Does smoothing a non-log-concave distribution make it more log-concave?

Question

Suppose that $p$ is a density on $\mathbb{R}^d$ that is $C^2$ and nonzero everywhere, and such that the Hessian of its negative logarithm is lower bounded: $$-\nabla^2 \ln p\succeq L$$ for some matrix $L$. (Here $A\succeq B$ means $A-B$ is positive semidefinite.) We say that $p$ is log-concave if this holds with $L=0$.

Let $q$ be the density of $N(0,\Sigma)$ (gaussian with covariance $\Sigma$). It is known that if $p$ is strongly log-concave with $L\succ 0$, then $p*q$ also satisfies a lower bound: Theorem 3.7b in [1] states that $$-\nabla^2 \ln (p*q) \succeq (L^{-1}+\Sigma)^{-1}.$$ Note this is tight in the case that $p$ is a gaussian with covariance $\Sigma'=L^{-1}$, in which case $p*q$ is the density of $N(0,\Sigma'+\Sigma)$.

My question is the following: What lower bound can we derive on $-\nabla^2\ln (p*q)$ when $p$ is not log-concave, that is, $L$ is not PSD? Intuitively I would expect that convolving with a gaussian makes it more log-concave, that is, increases the lower bound, though I am also interested in any lower bound that can be obtained (that works for all ranges of $L$ and $\Sigma$).

I would also be OK with assuming that $p$ satisfies a log-Sobolev inequality: $$\text{Ent}_p(f^2) \le 2C_{LS} \mathbb E_p [\|\nabla f\|^2]$$ for all smooth $f$, where $\text{Ent}_p(f^2):= \mathbb E_p[f^2 \ln (f^2/\mathbb E_p (f^2))]$. This may be helpful as it gives concentration properties for $p$.

[1] Adrien Saumard and Jon A Wellner. Log-concavity and strong log-concavity: a review. Statistics surveys, 8:45, 2014. https://arxiv.org/pdf/1404.5886.pdf

Answer 1

For log-concavity in the strict sense, the examples given in the comments show that the heat flow need not induce or improve the log-concavity property; the smoothing of a measure with heavier-than-Gaussian tails will still generally have heavier-than-Gaussian tails, and hence strong log-concavity is precluded.

For 'moral' log-concavity in the sense of functional inequalities, Chafaï's work here demonstrates that a fairly general class of Sobolev-type inequalities are transformed nicely by the heat flow; see Corollary 13 in Section 3.2. I believe that the reasoning is something like the following: functional inequalities behave nicely under the operations of { tensorisation, orthogonal change of variables, marginalisation }, and convolution can be expressed as a composition of these operations.

(I am fairly sure that the above is true, but have not re-read the paper recently; hopefully I am not missing some implicit restriction to log-concave measures!)

N.B. In the results of Chafaï, the constants appear to get worse in absolute terms. This is to be expected, as the resulting measures are genuinely more 'spread out', by virtue of the convolution. I expect that if one rescales the inequality by the 'effective diameter' of the resulting measure, then the concentration properties should look 'better' in relative terms as the level of smoothing increases.