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Quantitative correlation inequalities via extremal power series

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

  1. We note that both Eqs. (1) and (2) are sharp, and neither implies the other; see Sect. 4.1 of [23] or the discussion following Theorem 1.4 in [20].

  2. 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.

  3. Looking ahead (see Eq. (9)), it will be immediate from the monotone compatibility of \(\mathcal {F}\) with \(\mathrm {T}_\rho \) that \(a_{j^*}\ge 0\).

  4. 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.

  5. 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.

  6. Note that the 0/1 indicator function of a convex set is not a convex function.

  7. A function \(f:\mathbb {R}^n \rightarrow \mathbb {R}\) is quasiconcave if for all \(\lambda \in [0,1]\) we have \(f{\left( \lambda x + (1-\lambda )y \right) }\ge \min {\left\{ f(x), f(y) \right\} }\).

References

  1. Aizenman, M., Germinet, F., Klein, A., Warzel, S.: On Bernoulli decompositions for random variables, concentration bounds, and spectral localization. Probab. Theory Relat. Fields 143(1–2), 219–238 (2009)

    Article  MathSciNet  Google Scholar 

  2. Borwein, P., Erdélyi, T.: Littlewood-type polynomials on subarcs of the unit circle. Indiana Univ. Math. J. 46(4), 1323–1346 (1997)

    Article  MathSciNet  Google Scholar 

  3. Borwein, P., Erdélyi, T., Kós, G.: Littlewood-type problems on \([0,1]\). In: Proceedings of the London Mathematical Society, vol. 3, no. 79, pp. 22–46 (1999)

  4. Berry, A.C.: The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49(1), 122–136 (1941)

    Article  MathSciNet  Google Scholar 

  5. Bakry, D., Gentil, I., LeDoux, M.: Analysis and Geometry of Markov Diffusion Operators. Springer, New York (2013)

    MATH  Google Scholar 

  6. Cordero-Erausquin, D., Ledoux, M.: Hypercontractive measures, Talagrand’s inequality, and influences. In: Klartag, B., Mendelson, S., Milman, V. (Eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg, pp. 169–189 (2012)

  7. De, A., Diakonikolas, I., Servedio, R.A.: The inverse Shapley value problem. Games Econ. Behav. 105, 122–147 (2017)

    Article  MathSciNet  Google Scholar 

  8. De, A., Nadimpalli, S., Servedio, R.A.: Convex influences. In preparation. (2021)

  9. Eldan, R.: Second-order bounds on correlations between increasing families. arXiv:1912.11641 (2019)

  10. Esseen, C.-G.: On the Liapunoff limit of error in the theory of probability. Ark. Mat. Astron. Fys. A, 1–19 (1942)

  11. Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22(2), 89–103 (1971)

    Article  MathSciNet  Google Scholar 

  12. Gallavotti, G.: A proof of the Griffiths inequalities for the XY model. Stud. Appl. Math 50(1), 89–92 (1971)

    Article  Google Scholar 

  13. Griffiths, R.: Correlations in Ising ferromagnets. I. J. Math. Phys. 8(3), 478–483 (1967)

    Article  Google Scholar 

  14. Harris, T.E.: A lower bound for the critical probability in a certain percolation process. In: Proceedings of the Cambridge Philosophical Society, vol. 56, pp. 13–20 (1960)

  15. Hargé, G.: Characterization of equality in the correlation inequality for convex functions, the U-conjecture. Ann. Inst. Henri Poincaré Probabilités et Stat. 41(4), 753–765 (2005)

  16. Hu, Y.: Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities. J. Theor. Probab. 10(4), 835–848 (1997)

    Article  Google Scholar 

  17. Keller, N.: Improved FKG inequality for product measures on the discrete cube. (2008)

  18. Keller, N.: Influences of variables on Boolean functions. PhD thesis, Hebrew University of Jerusalem, (2009)

  19. Keller, N.: A simple reduction from a biased measure on the discrete cube to the uniform measure. Eur. J. Comb. 33, 1943–1957 (2012)

    Article  MathSciNet  Google Scholar 

  20. Kalai, G., Keller, N., Mossel, E.: On the correlation of increasing families. J. Comb. Theory Ser. A 144, 11 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Kleitman, D.J.: Families of non-disjoint subsets. J. Comb. Theory 1(1), 153–155 (1966)

    Article  MathSciNet  Google Scholar 

  22. Keller, N., Mossel, E., Sen, A.: Geometric influences. Ann. Probab. 40(3), 1135–1166 (2012)

    Article  MathSciNet  Google Scholar 

  23. Keller, N., Mossel, E., Sen, A.: Geometric influences II: correlation inequalities and noise sensitivity. Ann. Inst. l’IHP Poincaré Probab. Stat. 50(4), 1121–1139 (2014)

  24. Kelly, D., Sherman, S.: General Griffiths’ inequalities on correlations in Ising ferromagnets. J. Math. Phys. 9(3), 466–484 (1968)

  25. Latała, R., Matlak D.: Royen’s Proof of the Gaussian Correlation Inequality. In: Klartag B., Milman E. (Eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2169. Springer, Cham, pp. 265–275(2017)

  26. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press, Boca Raton (2002)

    Book  Google Scholar 

  27. Mossel, E.: Probabilistic aspects of voting, intransitivity and manipulation. (2020)

  28. O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)

  29. Royen, T.: A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions. arXiv preprint arXiv:1408.1028, (2014)

  30. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Inc., New York (1987)

    MATH  Google Scholar 

  31. Talagrand, M.: How much are increasing sets positively correlated? Combinatorica 16(2), 243–258 (1996)

    Article  MathSciNet  Google Scholar 

Download references

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:

$$\begin{aligned} a_d(t) := \frac{T_{\sqrt{d}}\left( t\left( 1+\frac{3}{d}\right) \right) }{ T_{\sqrt{d}}\left( 1+\frac{3}{d}\right) } \end{aligned}$$

where d is a parameter (a perfect square) that we will set later. We make the following simple observations:

  • \(|a_d(t)| \le 1\) for all \(t \in [0,1]\), and \(a(1)=1.\)

  • 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 \(t \ge 1.\)

  • 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

$$\begin{aligned} b(t) := a_1(1-t)\cdot a_4(1-t) \cdot a_{16}(1-t) \cdots a_{\log ^2M}(1-t). \end{aligned}$$

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

$$\begin{aligned} 2^{O{\left( \sqrt{1} \right) } + O{\left( \sqrt{4} \right) } + \ldots + O{\left( \sqrt{\log ^2 M} \right) }} = 2^{O(\log M)}. \end{aligned}$$

Finally, we define

$$\begin{aligned} p(t) := t\cdot b(t). \end{aligned}$$

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

$$\begin{aligned}&{|}a_{4^j}(1-t){|} \le \frac{1}{4} \qquad \text {which implies that}\qquad |a_{4^{(i+1)}}(t)| \cdot \\&\quad |a_{4^{(i+2)}}(t)| \cdots |a_{4^k}(t)| \le \frac{1}{4^{k-i}}. \end{aligned}$$

As \(t \le \frac{1}{4^{i-1}}\), it follows that

$$\begin{aligned} |p(t)| = |t\cdot b(t)| \le \frac{1}{4^{i-1}}\cdot \frac{1}{4^{k-i}} = \frac{1}{4^{k-1}} = \Theta {\left( \frac{1}{\log ^2 M} \right) }, \end{aligned}$$

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.

$$\begin{aligned} f(x) := \mathrm {sign}(x_1 + \cdots + x_n - n(1-2p)), \end{aligned}$$

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

$$\begin{aligned} \sum _{i=1}^n \widehat{f_p}(i)&= {\mathbf{E}}\left[ f({\varvec{x}}) \cdot \sum _{i=1}^n {\frac{{\varvec{x}}_i - (1-2p)}{2\sqrt{p(1-p)}}}\right] \nonumber \\&= {\frac{1}{2\sqrt{p(1-p)}}} {} \mathbf{E}\left[ \left| \left( \sum _{i=1}^n {\varvec{x}}_i\right) - n(1-2p) \right| \right] . \end{aligned}$$
(15)

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

$$\begin{aligned} \Pr \left[ \left| \left( \sum _{i=1}^n {\varvec{x}}_i\right) - n(1-2p) \right| \ge c\sqrt{np(1-p)}\right] \ge {\frac{1}{2}}. \end{aligned}$$

Consequently

$$\begin{aligned} {} \mathbf{E}\left[ \left| \left( \sum _{i=1}^n {\varvec{x}}_i\right) - n(1-2p) \right| \right] \ge {\frac{c\sqrt{np(1-p)}}{2}}, \end{aligned}$$

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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Mathematics Subject Classification

  • 52 (Convex and discrete geometry)
  • 60 (Probability theory and stochastic processes)