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Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a positively-homogeneous $1$-Lipschitz continuous function, and let $C \in \mathbb R^{k \times k}$ be the covariance matrix of random vector $y:=f(Ax) := (f(a_1^\top x),\ldots,f(a_k^\top x)) \in \mathbb R^k$.

Question. Is it possible to give lower-bounds on the smallest eigenvalue of $C$ which only depend on spectral properties of $A$ ?

N.B.: I'm particularly interested in the cases (i) $f(t):=\max(t,0)$, and (ii) $f(t) = |t|$.

Example

Consider the the case where $f(t) := t$, so that $C = cov(Wx) = Wcov(x)W^\top = (1/d)WW^\top$. Thus, $\lambda_{\min}(C) = (1/d)\lambda_{\min}(WW^\top)$.

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    $\begingroup$ arxiv.org/abs/1702.05419 could be useful. In particular check Table 1 there which gives $E[f(a^Tx) f(b^Tx)]$ for various example including max(0,t) and |t|. Some results there should also give you some asymptotic distributions of the eigenvalues. Various works with similar results in the last few years are cited there or cite this work. $\endgroup$
    – jlewk
    Commented May 28, 2021 at 8:55
  • $\begingroup$ Thanks for the input. I've solved my problem since then, and should writeup and answer asap. Indeed, in certain cases (when $f$ is positive homogeneous), C is a dot-product kernel and there are lots of papers (pioneered by El Karoui's projecteuclid.org/journals/annals-of-statistics/volume-38/…) on the eigenvalues of such random matrices. $\endgroup$
    – dohmatob
    Commented May 28, 2021 at 9:15

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