplus B^i\right)=\oplus \Delta \left(A^i,B^i\right)$$ (5.2)

for any orthogonal matrix O of the appropriate dimension.

Matrix pivots can be symmetric-matrix valued, for example Δ(A, B) = A – B, but need not be, for example Δ(A, B) = A⁻¹B − I.

Definition 5. Pivot-Losses. Let g be a non-negative function of a symmetric matrix variable that is definite: g(A) = 0 if and only if A = 0, and orthogonally invariant: g(OΔO‵) = g(Δ) for any orthogonal matrix O. A symmetric-matrix valued pivot Δ induces an orthgonally-invariant pivot loss

$$L\left(A,B\right)=g\left(\Delta \left(A,B\right)\right).$$ (5.3)

More generally, for any matrix pivot Δ, set $|\Delta |={\left({\Delta }^{\prime }\Delta \right)}^{1/2}$ and define

$$L\left(A,B\right)=g\left(|\Delta |\left(A,B\right)\right).$$ (5.4)

An orthogonally invariant function g depends on its matrix argument Δ or |Δ| only through its eigenvalues or singular values δ₁, ..., δ_p. We abuse notation to write g(Δ) = g(δ₁, ..., δ_p). Observe that if g has either of the forms

$$g\left({\delta }_1,\dots ,{\delta }_p\right)={\displaystyle \sum _j{g}_1\left({\delta }_j\right)} \text{or} g\left({\delta }_1,\dots ,{\delta }_p\right)=\underset{j}{\max }{g}_1\left({\delta }_j\right),$$

for some univariate g₁, then the pivot loss L(A, B) = g(Δ(A, B)) (symmetric pivot) or L(A,B)=g(|Δ|(A,B)) (general pivot) is respectively sum- or max-decomposable. In case Δ is symmetric, the two definitions agree so long as g₁ is an even function of δ.

5.1. Examples of Sum-Decomposable Losses

There are different strategies to derive sum-decomposable pivot-losses. First, we can use statistical discrepancies between the Normal distributions $\mathcal{N}(0,A)$ and $\mathcal{N}(0,B)$:

1. Stein Loss [1, 9, 39]: Stein’s Loss is defined as

   $$L^{st}\left(A,B\right)=\text{tr}\left(A^{-1},B-I\right)-\log \left(\det \left(B\right)/\det \left(A\right)\right).$$

   This is just twice the Kullback distance $D_{KL}\left(\mathcal{N}\left(0,B\right)\left|\right|\mathcal{N}\left(0,A\right)\right).$ Stein’s loss is a pivot-loss with respect to Δ(A, B) = A^−1/2BA^−1/2 and $g(\Delta )=\text{tr}\left(\Delta -I\right)-\log \det (\Delta )={\Sigma }_i{g}_1\left({\delta }_i\right),$ where g₁(δ) = δ − 1 − log δ.

2. Entropy/Divergence Losses: Because the Kullback discrepancy is not symmetric in its arguments, we may consider two other losses: reversing the arguments we get Entropy loss L^ent(A, B) = L^st(B, A) [11, 15] and summing the Stein and Entropy losses gives divergence loss:

   $$L^{di\upsilon }\left(A,B\right)=L^{st}\left(A,B\right)+L^{st}\left(B,A\right)=\text{tr}\left(A^{-1},B-I\right)+\text{tr}\left(B^{-1}A-I\right),$$

   see [43, 18]. Each can be shown to be sum-decomposable, following the same argument as above.
3. Bhattarcharya/Matusita Affinity [44, 45]: Let

   $$L^{aff}\left(A,B\right)=\frac{1}{2}\log \frac{\left|A+B\right|/2}{{\left|A\right|}^{1/2}{\left|B\right|}^{1/2}}.$$
   This measures the statistical distinguishability of $\mathcal{N}\left(0,A\right)$ and $\mathcal{N}\left(0,B\right)$ based on independent observations, since $L^{aff}=\frac{1}{2}\log \left(\int \sqrt{\varphi A}\sqrt{\varphi B}\right)$ with ϕ_A and ϕ_B the densities of $\mathcal{N}\left(0,A\right)$ and $\mathcal{N}\left(0,B\right).$ Hence convergence of affinity loss to zero is equivalent to convergence of the underlying densities in Hellinger or Variation distance. This is a pivot-loss w.r.t Δ(A, B) = A^−1/2BA^−1/2 and

   $$g(\Delta )=\frac{1}{4}\log \left(\det \left(2I+\Delta +{\Delta }^{-1}\right)/4\right)={\displaystyle \sum _i{g}_1\left({\delta }_i\right)},$$

   as is seen by setting C = A^−1/2(A + B)B^−1/2 and noting that C^′C = (2I + Δ + Δ⁻¹). Here,$g_1\left(\delta \right)=\frac{1}{4}\log \left(2+\delta +{\delta }^{-1}\right)/4.$

4. Fréchet Discrepancy [46, 47]: Let L^fre(A, B) = tr(A + B − 2A^1/2B^1/2). This measures the minimum possible mean-squared difference between zero-mean random vectors with covariances A and B respectively. This is a pivot-loss w.r.t Δ(A, B) = A^1/2 − B^1/2, and $g(\Delta )=\text{tr}\left({\Delta }^2\right)={\Sigma }_i{g}_1\left({\delta }_i\right)$ with g₁(δ) = δ².

Second, we may obtain sum-decomposable pivot-losses L(A, B) = g(Δ(A, B)) by simply taking g to be one of the standard matrix norms:

1. Squared Error Loss [3, 28, 25, 26]: Let $L^{F,1}\left(A,B\right)=\Vert A-B{\Vert }_F^2.$ This is a pivot-loss w.r.t Δ(A, B) = A − B and $g(\Delta )=\text{tr}{\Delta }^{\prime }\Delta ={\Sigma }_i{g}_1\left({\delta }_i\right)$ with g₁(δ) = δ².

2. Squared Error Loss on Precision [8]: Let $L^{F,2}\left(A,B\right)=\Vert A^{-1}-B^{-1}{\Vert }_F^2.$ This is a pivot-loss w.r.t Δ(A, B) = A⁻¹ − B⁻¹ and g(Δ)=tr Δ’Δ.

3. Nuclear Norm Loss. Let $L^{N,1}\left(A,B\right)=\Vert A-B{\Vert }_{*}$ where $\Vert \Delta {\Vert }_{*}$ denotes the nuclear norm of the matrix Δ, i.e. the sum of its singular values. This is a kpivot-loss k w.r.t Δ(A, B) = A – B and $g(\Delta )={\Sigma }_i\left|{\delta }_i\right|.$

4. Let $L^{F,3}\left(A,B\right)=\Vert A^{-1}B-I{\Vert }_F^2.$ This is a pivot-loss w.r.t Δ(A, B) = A⁻¹B – I. It was studied in [48, 6, 10] and later work.

5. Let $L^{F,7}\left(A,B\right)=\Vert \log A^{-1/2}B{A}^{-1/2}{\Vert }_F^2,$ where log() denotes the matrix logarithm¹ [51, 49]. This is a pivot-loss w.r.t

$$\Delta \left(A,B\right)=\log \left(A^{-1/2}B{A}^{-1/2}\right).$$

5.2. Examples of Max-Decomposable Losses

Max-decomposable losses arise by applying the operator norm (the maximal singular value or eigenvalue of a matrix) to a suitable pivot. Here are a few examples:

1. Operator Norm Loss [52]: Let $L^{O,1}\left(A,B\right)=\Vert A-B{\Vert }_{op}.$This is a pivot-loss w.r.t Δ(A, B) = A – B and $g(\Delta )=\Vert \Delta {\Vert }_{op}={\max }_i{\delta }_i.$

2. Operator Norm Loss on Precision: Let $L^{O,2}\left(A,B\right)=\Vert A^{-1}-B^{-1}{\Vert }_{op}.$ This is a pivot-loss w.r.t. Δ(A, B) = A⁻¹ − B⁻¹.

3. Condition Number Loss: Let $L^{O,7}\left(A,B\right)=\Vert \log \left(A^{-1/2}B{A}^{-1/2}\right){\Vert }_{op}.$ This is a pivot-loss w.r.t Δ(A, B) = log(A^−1/2BA^−1/2), related to [29]. In the spiked kmodel discussed below, L^O,7 effectively measures the condition number of A^−1/2BA^−1/2.

We adopt the systematic naming scheme L^norm,pivot where norm ∈ {F, O, N}, and pivot ∈ {and 1, ..., 7}. This set of 21 combinations covers the previous matrix norm examples and adds some more. Together with Stein’s loss and the others based on statistical discrepancy mentioned above, we arrive at a set of 26 loss functions, Table 1, to be studied in this paper.

Table 1:

Systematic notation for the 26 loss functions considered in this paper.

	MatrixNorm
Pivot	Frobenius	Operator	Nuclear
A − B	LF,1	LO,1	LN,1
A−1 − B−1	LF,2	LO,2	LN,2
A−1B − I	LF,3	LO,3	LN,3
B−1A − I	LF,4	LO,4	LN,4
A−1 B + B−1 A − 2I	LF,5	LO,5	LN,5
A−1/2 BA−1/2 − I	LF,6	LO,6	LN,6
Log(A−1/2 BA−1/2)	LF,7	LO,7	LN,7
	Statistical Measures
	St	Ent	Div
Stein	Lst	Lent	Ldiv
Affinity	Laff
Fréchet	Lfre

6. Optimal Shrinkage for Decomposable Losses

6.1. Formally Optimal Shrinker

Formula (4.2) for the asymptotic loss has only been shown to hold in the single spike model and only for a certain class of nonlinearities η. In fact, the same is true in the r-spike model and for a much broader class of nonlinearities η. To preserve the narrative flow of the paper, we defer the proof, which is more technical, to Section 7. Instead, we proceed under the single spike model, and simply assume that $L_{\infty }\left(\ell |\eta \right)$ from (4.2) is the correct limiting loss, and draw conclusions on the optimal shape of the shrinker η.

Definition 6. Optimal Shrinker. Let $L={\left\{L_p\right\}}_{p=1}^{\infty }$ be a given loss family and let $L_{\infty }\left(\ell |\eta \right)$ be the asymptotic loss corresponding to a nonlinearity η, as defined in (4.2), under assumption [Asy(γ)] . If η^∗ satisfies

$$L_{\infty }\left(\ell |{\eta }^{\ast }\right)=\underset{\eta }{\min }{L}_{\infty }\left(\ell |\eta \right),\forall \ell \ge 1,$$ (6.1)

and for any η ≠ η^∗ there exists ℓ ≥ 1 with L_∞(ℓ, η*) < L_∞(ℓ, η), then we say that η^∗ is the formally optimal shrinker for the loss family L and shape factor γ, and denote the corresponding shrinkage rule by $\text{λ}\mapsto {\eta }^{*}\left(\text{λ};\gamma ,L\right).$

Below, we call formally optimal shrinkers simply “optimal”. By definition, the optimal shrinkage rule η^∗(λ ; γ, L) is the unique admissible rule, in the asymptotic sense, among rules of the form ${\widehat{\Sigma }}_{\eta }\left(S_{n,p}\right)=V\eta \left(\Lambda \right)V^{\prime }$ in the single-spike model. In the single spiked model (and as we show later, generally in the spiked model) one never regrets using the optimal shrinker over any other (reasonably regular) univariate shrinker. In light of our results so far, an obvious characterization of an optimal shrinker is as follows.

Theorem 1. Characterization of Optimal Shrinker. Let $L={\left\{L_p\right\}}_{p=1}^{\infty }$ be a loss family. Define

$$F\left(\ell ,\eta \right)=L_2\left(\left[\begin{array}{cc}\ell & 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1+\left(\eta -1\right)c^2& \left(\eta -1\right)cs\\ \left(\eta -1\right)cs& 1+\left(\eta -1\right)s^2\end{array}\right]\right).$$ (6.2)

Here, c = c(ℓ) and s = s(ℓ) satisfy $c^2\left(\ell \right)=\frac{1-\gamma /{\left(\ell -1\right)}^2}{1+\gamma /\left(\ell -1\right)}$ and s²(ℓ) = 1 − c²(ℓ). Suppose that for any ℓ > ℓ₊(γ), there exists a unique minimizer

$${\eta }^{\ast }\left(\ell \right):=argmi{n}_{\eta \ge 1}F\left(\ell ,\eta \right).$$ (6.3)

Further suppose that for every $1\le \ell \le {\ell }_{+}\left(\gamma \right)$ we have argmin ${}_{\eta \ge 1}G\left(\eta \right)=1,$ where

$$G\left(\ell ,\eta \right)=L_2\left(\left[\begin{array}{cc}\ell & 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \eta \end{array}\right]\right).$$ (6.4)

Then the shrinker

$${\eta }^{\ast }\left(\text{λ}\right)=\{\begin{array}{ll}{\eta }^{\ast }\left(\ell \left(\text{λ}\right)\right)\hfill & \ell >\text{λ}+\left(\text{γ}\right)\hfill \\ 1\hfill & 1\le \ell \le \text{λ}+\left(\text{γ}\right)\hfill \end{array},$$

where ℓ(λ) is given by (1.10), is the optimal shrinker of the loss family L.

Many of the 26 loss families discussed in Section 3 admit a closed form expression for the optimal shrinker; see Table 2. For others, we computed the optimal shrinker numerically, by implementing in software a solver for the simple scalar optimization problem (6.3). Figure 3 portrays the optimal shrinkers for our 26 loss functions. We refer readers interested in computing specific individual shrinkers to our reproducibility advisory at the bottom of this paper, and invite the reader to explore the code supplement [41], consisting of online resources and code we offer.

6.2. Optimal Shrinkers Collapse the Bulk

We first observe that, for any of the 26 losses considered, the optimal shrinker collapses the bulk to 1. The following lemma is proved in the supplemental article [40]:

Lemma 4. Let L be any of the 26 losses mentioned in Table 1. Then the rule η**(ℓ) = 1 is unique asymptotically admissible on [1,ℓ₊(γ)] namely, for every ℓ ∈[1,ℓ₊(γ)] we have $\mathbb{E}L\left(\ell ,\eta \right)\ge L\left(\ell ,{\eta }^{**}\right),$ with strict inequality for at least one point in [1,ℓ₊(γ)]

As part of the proof of Lemma 4, in Table 6 in the supplemental article [40], we explicitly calculate the fundamental loss function G(ℓ, η) of (6.4) for many of the loss families discussed in this paper.

To determine the optimal shrinker η^∗(λ ; γ, L) for each of our loss functions L, it therefore remains to determine the map $\text{λ}\mapsto {\eta }^{*}\left(\text{λ}\right)$ or equivalently $\ell \mapsto {\eta }^{*}\left(\text{λ}\left(\ell \right)\right)$ only for ℓ > ℓ₊(γ) This is our next task.

6.3. Optimal Shrinkers by Computer

The scalar optimization problem (6.3) is easy to solve numerically, so that one can always compute the optimal shrinker at any desired value λ. In the code supplement [41] we provide Matlab code to compute the optimal nonlinearity for each of the 26 loss families discussed. In the sibling problem of singular value shrinkage for matrix denoising, [53] demonstrates numerical evaluation of optimal shrinkers for the Schatten-p norm, where analytical derivation of optimal shrinkers appears to be impossible.

6.4. Optimal Shrinkers in Closed Form

We were able to obtain simple analytic formulas for the optimal shrinker η^∗ in each of 18 loss families from Section 3. While the optimal shrinkers are of course functions of the empirical eigenvalue λ, in the interest of space, we state the lemmas and provide the formulas in terms of the quantities $\ell ,$ c and s. To calculate any of the nonlinearities below for a specific empirical eigenvalue λ, use the following procedure:

1. If λ ≤ λ₊(γ) set η^∗(λ) = 1. Otherwise:

2. Calculate ℓ(λ) using (1.10).

3. Calculate c(λ) = c(ℓ(λ)) using (1.6) and (1.10).

4. Calculate s(λ) = s(ℓ(λ)) using $s\left(\ell \right)=\sqrt{1-c^2\left(\ell \right)}.$

5. Substitute ℓ(λ) c(λ) and s(λ) into the formula provided to get η^∗(λ).

The closed forms we provide are summarized in Table 2. Note that ℓ, c and s refer to the functions ℓ(λ), c(ℓ(λ)) and s(ℓ(λ)) These formulae are formally derived in a sequence of lemmas that are stated and proved in the supplemental article [40]. The proofs also show that these optimal shrinkers are unique, as in each case the optimal shrinker is shown to be the unique minimizer, as in (6.3), of (6.2). We make some remarks on these optimal shrinkers by focusing first on operator norm loss for covariance and precision matrices:

$${\eta }^{\ast }\left(\text{λ;γ}.L^{O,1}\right)={\eta }^{\ast }\left(\text{λ;γ}.L^{O,2}\right)=\{\begin{array}{c}\ell ,\ell >{\ell }_{+}\left(\text{γ}\right)\\ 1,\ell \le {\ell }_{+}\left(\text{γ}\right)\end{array}.$$ (6.5)

This asymptotic relationship reflects the classical fact that in finite samples, the top empirical eigen value is always biased upwards of the underlying population eigenvalue [54, 55]. Formally defining the (asymptotic) bias as

$$bias\left(\eta ,\ell \right)=\eta \left(\text{λ}\left(\ell \right)\right)-\ell ,$$

we have bias(λ(ℓ),ℓ) > 0. The formula η^∗(λ) = ℓ shows that the optimal nonlinearity for operator norm loss is what we might simply call a debiasing transformation, mapping each empirical eigenvalue back to the value of its “original” population eigenvalue, and the corresponding shrinkage estimator ${\widehat{\Sigma }}_{\eta }$ uses each sample eigenvectors with its corresponding population eigenvalue. In words, within the top branch of (6.5), the effect of operator-norm optimal shrinkage is to debias the top eigenvalue:

$$bias=\left({\eta }^{\ast }\left(\cdot \text{;γ}.L^{O,1}\right),\ell \right)=bias=\left({\eta }^{\ast }\left(\cdot \text{;γ}.L^{O,2}\right),\ell \right)=0,\forall \ell >{\ell }_{+}\left(\text{γ}\right).$$

On the other hand, within the bottom branch, the effect is to shrink the bulk to 1. In terms of Definition 3 we see that η^∗ is a bulk shrinker, but not a neighborhood bulk shrinker.

One might expect asymptotic debiasing from every loss function, but, perhaps surprisingly, precise asymptotic debiasing is exceptional. In fact, none of the other optimal nonlinearities in Table 2 is precisely debiasing.

In the supplemental article [40] we also provide a detailed investigation of the large-λ asymptotics of the optimal shrinkers, including their asymptotic slopes, asymptotic shifts and asymptotic percent improvement.

7. Beyond Formal Optimality

The shrinkers we have derived and analyzed above are formally optimal, as in Definition 6, in the sense that they minimize the formal expression L_∞(ℓ|η) So far we have only shown that formally optimal shrinkers actually minimize the asymptotic loss (namely, are asymptotically unique admissible) in the single-spike case, under assumptions [Asy(γ)] and [Spike(ℓ)], and only over neighborhood bulk shrinkers.

In this section, we show that formally optimal shrinkers in fact minimize the asymptotic loss in the general Spiked Covariance Model, namely under assumptions [Asy(γ)] and [Spike(ℓ₁,…,ℓ_r)], and over a large class of bulk shrinkers, which are possibly not neighborhood bulk shrinkers.

We start by establishing the rank r analog of Lemma 1. For a vector $\ell \in {ℝ}^r,$ let Δ_r(ℓ) = diag(ℓ₁,…,ℓ_r)

Lemma 5. Assume that Ʃ and $\widehat{\Sigma }$ are fixed matrices with

$$\begin{array}{l}spec(\Sigma )=\left[\left({\ell }_1\dots {\ell }_r,1\dots ,1\right),\left(u_1,\dots ,u_p\right)\right]\hfill \\ spec\left(\widehat{\Sigma }\right)=\left[\left({\eta }_1\dots {\eta }_r,1\dots ,1\right),\left({\upsilon }_1,\dots ,{\upsilon }_p\right)\right].\hfill \end{array}$$

Let U_r and V_r denote the p-by-r matrices consisting of the top r eigenvectors of Ʃ and $\widehat{\Sigma }$ respectively. Suppose that [U_r V_r] has full rank 2r, and consider the QR decomposition

$$\left[U_r{V}_r\right]=QR,$$

where Q has 2r orthonormal columns and the 2r × 2r matrix R is upper triangular. Let R₂ denote the 2r × r submatrix formed by the last r columns of R. Fill out Q to an orthogonal matrix W = [Q Q^⊥]. Then in the transformed basis we have the simultaneous block decompositions

$$W^{\prime }\Sigma W={\Sigma }_{2r}^{\circ }\oplus I_{p-2r}, {\Sigma }_{2r}^{\circ }={\Delta }_r\left(\ell \right)\oplus I_r$$ (7.1)

$$W^{\prime }\widehat{\Sigma }W={\widehat{\Sigma }}_{2r}^{\circ }\oplus I_{p-2r}, {\widehat{\Sigma }}_{2r}^{\circ }=I_{2r}+R_2{\Delta }_r\left(\eta -1\right)R_2^{\prime }.$$ (7.2)

Proof. We start with observations about the structure of Q and R. Since the first r columns of Q are identically those of U_r, we let Z_r be the n-by-r matrix such that Q = [U_r Z_r]. For the same reason, R has the block structure

$$R=\left[\begin{array}{cc}{I}_{r\times r}& R_{12}\\ 0_{r\times r}& R_{22}\end{array}\right],$$

where the matrices R₁₂ and R₂₂ satisfy V_r = U_rR₁₂ + Z_rR₂₂ , so that

$$R_{12}={U^{\prime }}_r{V}_r R_{22}={Z^{\prime }}_r{V}_r.$$ (7.3)

Since V_r has orthogonal columns, we have

$$\begin{gathered} I_r={V^{\prime }}_r{V}_r={R^{\prime }}_{12}{R}_{12}+{R^{\prime }}_{22}{R}_{22} \\ {R^{\prime }}_{22}{R}_{22}=I-{R^{\prime }}_{12}{R}_{12}. \end{gathered}$$ (7.4)

Let H be a p×r matrix whose columns lie in the column span of Q and let Δ be an r × r diagonal matrix. Observe that

$$\begin{array}{lll}{W}^{\prime }\left(I+H\Delta H^{\prime }\right)W\hfill & =\hfill & I+W^{\prime }H\Delta H^{\prime }W\hfill \\ \hfill & =\hfill & \left(I_{2r}+Q^{\prime }H\Delta H^{\prime }Q\right)\oplus I_{p-2r}=C_{2r}\oplus I_{p-2r},\hfill \end{array}$$

say, since the columns of Q^⊥ are orthogonal to those of H. By analogy to (2.6), we may write

$$\Sigma =I+U_r\left({\Delta }_r\left(\ell \right)-I_r\right){U^{\prime }}_r, \widehat{\Sigma }=I+V_r\left({\Delta }_r\left(\eta \right)-I_r\right){V^{\prime }}_r$$ (7.5)

and so both of the form I + HΔH^′, with H = U_r and V_r respectively. We find that

$$Q^{\prime }{U}_r=\left[\begin{array}{c}{I}_r\\ 0\end{array}\right], Q^{\prime }{V}_r=\left[\begin{array}{c}{R}_{12}\\ R_{22}\end{array}\right]=R_2,$$

We can then compute the value of C_2r in the two cases to be given by ${\Sigma }_{2r}^{\circ }$ and ${\widehat{\Sigma }}_{2r}^{\circ }$ respectively, which establishes (7.1) and (7.2), and hence the lemma.

We intend to apply Lemma 5 to Ʃ and $\widehat{\Sigma }={\widehat{\Sigma }}_{\eta ,r},$ the “rank-aware” modification (1.13) of the estimator ${\widehat{\Sigma }}_{\eta }$ in (1.7). Assume now that $\widehat{\Sigma }$ and the p × r matrix V_r,n formed by the top eigenvectors of V are random.

Lemma 6. The rank of [U_r V_r,n] equals 2r almost surely.

Proof. Let ∏_r(V) be the projection that picks out the first r columns of an orthogonal matrix V. For a fixed r-frame U_r, we consider the event

$$A=\left\{V\in O_p:\text{\hspace{0.17em}rank}\left(\left[U_r{\prod }_r\left(V\right)\right]\right)<2r\right\},$$

where O_p is the group of orthogonal p-by-p matrices. Let $P_{\Sigma }\left(d\Lambda ,dV\right)$ denote the joint distribution of eigenvalues Λ = diag(λ₁,...,λ_p) and eigenvectors V when S ~ W_p(n, Ʃ). As shown by [56], P_Ʃ is absolutely continuous with respect to v_p × μ_p’, the product of Lebesgue measure on ${ℝ}^p$ and Haar measure on O(p). Since μ_p(A) = 0, it follows that P_Ʃ(A) = 0.

Lemma 7. Adopt models [Asy(γ)] [Spike(ℓ₁,…,ℓ_r)]and with ℓ₁,…,ℓ_r > ℓ_r > ℓ₊(γ). Suppose the scalar nonlinearity η is continuous on (λ₊(γ), ∞). For each p there exists w.p. 1 an orthogonal change of basis W such that

$$W^{\prime }\Sigma W={\Sigma }_{2r}\oplus I_{p-2r}, W^{\prime }{\widehat{\Sigma }}_{\eta ,r}W={\widehat{\Sigma }}_{2r}\oplus I_{p-2r},$$ (7.6)

where the 2r ×2r matrices ${\Sigma }_{2r},$ ${\widehat{\Sigma }}_{2r}$ satisfy

$${\Sigma }_{2r}={\oplus }_{i=1}^r,A\left({\ell }_i\right),{\widehat{\Sigma }}_{2r}\stackrel{a.s.}{\to }{\oplus }_{i=1}^p,B\left({\ell }_i,\eta \right),$$ (7.7)

and the 2 × 2 matrices A(ℓ), B(ℓ, η) are defined at (2.5).

Suppose also that the family L = {L_p} of loss functions is orthogonally invariant and sum- or maxdecomposable, and that B → L_2r (A, B) is continuous. Then

$$L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta ,r}\right)\stackrel{a.s.}{\to }{\left(\sum /\max \right)}_{i=1,\dots r}{L}_2\left(A\left({\ell }_i\right),B\left({\ell }_i,\eta \right)\right)).$$ (7.8)

If η is a neighborhood bulk shrinker, then $L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta }\right)$ also has this limit a.s.

This is the rank r analog of Lemma 3. The optimal nonlinearity η^∗ is continuous on [0,∞) for all losses except the operator norm ones, for which η^∗ is continuous except at λ = λ (γ)₊. Our result (7.7) requires only continuity on (λ₊(γ), ∞) and so is valid for all 26 loss functions, as is the deterministic limit (7.8) for the rank-aware ${\widehat{\Sigma }}_{\eta ,r}.$ However, as we saw earlier, only the nuclear norm based loss functions yield optimal functions that are neighborhood bulk shrinkers. To show that (7.8) holds for $L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta }\right)$ for most other important shrinkage functions, some further work is needed – see Section 7.1 below.

Proof. We apply Lemma 5 to ∑ and ${\widehat{\Sigma }}_{\eta ,r}$ on the set of probability 1 provided by Lemma 6. First, we rewrite (7.2) to show the subblocks of R:

$${\widehat{\Sigma }}_{2r}^{\circ }=I_{2r}+\left[\begin{array}{c}{R}_{12}\\ R_{22}\end{array}\right]{\Delta }_r\left({\eta }^{\left(n\right)}-1\right)\left[{R^{\prime }}_{12}{R^{\prime }}_{22}\right],$$

where we write η⁽ⁿ⁾ = (η(λ_1,n),…, η(λ_r,n)) to show explicitly the dependence on n. The limiting behavior of R may be derived from (7.3) and (7.4) along with spiked model properties (1.2) and (1.5), so we have², as n → ∞,

$$\begin{array}{r}\hfill R_{12}={U^{\prime }}_r{V}_{r,n}{\to }_{a.s.}{\Delta }_r\left(c\right)\\ \hfill \text{}{R}_{22}{R^{\prime }}_{22}=I-R_{12}{R^{\prime }}_{12}{\to }_{a.s.}{\Delta }_r\left(s^2\right)\text{}\\ \hfill R_{22}{\to }_{a.s.}{\Delta }_r\left(s\right).\end{array}$$ (7.9)

Here c = (c(ℓ₁),…, c(ℓ_r)) and s = (s(ℓ₁),…, s(ℓ_r)).

Again by (1.2) λ_i,n →_a.s. λ(ℓ_i) > λ₊(γ) and so continuity of η above λ₊ (γ) assures that Δ_r (η^(η)–1) → Δ_r(η −1), where η = (η_i) and η_i = η(λ(ℓ_i)). Together with (1.5), we obtain simplified structure in the limit,

$${\widehat{\Sigma }}_{2r}^{\circ }{\to }_{a.s.}{I}_{2r}+\left[\begin{array}{cc}{\Delta }_r\left(\left(\eta -1\right)c^2\right)& {\Delta }_r\left(\left(\eta -1\right)cs\right)\\ {\Delta }_r\left(\left(\eta -1\right)cs\right)& {\Delta }_r\left(\left(\eta -1\right)s^2\right)\end{array}\right].$$ (7.10)

To rewrite the limit in block diagonal form, let Π_2r be the permutation matrix corresponding to the permutation defined by

(1,…,2_r) ⟼ (1,r + 1, 2, r + 2, 3,…, 2_r).

Permuting rows and columns in (7.1) and (7.10) using Π_2r to obtain

$$\begin{array}{ll}{\Sigma }_{2r}\hfill & :={\prod }_{2r}^{\prime }{\Sigma }_{2r}^{\circ }{\prod }_{2r}={\oplus }_{i-1}^{r}A\left({\mathcal{l}}_i\right),\text{}\hfill \\ {\widehat{\Sigma }}_{2r}\hfill & :={\prod }_{2r}^{\prime }{\widehat{\Sigma }}_{2r}^{\circ }{\prod }_{2r}{\to }_{a.s.}{\oplus }_{i-1}^{p}B\left({\mathcal{l}}_i,\eta \right),\hfill \end{array}$$

we obtain (7.7). Using (7.6), the orthogonal invariance and sum/max decomposability, along with the continuity of L_2r(A,·), we have

$$\begin{array}{lll}{L}_p\left({\Sigma }_p,{\widehat{\Sigma }}_{\eta ,r}\right)\hfill & =\hfill & L_p\left({\Sigma }_{2r}\oplus I_{p-2r},{\widehat{\Sigma }}_{2r}\oplus I_{p-2r}\right)\hfill \\ \hfill & =\hfill & L_{2r}\left({\Sigma }_{2r},{\widehat{\Sigma }}_{2r}\right)\hfill \\ \hfill & =\hfill & L_{2r}\left({\prod }_{2r}^{\prime }{\Sigma }_{2r}{\prod }_{2r},{\prod }_{2r}^{\prime }{\widehat{\Sigma }}_{2r}{\prod }_{2r}\right){\to }_{a.s.}{L}_{2r}\left({\oplus }_{i=1}^{r}A\left({\ell }_i\right),{\oplus }_{i=1}^{p}B\left({\ell }_i,\eta \right)\right)\hfill \\ \hfill & =\hfill & {\left(\sum /\text{max}\right)}_{i=1,\dots r}{L}_2\left(A\left({\ell }_i\right),B\left({\ell }_i,\eta \right)\right)),\hfill \end{array}$$

which completes the proof of Lemma 7.

7.1. Removing the rank-aware condition

In this section we prove Proposition 2 below, whereby the asymtotic losses coincide for a given estimator sequence ${\widehat{\Sigma }}_{\eta }$ and the rank-aware versions ${\widehat{\Sigma }}_{\eta ,r}.$ This result is plausible because of two observations:

1. Null eigenvalues stick to the bulk, i.e. for i ≥ r + 1 exceptions , most eigenvalues λ_in λ₊(γ) and the few exceptions are not much larger. Hence, if η is a continuous bulk shrinker, we expect ${\widehat{\Sigma }}_{\eta }$ to be close to ${\widehat{\Sigma }}_{\eta ,r},$

2. under a suitable continuity assumption on the loss functions L_p, $L\left(\Sigma ,{\widehat{\Sigma }}_{\eta }\right)$should then be close to $L\left(\Sigma ,{\widehat{\Sigma }}_{\eta ,r}\right).$

Observation 1 is fleshed out in two steps. The first step is eigenvalue comparison: The sample eigenvalue λ_in arise as eigenvalues of XX^′/n when X is a p_n-by-n matrix whose rows are i.i.d draws from $\mathcal{N}\left(0,{\Sigma }_{p_n}\right).$ (0, ∑_pn). Let $\prod :{ℝ}^{p_n}\to {ℝ}^{p_n-r}$ denote the projection on the last p_n – r coordinates in ${ℝ}^{p_n}$ and let μ_1n ≥ … ≥ μ_pn−r,n denote the eigenvalues of ∏X(∏X)^′/n. By the Cauchy interlacing Theorem (e.g. [57, p. 59]), we have

$${\text{λ}}_{in}\le {\mu }_{i-r,n} \text{for }r+1\le i\le p_n,$$ (7.11)

where the (μ_in) are the eigenvalues of a white Wishart matrix W_pn−r(n, I).

The second step is a bound on eigenvalues of a white Wishart that exit the bulk. Before stating it, we return to an important detail introduced in the Remark concluding Section 1.1.

Definition 3 of a bulk shrinker depends on the parameter γ = lim p/n through λ₊(γ). Making that dependence explicit, we obtain a bivariate function η(λ, c). In model [Asy(γ)]and in the n-th problem, we might use η(λ, c_n) either with c_n = γ or c_n = p/n. For Proposition 1 below, it will be more natural to use the latter choice. We also modify Definition 3 as follows.

Definition 7. We call η : [0, ∞) × (0, 1] → [1, ∞) a jointly continuous bulk shrinker if η(λ, c) is jointly continuous in λ and c, satisfies η(λ, c) = 1 for λ ≤ λ₊(c) and is dominated: η(λ, c) ≤ Mλ for some M and all λ.

The following result is proved in [58, Theorem 2(a)].

Proposition 1. Let ${\left({\mu }_{in}\right)}_{i=1}^N$ denote the sample eigenvalues of a matrix distributed as W_N(n, I), with N/n → γ > 0. Suppose that η(λ, c) is a jointly continuous bulk shrinker and that c_n − N/n = O(n−2/3). Then for q > 0,

$${\Vert \eta \left({\mu }_{in},c_n\right)-1\Vert }_{{\ell }_q\left({ℝ}^N\right)}\to P0.$$ (7.12)

The continuity assumption on the loss functions may be formulated as follows. Suppose that A, B₁, B₂ are p-by-p positive definite matrices, with A satisfying assumption [Spike(ℓ₁,…, ℓ_r)] and spec(B_k) = [(η_ki), (v_i)], thus B₁ and B₂ have the same eigenvectors. Set η₁ = max {η₁₁, η₂₁} . We assume that for some q ∈ [1, ∞] and some continuous function C(ℓ₁, η₁ ) not depending on p, we have

$$\left|L_p\left(A,B_1\right)-L_p\left(A,B_2\right)\right|\le C\left({\ell }_1,{\eta }_1\right){\Vert {\eta }_1-{\eta }_2\Vert }_{{\ell }_q\left({ℝ}^p\right)}$$ (7.13)

whenever $\Vert {\eta }_1-{\eta }_2{\Vert }_{{\ell }_q\left({ℝ}^p\right)}\le 1.$ Condition (7.13) is satisfied for all 26 of the loss functions of Section 3, as is verified in Proposition 1 in SI.

In the next proposition we adopt the convention that estimators ${\widehat{\Sigma }}_{\eta }$ of (1.7) and ${\widehat{\Sigma }}_{\eta ,r}$ of (1.13) are constructed with a jointly continuous bulk shrinker, which we denote η(λ, c_n).

Proposition 2. Adopt models [Asy(γ)] and [Spike(ℓ₁,…, ℓ_r)]. Suppose also that the family L = {L_p} of loss functions is orthogonally invariant and sum- or max- decomposable, and satisfies continuity condition (7.13). If η(λ, c_n) is a jointly continuous bulk shrinker with c_n = p_n/n, then

$$L_p\left(\sum ,{\widehat{\sum }}_{\eta }\right)-L_p\left(\sum ,{\widehat{\sum }}_{\eta ,r}\right)\to P0,$$

and so $L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta }\right)$ converges in probability to the deterministic asymptotic loss (7.8).

proof in the left side of (7.13), substitute A=∑, $B_1={\widehat{\Sigma }}_{\eta }$ and $B_2={\widehat{\Sigma }}_{\eta ,r}.$ By definition ,${\widehat{\Sigma }}_{\eta }$ and ${\widehat{\Sigma }}_{\eta ,r}$ share the same eigenvectors. The components of η₁‒η₂ then satisfy

$${\eta }_{1i}-{\eta }_{2i}=\{\begin{array}{ll}\eta \left({\text{λ}}_{in},c_n\right)-1\hfill & i\ge r+1\hfill \\ 0\hfill & 1\le i\le r.\hfill \end{array}$$

We now use (7.11)to compare the eigenvectors λ_in of the spiked model to those of a suitable white Wishart matrix to which Proposition 1 applies. The fuction η^↑(μ,c) = max{η(λ,c),1 ≤ λ ≤ μ} and is non-decreasing and jointly continuos. Hence η(λ_in, c_n) ≤ η^↑(λ_in, c_n) ≤ η^↑(μ_i‒r,nin,c_n), and so

$${{\displaystyle \underset{i=r+1}{\overset{p}{\Sigma }}\left[\eta \left({\text{λ}}_{in},c_n\right)-1\right]}}^q\le {{\displaystyle \underset{j=1}{\overset{p-r}{\Sigma }}\left[{\eta }^{\uparrow }\left({\mu }_{jn},c_n\right)-1\right]}}^q,$$

with a corresponding bound for q = ∞. From continuity condition (7.13),

$$\left|L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta }\right)-L_p\left(\Sigma ,{\widehat{\Sigma }}_{\eta ,r}\right)\right|\le C\left({\ell }_1,\eta \left({\text{λ}}_{1n},c_n\right)\right){\Vert {\eta }^{\uparrow }\left({\mu }_{jn},c_n\right)-1\Vert }_{{\ell }_q\left({ℝ}^{p-r}\right).}$$

The constant C(ℓ₁,η(λ_1n,c_n)) remains bounded by (1.2). The ℓ_q norm converges to 0 in probability, applying Proposition 1 to the eigenvalues of Wpn−r(n, I), with N = p_n − r, noting that c_n − N/n = r/n = O(n^−2/3).

7.2. Asymptotic loss for discontinuous optimal shrinkers

Formula (6.5) showed that the optimal shrinker η^*(λ, γ) for operator norm losses L^O,1,L^O,2 is discontinuous at $\ell ={\ell }_{+}\left(\gamma \right)=1+\sqrt{\gamma }.$ In this section, we show that when η^* is used, a deterministic asymptotic loss exists for L^O,1, but not for L^O,2. The reason will be seen to lie in the behavior of the optimal component loss F_*(ℓ) = L₂[A(ℓ),B(ℓ, η^*)]. Indeed, calculation based on (6.2), (6.5) shows that for ℓ ≥ ℓ₊,

$$F_{*}\left(\ell \right)={\left[\frac{{\ell }^a\text{γ}\left(\ell -1\right)}{\ell -1+\text{γ}}\right]}^{1/2}\to F_{*}\left({\ell }_{+}\right)=\{\begin{array}{ll}\sqrt{\text{γ}}\hfill & a=1\hfill \\ \frac{\sqrt{\text{γ}}}{1+\sqrt{\text{γ}}}\hfill & a=-1\hfill \end{array}$$

as ℓ ↓ ℓ₊, where indices a = 1 and −1 correspond to $F_{*}^{O,1}$ and $F_{*}^{O,2}$ respectively. Importantly, $F_{*}^{O,1}$ is strictly increasing on [ℓ₊,∞) while $F_{*}^{O,2}$ is strictly decreasing there.

Proposition 3. Adopt models [Asy(γ)] and [Spike(ℓ₁,…, ℓ_r)] with ℓ_r > ℓ₊(γ). Consider the optimal shrinker η^*(λ, γ ) with γ_n = p_n/n given by (6.5) for both L^O,1 and L^O,2. For L^O,1, the asymptotic loss is well defined:

$${\left|\right|{\widehat{\Sigma }}_{\eta }-\Sigma \left|\right|}_{\infty }-{\left|\right|{\widehat{\Sigma }}_{\eta ,r}-\Sigma \left|\right|}_{\infty }\to P0.$$ (7.14)

However, for L^O,2,

$${\Vert {\widehat{\Sigma }}_{{}_{\eta }}^{-1}-{\Sigma }^{-1}\Vert }_{\infty }-{\Vert {\widehat{\Sigma }}_{{}_{\eta ,r}}^{-1}-{\Sigma }^{-1}\Vert }_{\infty }\stackrel{\mathcal{D}}{\to }W.$$ (7.15)

where W has a two point distribution in which

$$W=\{\begin{array}{ll}{F}_{*}^{O,2}\left({\ell }_{+}\right)-F_{*}^{O,2}\left({\ell }_r\right)\hfill & with\hspace{0.17em}prob\hspace{0.17em}1-F_1\left(0\right)\hfill \\ 0\hfill & otherwise,\hfill \end{array}$$

and $F_1\left(0\right)=\mathbb{P}\left\{T{W}_1\le 0\right\}$ for a real Tracy-Widom variate TW₁ [59].

Roughly speaking, there is positive limiting probability that the largest noise eigenvalue will exit the bulk distribution, and in such cases the corresponding component loss F_*(ℓ₊) – which is due to noise alone – exceeds the largest component loss due to any of the r spikes, namely F_*(ℓ_r). Essentially, this occurs because precision losses $L^{\left\{O,F,N\right\},2}\left(a\Sigma ,a\widehat{\Sigma }\right)$ decrease as signal strength a increases. The effect is not seen for L^{F,N},2 because the optimal shrinkers in those cases are continuous at ℓ₊ !

Proof. For the proof, write ∥·∥ for ∥·∥_∞. Let W = [W₁ W₂] be the orthogonal change of basis matrix constructed in Lemma 7, with W₁ containing the first 2r columns. We treat the two losses L^O,1 and L^O,2 at once using an exponent a = ±1, and write η^a(λ) for η^a(λ, γ_n). Thus, let

$$\Delta ={\Delta }_n={\widehat{\Sigma }}_{{}_{\eta }}^a-{\widehat{\Sigma }}_{{}_{\eta ,r}}^a={\displaystyle \sum _{i=r+1}^p\left[{\eta }^a\left({\text{λ}}_i\right)-1\right]}{\upsilon }_i{\upsilon }_i^{\prime },$$

and observe that the loss of the rank-aware estimator

$$\Psi ={\Psi }_n={\widehat{\Sigma }}_{{}_{\eta ,r}}^a-{\widehat{\Sigma }}^a={\displaystyle \sum _{i=1}^r\left[{\eta }^a\left({\text{λ}}_i\right)-1\right]}{\upsilon }_i{\upsilon }_i^{\prime }-{\displaystyle \sum _{i=1}^r\left[{\ell }_i^a-1\right]}{u}_i{u}_i^{\prime }$$

lies in the column span of W₁. We have ${\widehat{\Sigma }}_{\eta }^a-{\Sigma }^a={\Psi }_n+{\Delta }_n,$ and the main task will be to show that for a = ±1,

$$\Vert {\Psi }_n+{\Delta }_n\Vert =\text{max}\left(\Vert {\Psi }_n\Vert ,\Vert {\Delta }_n\Vert \right)+o_P\left(1\right).$$ (7.16)

Assuming the truth of this for now, let us derive the proposition. The quantities of interest in(7.14), (7.15) become First, note from Lemma 7 that

$$\begin{array}{lll}\Vert {\widehat{\Sigma }}_{{}_{\eta }}^a-{\widehat{\Sigma }}^a\Vert -\Vert {\widehat{\Sigma }}_{{}_{\eta ,r}}^a-{\widehat{\Sigma }}^a\Vert \hfill & =\hfill & \Vert {\Psi }_n+{\Delta }_n\Vert -\Vert {\Psi }_n\Vert \text{}\hfill \\ \hfill & =\hfill & \text{max}\left(\Vert {\Delta }_n\Vert -\Vert {\Psi }_n\Vert ,0\right)+o_P\left(1\right).\hfill \end{array}$$

First, note from Lemma 7 that

$$\Vert {\Psi }_n\Vert {\to }_{a.s.}\underset{1\le i\le r}{\text{max}}{F}_{*}\left({\ell }_i\right).$$ (7.17)

Observe that for both a = 1 and −1,

$$\Vert {\Delta }_n\Vert =\underset{i\ge r+1}{\text{max}}⏐{\eta }^{*a}\left({\text{λ}}_{in}\right)-1⏐=⏐{\eta }^a\left({\text{λ}}_{r+1,n}\right)-1⏐.$$

The rescaled noise eigenvalue $p^{2/3}\left({\text{λ}}_{r+1,n}-{\text{λ}}_{+}\left({\gamma }_n\right)\right)\stackrel{\mathcal{D}}{\to }\sigma \left(\gamma \right)W$ has a limiting real Tracy-Widom distribution with scale factor σ(γ) > 0 [60, Prop. 5.8]. Hence, using the discontinuity of the optimal shrinker η^*, and the square root singularity from above

$${\eta }^{*}\left({\text{λ}}_{r+1,n},{\text{γ}}_n\right)=\{\begin{array}{ll}{\ell }_{+}\left({\text{γ}}_n\right)+O_P\left(p^{-1/3}\right)\hfill & {\text{λ}}_{r+1,n}>{\lambda }_{+}\left({\text{γ}}_n\right)\hfill \\ 1\hfill & {\text{λ}}_{r+1,n}\le {\lambda }_{+}\left({\text{γ}}_n\right).\hfill \end{array}$$

Consequently, recalling that $F_{*}\left({\ell }_{+}\right)=\left|{\left(1+\sqrt{\gamma }\right)}^a-1\right|,$ we have

$$\Vert {\Delta }_n\Vert {\to }_P{F}_{*}\left({\ell }_{+}\right)I\left(TW>0\right).$$ (7.18)

For L^O,1, with a = 1, F_*(ℓ) is strictly increasing and so from (7.17) and (7.18), we obtain $\Vert {\Psi }_n\Vert \ge \Vert {\Delta }_n\Vert +o_P\left(1\right)$ and hence (7.14).For L^O,2, with a = −1, F_* (ℓ) is strictly so on the event TW > 0,

$$\Vert {\Delta }_n\Vert -\Vert {\Psi }_n\Vert \stackrel{\mathcal{D}}{\to }{F}_{*}\left({\ell }_{+}\right)-F_{*}\left({\ell }_r\right)>0,$$

which leads to (7.15) and hence the main result.

It remains to prove (7.16). For a symmetric block matrix,

$$\text{max}\left(\Vert A\Vert ,\Vert C\Vert \right)\le \Vert \left(\begin{array}{ll}A\hfill & B\hfill \\ B^{\prime }\hfill & C\hfill \end{array}\right)\Vert \le \text{max}\left(\Vert A\Vert ,\Vert C\Vert \right)+\Vert B\Vert .$$ (7.19)

Apply this to $W^{\prime }\left(\Psi +\Delta \right)W$ with

$$\begin{array}{lll}{A}_n\hfill & =\hfill & W_1^{\prime }\left(\Psi +\Delta \right)W_1,\text{}\hfill \\ B_n\hfill & =\hfill & W_1^{\prime }\left(\Psi +\Delta \right)W_2=W_1^{\prime }\Delta W_2\hfill \\ C_n\hfill & =\hfill & W_2^{\prime }\left(\Psi +\Delta \right)W_2=W_2^{\prime }\Delta W_2,\hfill \end{array},$$

since $\Psi W_2=0.$ Hence

$$\Vert {\Psi }_n+{\Delta }_n\Vert =\text{max}\left(\Vert A_n\Vert ,\Vert C_n\Vert \right)+O_P\left(\Vert B_n\Vert \right).$$ (7.20)

We now show that ∥ΔW₁∥ →_P 0. Using notation from Lemma 5,

$$W_1=\left[U_r\hspace{0.17em}{V}_r\right]R^{-1}=\left[U_r\left(V_r-U_r{R}_{12}\right)R_{22}^{-1}\right].$$

Since Δv_k = 0 for k = 1,…, r,

$$||\Delta W_1||\le ||\Delta U_r||\left(1+||R_{12}{R}_{22}^{-1}||\right).$$

From (7.9), we have $||R_{12}{R}_{22}^{-1}||\to ||{\Delta }_r\left(c/s\right)||=c\left({\ell }_1\right)/s\left({\ell }_1\right),$ and hence is bounded. Observe that ${\Delta }_{u_k}={\displaystyle \sum }_{i=r+1}^p{\delta }_{in}^a\left(v_i^{\prime }{u}_k\right)v_i,$ where we have set δ_in = η(λ_i, γ_n)−1. Note from (6.5) that δ_in = 0 unless λ_i > λ₊(γ_n). With N_n = #{i ≥ r + 1 : λ_in > λ₊(γ_n)}, we then have

$$\Vert \Delta U_r\Vert \le \sqrt{r}\underset{k=1,\dots ,r}{\text{max}}\Vert {\Delta }_{u_k}\Vert 2\le \sqrt{r}\Vert \Delta \Vert N_n\underset{k\le r;i>r}{\text{max}}\left|{\upsilon }_i^{\prime }{u}_k\right|.$$ (7.21)

From (7.18) we have ∥Δ_n∥ = O_P (1). Since each v_i, i > r is uniformly distributed on S^p−1, a simple union bound based on (7.23) below yields

$$\underset{i>r,k\le r}{\text{max}}{\left(v_i^{\prime }{u}_k\right)}^2=O_P\left(\frac{\text{log\hspace{0.17em}}p}{p}\right).$$ (7.22)

It remains to bound N_n. From the interlacing inequality (7.11),

$$N_n\le {\tilde{N}}_n=\#\left\{j\ge 1\hspace{0.17em}:\hspace{0.17em}{\mu }_{jn}>{\lambda }_{+}\left({\text{γ}}_n\right)\right\},$$

where {μ_jn} are the eigenvalues of a white Wishart matrix W_pn^−r(n, I). This quantity is bounded in [58, Theorem 2(c)], which says that Ñ_n = O_p(1). In more detail, we make the correspondences N ← p_n − r, γ_N ← (p_n − r)/n and c_N ← p_n/n so that c_N − γ_N = r/n = o(n^−2/3) and obtain EÑ_n→ c₀ = 0.17.

From (7.21) and the preceding two paragraphs, we conclude that $\Vert \Delta U_r\Vert =O_P\left(p^{-1/2}\sqrt{\text{log}p}\right)$ and so $\Vert \Delta W_1\Vert {\to }_{P}0.$

Returning to (7.20), we deduce now that ∥Bn∥ ≤ ∥ΔW₁∥ →_P 0. From the definition of W₁ we have $\Vert W_1^{\prime }\Psi W_1\Vert =\Vert \Psi \Vert$ and hence the inequalities

$$\left|\Vert A_n\Vert -\Vert {\Psi }_n\Vert \le \Vert W_1^{\prime }\Delta W_1\Vert \right|\to P0.$$

Now observe that ∥C_n∥ ≤ ∥Δ_n∥. Apply (7.19) to W^′ΔW to get

$$\Vert {\Delta }_n\Vert \le \Vert C_n\Vert +\Vert W_1^{\prime }\Delta W_1\Vert +\Vert W_2^{\prime }\Delta W_1\Vert ,$$

and hence that ∥C_n∥ ≥ ∥Δ_n∥ − o_P (1). Thus ∥C_n∥ = ∥Δ_n∥ + o_P (1). Inserting these results into (7.20), we obtain

$$\Vert {\Psi }_n+{\Delta }_n\Vert =\text{max}\left(\Vert A_n\Vert ,\Vert C_n\Vert \right)+o_P\left(1\right)=\text{max}\left(\Vert {\Psi }_n\Vert ,\Vert {\Delta }_n\Vert \right)+o_P\left(1\right),$$

which completes the proof of (7.16), and hence of Proposition 3.

Finally, we record a concentration bound for the uniform distribution on spheres. While more sophisticated results are known [61], an elementary bound suffices for us.

Lemma 8. If U is uniformly distributed on S^n-1 and u ∈ S^n-1 if fixed, then for M < 0 and n ≥ 4,

$$P\left(\left|\langle U,u\rangle \right|\ge 2\sqrt{M{n}^{-1}\text{log\hspace{0.17em}}n}\right)\le \sqrt{\pi /2}\cdot n^{1/2-M}.$$ (7.23)

Proof. Since $U_1^2≔{\langle U,u\rangle }^2$ has the Beta $\left(\frac{1}{2},\frac{n-1}{2}\right)$ distribution,

$$P\left(U_1^2\ge a\right)\le B{\left(\frac{1}{2},\frac{n-1}{2}\right)}^{-1}{\displaystyle {\int }_a^1{t}^{-\frac{1}{2}}{\left(1-t\right)}^{\frac{n-3}{2}}dt\le {\gamma }_n}{\left(1-a\right)}^{\frac{n}{2}-1},$$

where by Gautschi’s inequality [62, 63, (5.6.4)]

$${\text{γ}}_n=B\left(\frac{1}{2},\frac{1}{2}\right)/B\left(\frac{1}{2},\frac{n-1}{2}\right)=\sqrt{\pi }\Gamma \left(\frac{n}{2}\right)/\Gamma \left(\frac{n-1}{2}\right)<\sqrt{\pi n/2}$$

Since (1 − x/m)^m < e^−x for x, m > 0, and 4/n ≥ 2/(n − 2) for n ≥ 4,

$$P\left(U_1^2\ge 4M{n}^{-1}\text{log}n\right)<\sqrt{\pi n/2}{\left(1-\frac{M\text{log}n}{n/2-1}\right)}^{n/2-1}<\sqrt{\pi /2}\cdot n^{1/2-M}.$$

8. Optimality Among Equivariant Procedures

The notion of optimality in asymptotic loss, with which we have been concerned so far, is relatively weak. Also, the class of covariance estimators we have considered, namely procedures that apply the same univariate shrinker to all empirical eigenvalues, is fairly restricted.

Consider the much broader class of orthogonally-equivariant procedures for covariance estimation [2, 19, 64], in which estimates take the form $\widehat{\Sigma }=V\Delta V^{\prime }$ Here, Δ = Δ (Λ) is any diagonal matrix that depends on the empirical eigenvalues ∧ in possibly a more complex way than the simple scalar element-wise shrinkage η(∧) we have considered so far. One might imagine that the extra freedom available with more general shrinkage rules would lead to improvements in loss, relative to our optimal scalar nonlinearity; certainly the proposals of [2, 19, 26] are of this more general type.

The smallest achievable loss by any orthogonally equivariant procedure is obtained with the “oracle” procedure ${\widehat{\Sigma }}^{oracle}=V{\Delta }^{oracle}{V}^{\prime }$, where

$${\Delta }^{oracle}={\text{argmin}}_{\Delta }L\left(\Sigma ,V\Delta V^{\prime }\right),$$ (8.1)

the minimum being taken over diagonal matrices with diagonal entries ≥1. Clearly, this optimal performance is not attainable, since the minimization problem explicitly demands perfect knowledge of Σ, precisely the object that we aim to recover. This knowledge is never available to us in practice – hence the label oracle³. Nevertheless, this optimal performance is a legitimate benchmark.

Interestingly, at least for the popular Frobenius and Stein losses, our optimal nonlinearities η^* deliver oracle-level performance – asymptotically. To state the result, recall expression (6.2) for these losses: $F\left(\ell ,\Delta \right)=L_2\left(A\left(\ell \right),B\left(\ell ,\Delta \right)\right)$.

Theorem 2. (Asymptotic optimality among all equivariant procedures.) Let L denote either the direct Frobenius loss L^F,1 or the Stein loss L^st. Consider a problem sequence satisfying assumptions [Asy(γ)] and [Spike(ℓ₁,…, ℓ_r)]. We have

$$\underset{n\to \infty }{\text{lim}}{L}_{p_n}\left(\Sigma ,{\widehat{\Sigma }}^{oracle}\right){=}_P{L}_{\infty }\left({\mathcal{l}}_1\dots ,{\mathcal{l}}_r|{\eta }^{*}\right)={\displaystyle \sum }_{i=1}^{r}F\left({\mathcal{l}}_i,{\eta }^{*}\right),$$

where η^* is the optimal shrinker for the losses L^F,1 or L^st in Table 2.

In short, the shrinker η^*(), which has been designed to minimize the limiting loss, asymptotically delivers the same performance as the oracle procedure, which has the lowest possible loss, in finite-n, over the entire class of covariance estimators by arbitrary high-dimensional shrinkage rules. On the other hand, by definition, the oracle procedure outperforms every orthogonally-equivariant statistical estimator. We conclude that η^* – as one such orthogonally-invariant estimator – is indeed optimal (in the sense of having the lowest limiting loss) among all orthogonally invariant procedures. While we only discuss the cases L^F,1 and L^st, we suspect that this theorem holds true for many of the 26 loss functions considered.

Proof. We first outline the approach. We can write Σ and Σ⁻¹ in the form I + F , and ${\widehat{\Sigma }}_{\Delta }=I+\tilde{\Delta }$ with

$$F={\displaystyle \sum _{k=1}^r{\beta }_k{u}_k{u}_k^{\prime },}\tilde{\Delta }={\displaystyle \sum _{i=1}^p{\tilde{\Delta }}_i{v}_i{v}_i^{\prime }}$$

where β_k = ℓ_k − 1 for L^F,1 and ${\ell }_k^{-1}-1$ for L^St and ${\tilde{\Delta }}_i={\Delta }_i-1$. Write

$$\text{tr\hspace{0.17em}}F\tilde{\Delta }={\displaystyle \sum _{i=1}^p{\tilde{\Delta }}_i{b}_i,}{b}_i≔{\displaystyle \sum _{k=1}^r{\beta }_k{\left(u_k^{\prime }{v}_i\right)}^2.}$$ (8.2)

For both L = L^F,1 and L^St, we establish a decomposition

$$L_p\left(\Sigma ,{\widehat{\Sigma }}_{\Delta }\right)={\displaystyle \sum _{i=1}^{r}F\left({\ell }_i,{\Delta }_i\right)+a\left({\Delta }_i-1\right){ϵ}_i+{\displaystyle \sum _{i=r+1}^{p}H\left(b_i,{\Delta }_i\right)}}.$$ (8.3)

Here, a is a constant depending only on the loss function,

$${ϵ}_i=b_i-{\beta }_{i}c{\left({\ell }_i\right)}^2,$$ (8.4)

and

$$H\left(b,\Delta \right)=\{\begin{array}{l}{\left(\Delta -1\right)}^2-2\left(\Delta -1\right)b\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for }{L}^{F,1}\\ \left(\Delta -1\right)\left(1+b\right)-\text{log}\Delta \text{\hspace{0.17em}\hspace{0.17em}for }{L}^{St}.\end{array}$$ (8.5)

Decomposition (8.3) shows that the oracle estimator (8.1) may be found term by term, using just univariate minimization over each Δ_i. Consider the first sum in (8.3), and let $\tilde{F}\left({\ell }_i,{\Delta }_i\right)$ denote the summand. We will show that

$$\underset{{\Delta }_i}{\text{min }}\tilde{F}\left({\ell }_i,{\Delta }_i\right)\stackrel{P}{\to }\underset{{\Delta }_i}{\text{min }}F\left({\ell }_i,{\Delta }_i\right),$$ (8.6)

and that

$${\displaystyle \sum _{i=r+1}^p\underset{{\Delta }_i}{\text{min }}H\left(b_i,{\Delta }_i\right)}=O_P\left(\frac{{\text{log}}^{2}p}{p}\right).$$ (8.7)

Together (8.6) and (8.7) establish the Theorem.

Turning to the details, we begin by showing (8.3). For Frobenius loss, we have from our definitions and (8.2) that

$${\Vert {\widehat{\Sigma }}_{\Delta }-\Sigma \Vert }_F^2=\text{tr}\left(\tilde{\Delta }-F\right){\left(\tilde{\Delta }-F\right)}^{\prime }={\displaystyle \sum _{i=1}^p{\left({\Delta }_i-1\right)}^2-2\left({\Delta }_i-1\right)b_i+{\displaystyle \sum _{i=1}^r{\left({\ell }_i-1\right)}^2}}.$$

For i ≥ r + 1, each summand in the first sum equals H(b_i, Δ_i) and for i ≤ r, we use the decomposition b_i = (ℓ_i − 1)c(ℓ_i)² + ∈_i. We obtain decomposition (8.3) with a = −2 and

$$F\left(\ell ,\Delta \right)={\left(\ell -1\right)}^2-2\left(\ell -1\right)\left(\Delta -1\right)c^2+{\left(\Delta -1\right)}^2.$$

For Stein’s loss, our definitions yield

$$\begin{array}{ll}{L}^{St}\left(\Sigma ,{\widehat{\Sigma }}_{\Delta }\right)\hfill & \text{=tr}\tilde{\Delta }+\text{tr}F+\text{tr}F\tilde{\Delta }-\text{log}\left(\left|{\widehat{\Sigma }}_{\Delta }\right|/\left|\Sigma \right|\right)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^p{\tilde{\Delta }}_i\left(1+b_i\right)-\text{log}{\Delta }_i}+{\displaystyle \sum _{k=1}^r{\beta }_k+\text{log }{\mathcal{l}}_k}.\hfill \end{array}$$

Again, for each i ≥ r + 1, each summand in the first sum equals H(b_i, Δ_i) and with b_i = (ℓ_i ‒ 1)c(ℓ_i)² + ∈_i we obtain (8.3) with a = 1 and

$$F\left(\ell ,\Delta \right)=\left({\ell }^{-1}-1\right)+\left(\Delta -1\right)\left(c^2/\ell +s^2\right)-\text{log}\left(\Delta /\ell \right).$$

It remains to verify (8.6) and (8.7). Theorem 1 says that for 1 ≤ i ≤ r,

$${ϵ}_i={\displaystyle \sum _{k=1}^r{\beta }_k\left[{\left(u_k^{\prime }{v}_i\right)}^2-{\delta }_{k,i}c{\left({\ell }_i\right)}^2\right]}\stackrel{P}{\to }0,$$

which yields (8.6). From (8.5), we observe that in our two cases

$$h\left(b\right)≔\underset{\Delta }{\text{min }}H\left(b,\Delta \right)=\{\begin{array}{l}-b^2\\ -b+\text{log}\left(1+b\right)\end{array}=O\left(b^2\right),$$ (8.8)

Now, using (8.2) and (7.22), we get

$$\underset{r+1\le i\le p}{\text{max}}\left|b_i\right|\le r\underset{1\le k\le r}{\text{max}}\text{|}{\beta }_k\text{|}\cdot \underset{i>r,k\le r}{\text{max}}{\left(u_k^{\prime }{v}_i\right)}^2=O_P\left(\frac{\text{log }p}{p}\right).$$

From the previous two displays, we conclude

$${\displaystyle \sum _{i=r+1}^p\underset{{\Delta }_i}{\text{min}}}H\left(b_i,{\Delta }_i\right)={\displaystyle \sum _{i=r+1}^{p}h\left(b_i\right)}=O_P\left(\frac{{\text{log}}^{2}p}{p}\right).$$

which is (8.7), and so completes the full proof.

9. Optimal Shrinkage with common variance σ² ≠ 1

Simply put, the Spiked Covariance Model is a proportional growth independent-variable Gaussian model, where all variables, except the first r, have common variance σ. Literature on the spiked model often simplifies the situation by assuming σ² = 1, as we have done in our assumption [Spike(ℓ₁,…, ℓ_r)] above. To consider optimal shrinkage in the case of general common variance σ² > 0, assumption [Spike(ℓ₁,…, ℓ_r)] has to be replaced by

[Spike(ℓ₁,…, ℓ_r|σ²)] The population eigenvalues in the n-th problem, namely the eigenvalues of Σ_pn, are given by (ℓ₁,…, ℓ_r, σ²,…, σ²), where the number of “spikes” r and their amplitudes ℓ₁ >…> ℓ_r ≥ 1 are fixed independently of n and p_n.

In this section we show how to use an optimal shrinker, designed for the spiked model with common variance σ² = 1, in order to construct an optimal shrinker for a general common variance σ², namely, under assumptions [Asy(γ)] and [Spike(ℓ₁,…, ℓ_r|σ²)].

9.1. σ² known

Let Σ_p and S_n,p be population and sample covariance matrices, respectively, under assumption [Spike(ℓ₁,…, ℓ_r|σ²)]. When the value of σ is known, the matrices ${\tilde{\Sigma }}_p={\Sigma }_p/{\sigma }^2$and the sample covariance matrix ${\tilde{S}}_{n,p}=S_{n,p}/{\sigma }^2$ constitute population and sample covariance matrices, respectively, under assumption [Spike(ℓ₁,…, ℓ_r)]. Let L be any of the loss families considered above and let η be a shrinker. Define the shrinker $\tilde{\eta }$ corresponding to η by

$$\tilde{\eta }:\lambda \mapsto {\sigma }^2\cdot \eta \left(\lambda /{\sigma }^2\right).$$ (9.1)

Observe that for each of the loss families we consider, L_p (σ²A, σ²B) = σ^2κL_p(A, B), where κ ∈ {−2, −1, 0, 1, 2} depends on the family {L_p} alone. Hence

$$L_p\left({\Sigma }_p,{\widehat{\Sigma }}_{\tilde{\eta }}\left(S_{n,p}\right)\right)={\sigma }^{2\kappa }{L}_p\left({\tilde{\Sigma }}_p,{\widehat{\Sigma }}_{\tilde{\eta }}\left({\tilde{S}}_{n,p}\right)\right)$$

It follows that if η^* is the optimal shrinker for the loss family L, in the sense of Definition 6, under Assumption [Spike(ℓ₁,…, ℓ_r)] , then ${\tilde{\eta }}^{*}$is the optimal shrinker for L under Assumption [Spike(ℓ₁,…, ℓ_r|σ²)]. Formula (9.1) therefore allows us to translate each of the optimal shrinkers given above to a corresponding optimal shrinker in the case of a general common variance σ² > 0.

9.2. σ² unknown

In practice, even if one is willing to assume a common variance σ² and subscribe to the spiked model, the value of σ² is usually unknown. Assume however that we have a sequence of estimators${\left\{{\widehat{\sigma }}_n\right\}}_{n=1,2\dots ,}$where for each n, ${\widehat{\sigma }}_n$ is a real function of a p_n-by-p_n positive definite symmetric matrix argument. Assume further that under the spiked model with general common variance σ², namely under assumptions [Asy(γ)] and [Spike(ℓ₁,…, ℓ_r |σ²)], the sequence of estimators is consistent in the sense that ${\widehat{\sigma }}_n\left(S_{n,p_n}\right)\to \sigma$, almost surely. For a continuous shrinker η, define a sequence of shrinkers ${\left\{{\tilde{\eta }}_n\right\}}_{n=1,2,\dots }$by

$${\tilde{\eta }}_n:\lambda \mapsto {\widehat{\sigma }}_n^2\cdot \eta \left(\lambda /{\widehat{\sigma }}_n^2\right).$$ (9.2)

Again for each of the loss families we consider, almost surely,

$$\underset{n\to \infty }{\text{lim}}{L}_{p_n}\left({\Sigma }_{p_n},{\widehat{\Sigma }}_{{\tilde{\eta }}_n}\left(S_{n,p_n}\right)\right)={\sigma }^{2\kappa }\underset{n\to \infty }{\text{lim}}{L}_{p_n}\left({\tilde{\Sigma }}_{p_n},{\widehat{\Sigma }}_{\eta }\left({\tilde{S}}_{n,p_n}\right)\right).$$

We conclude that, using (9.2), any consistent sequence of estimators ${\widehat{\sigma }}_n$ yields a sequence of shrinkers with the same asymptotic loss as the optimal shrinker for known σ². In other words, at least inasmuch as the asymptotic loss is concerned, under the spiked model, there is no penalty for not knowing σ².

Estimation of σ² under Assumption [Spike(ℓ₁,…, ℓ_r |σ²)] has been considered in [65, 66, 31] where several approaches have been proposed. As an simple example of a consistent sequence of estimators ${\widehat{\sigma }}_n$, we consider the following simple and robust approach based on matching of medians [32]. The underlying idea is that for a given value of n the sample eignevalues λ_r+1,…, λ_pn form an approximate Marčenko-Paster bulk inflated by σ², and that a median sample eigenvalue is well suited to detect this inflation as it is unaffected by the sample spikes λ₁,…, λ_r.

Define, for a symmetric p-by-p positive definite matrix S with eigenvalues λ₁,…, λ_p the quantity

$$\mu \left(S\right)=\frac{{\lambda }_{med}}{{\mu }_{\gamma }},$$ (9.3)

where λ_med is a median of λ₁,…, λ_p and μ_γ is the median of the Marčenko-Pastur distribution, namely, the unique solution in λ₋(γ) ≤ x ≤ λ₊(γ) to the equation

$${\displaystyle \underset{{\lambda }_{-}\left(\gamma \right)}{\overset{x}{\int }}\frac{\sqrt{\left({\lambda }_{+}\left(\gamma \right)-t\right)\left(t-{\lambda }_{-}\left(\gamma \right)\right)}}{2\pi \gamma t}}dt=\frac{1}{2},$$

where as before ${\lambda }_{\pm }\left(\gamma \right)={\left(1\pm \sqrt{\gamma }\right)}^2$. Note that the median μ_γ is not available analytically but can easily be obtained numerically, for example using remarks on the Marčenko-Pastur cumulative distribution function included in SI. Now for a sequence {S_n,pn} of sample covariance matrices,define the sequence of estimators

$${\widehat{\sigma }}_n:\left(S_{n,p_n}\right)\mapsto \sqrt{\mu \left(S_{n,p_n}\right)}.$$ (9.4)

Lemma 9. Let σ² > 0, and assume [Asy(γ)] and [Spike(ℓ₁, … , ℓ_r|σ²)]. Then almost surely

$$\underset{n\to \infty }{\text{lim}}{\widehat{\sigma }}_n\left(S_{n,p_n}\right)=\sigma .$$

In summary, using (9.1) (for σ² known) or (9.2) with (9.4) (for σ² unknown) one can use the optimal shrinkers for each of the loss families discussed above, designed for the case σ = 1, to construct a shrinker that is optimal, for the same loss family, under the spiked model with common variance σ² ≠ 1.

10. Discussion

In this paper, we considered covariance estimation in high dimensions, where the dimension p is comparable to the number of observations n. We chose a fixed-rank principal subspace, and let the dimension of the problem grow large. A different asymptotic framework for covariance estimation would choose a principal subspace whose rank is a fixed fraction of the problem dimension; i.e. the rank of the principal subspace is growing rather than fixed. (In the sibling problem of matrix denoising, compare the “spiked” setup [32, 31, 53] with the “fixed fraction” setup of [67].)

In the fixed fraction framework, some of underlying phenomena remain qualitatively similar to those governing the spiked model, while new effects appear. Importantly, the relationships used in this paper, predicting the location of the top empirical eigenvalues, as well as the displacement of empirical eigenvectors, in terms of the top theoretical eigenvalues, no longer hold. Instead, a complex nonlinear relation exists between the limiting distribution of the empirical eigenvalues and the limiting distribution of the theoretical eigenvalues, as expressed by the Marčenko-Pastur (MP) relation between their Stieltjes transforms [33, 68].

Covariance shrinkage in the proportional rank model should then, naturally, make use of the so-called MP Equation. Noureddine El Karoui [24] proposed a method for debiasing the empirical eigenvalues, namely, for estimating (in a certain specific sense) their corresponding population eigenvalues; Olivier Ledoit and Sandrine Peché [25] developed analytic tools to also account for the inaccuracy of empirical eigenvectors, and Ledoit and Michael Wolf [26] have implemented such tools and applied them in this setting.

The proportional rank case is indeed subtle and beautiful. Yet, the fixed-rank case deserves to be worked out carefully. In particular, the shrinkers we have obtained here in the fixed-rank case are extremely simple to implement, requiring just a few code lines in any scientific computing language. In comparison, the covariance estimation ideas of [24, 26], based on powerful and deep insights from MP theory, require a delicate, nontrivial effort to implement in software, and call for expertise in numerical analysis and optimization. As a result, the simple shrinkage rules we propose here may be more likely to be applied correctly in practice, and to work as expected, even in relatively small sample sizes.

An analogy can be made to shrinkage in the normal means problem, for example [69]. In that problem, often a full Bayesian model applies, and in principle a Bayesian shrinkage would provide an optimal result [70]. Yet, in applications one often wants a simple method which is easy to implement correctly, and which is able to deliver much of the benefit of the full Bayesian approach. In literally thousands of cases, simple methods of shrinkage - such as thresholding - have been chosen over the full Bayesian method for precisely that reason.

Reproducible Research.

In the code supplement [41] we offer a Matlab software library that includes:

1. A function to compute the value of each of the 26 optimal shrinkers discussed to high precision.

2. A function to test the correctness of each of the 18 analytic shrinker fomulas provided.

3. Scripts that generate each of the figures in this paper, or subsets of them for specified loss functions.