Abstract
Many correlation inequalities for high-dimensional functions in the literature, such as the Harris–Kleitman inequality, the Fortuin–Kasteleyn–Ginibre inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. Previous work has used Markov semigroup arguments to obtain quantitative extensions of some of these correlation inequalities. In this work, we augment this approach with a new extremal bound on power series, proved using tools from complex analysis, to obtain a range of new and near-optimal quantitative correlation inequalities. These new results include: A quantitative version of Royen’s celebrated Gaussian Correlation Inequality (Royen, 2014). In (Royen, 2014) Royen confirmed a conjecture, open for 40 years, stating that any two symmetric convex sets must be non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 Hermite coefficients of the two convex sets, conceptually similar to Talagrand’s quantitative correlation bound for monotone Boolean functions over \(\{0,1\}^n\) (Talagrand in Combinatorica 16(2):243–258, 1996). We show that our quantitative version of Royen’s theorem is within a logarithmic factor of being optimal. A quantitative version of the well-known FKG inequality for monotone functions over any finite product probability space. This is a broad generalization of Talagrand’s quantitative correlation bound for functions from \(\{0,1\}^n\) to \(\{0,1\}\) under the uniform distribution (Talagrand in Combinatorica 16(2):243–258, 1996). In the special case of p-biased distributions over \(\{0,1\}^n\) that was considered by Keller, our new bound essentially saves a factor of \(p \log (1/p)\) over the quantitative bounds given in Keller (Eur J Comb 33:1943–1957, 2012; Improved FKG inequality for product measures on the discrete cube, 2008; Influences of variables on Boolean functions. PhD thesis, Hebrew University of Jerusalem, 2009).
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Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
Since convexity is preserved under linear transformation, no loss of generality is incurred in assuming that the background measure is the standard normal distribution \(\mathcal{N}(0,1)^n\) rather than an arbitrary centered Gaussian.
Looking ahead (see Eq. (9)), it will be immediate from the monotone compatibility of \(\mathcal {F}\) with \(\mathrm {T}_\rho \) that \(a_{j^*}\ge 0\).
Note that this implies the following order property: if \(f \ge g\) almost everywhere, then \(\mathrm {P}_t f \ge \mathrm {P}_t g\) almost everywhere.
The “3/2” in the lemma below could be replaced by any constant bounded above 1; we use 3/2 because it is convenient in our later application of Lemma 11.
Note that the 0/1 indicator function of a convex set is not a convex function.
A function \(f:\mathbb {R}^n \rightarrow \mathbb {R}\) is quasiconcave if for all \(\lambda \in [0,1]\) we have \(f{\left( \lambda x + (1-\lambda )y \right) }\ge \min {\left\{ f(x), f(y) \right\} }\).
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Acknowledgements
A.D. is supported by NSF grants CCF 1910534 and CCF 1926872. S.N. is supported by NSF grants CCF-1563155 and by CCF-1763970. R.A.S. is supported by NSF grants CCF-1814873, IIS-1838154, CCF-1563155, and by the Simons Collaboration on Algorithms and Geometry. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF). This work was done while A.D. was participating in the “Probability, Geometry, and Computation in High Dimensions” program at the Simons Institute for the Theory of Computing.
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Appendices
Proof of Claim 12
For \(c \in \mathbb {N}\), let \(T_c(x)\) denote the degree-c Chebyshev polynomial of the first kind. Define the univariate polynomial:
where d is a parameter (a perfect square) that we will set later. We make the following simple observations:
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\(|a_d(t)| \le 1\) for all \(t \in [0,1]\), and \(a(1)=1.\)
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Let \(d\ge 4\). For \(t\in {\left[ 0, 1-\frac{3}{d} \right] }\), we have \(a_d(t) \in {\left[ -\frac{1}{4}, \frac{1}{4} \right] }\). This follows from the fact that \(\left( 1 - \frac{3}{d}\right) \left( 1 + \frac{3}{d}\right) < 1\), that \(|T_{\sqrt{d}}(t)| \le 1\) for \(|t| \le 1\), and that the derivative \(T'_{\sqrt{d}}(t)\) is at least d for all \(t \ge 1.\)
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The sum of the absolute values of the coefficients of \(a_d(t)\) is at most \(2^{O{\left( \sqrt{d} \right) }}\). This is an easy consequence of standard coefficient bounds for Chebyshev polynomials (see e.g. Sect. 2.3.2 of [26]).
For simplicity, assume \(\log ^2 M =4^k\) for some \(k\in \mathbb {N}\). We define b(t) as
Note that b(t) is a polynomial of degree \(\sqrt{1} + \sqrt{4} + \sqrt{16} + \ldots + \sqrt{\log ^2 M} = \Theta (\log M)\), and that \(|b(t)| \le 1\) for all \(t \in [0,1]\). It follows from the third item above that the sum of the absolute values of the coefficients of b(t) is at most
Finally, we define
In order to upper bound |p(t)| for \(t \in [0,1]\), we first observe that if \(t \le {\frac{1}{4^k}}\) then we have \(|p(t)| \le {\frac{1}{4^k}} |b(t)| \le {\frac{1}{4^k}} \le {\frac{1}{\log ^2 M}}\) as desired. Thus we may suppose that \(t \in {\left[ \frac{1}{4^i}, \frac{1}{4^{i-1}} \right] }\) for some \(i \in \{1,\dots ,k\}\); in particular, let \(t = \frac{1}{4^i} + \delta \) for \(\delta \in {\left[ 0, \frac{3}{4^i} \right] }\). Now, for each \(j \ge i+1\), we have
As \(t \le \frac{1}{4^{i-1}}\), it follows that
and Claim 12 is proved. It follows that Lemma 11 is tight up to constant factors.
Comparison of Theorem 28 and Theorem 29
Let \(\omega (1)/n \le p \le 1/2\). Observe that under \(\{-1,1\}^n_p\) we have \({\mathbf{E}}[{\varvec{x}}_1 + \cdots + {\varvec{x}}_n] = n(1-2p)\). We define \(f: \{-1,1\}^n\rightarrow \{-1,1\}\) to be the “p-biased analogue of the majority function,” i.e.
and we take \(g=f.\)
Since (as is well known) the median of the Binomial distribution \(\mathrm {Bin}(n,p)\) differs from the mean by at most 1, it follows (using the Littlewood-Offord anticoncentration inequality described below) that \({\mathbf{E}}[f] = o(1)\), and hence we have (i): \({\mathbf{E}}[fg] - {\mathbf{E}}[f]{\mathbf{E}}[g] \ge 1 - o(1).\) To establish (ii) and (iii) it remains only to show that for any fixed \(i \in [n]\) we have that the p-biased degree-1 Fourier coefficient \(\widehat{f_p}(i)\) is at least \(\Omega (1/\sqrt{n})\), or equivalently, that \(\widehat{f_p}(1) + \cdots + \widehat{f_p}(n) = \Omega (\sqrt{n}).\) To see this, we observe that this sum of degree-1 Fourier coefficients is
We now recall the Littlewood-Offord anticoncentration inequality for the p-biased Boolean hypercube (see e.g. Theorem 5 of [7] or [1]). Specialized to our context, this says that for any real interval I of length at least 1, it holds that \(\Pr \left[ \sum _{i=1}^n {\varvec{x}}_i \in I\right] \le O(|I|)/\sqrt{np(1-p)}.\) Taking I to be the interval of length \(c \sqrt{np(1-p)}\) centered at \(n(1-2p)\) for a suitably small positive constant c, it holds that
Consequently
which together with Eq. (15) gives that \(\sum _{i=1}^n \widehat{f_p}(i) \ge c\sqrt{n}/4\) as desired.
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De, A., Nadimpalli, S. & Servedio, R.A. Quantitative correlation inequalities via extremal power series. Probab. Theory Relat. Fields 183, 649–675 (2022). https://doi.org/10.1007/s00440-022-01120-5
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DOI: https://doi.org/10.1007/s00440-022-01120-5
Mathematics Subject Classification
- 52 (Convex and discrete geometry)
- 60 (Probability theory and stochastic processes)