arXiv subjects

Lampros Gavalakis

Publications and source records attributed to Lampros Gavalakis.

At least 19 recordsLinked to original sources

arXiv2026

Entropy lower bounds and sum-product phenomena

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

math.CO↗

arXiv2026

Entropy power inequalities in compact groups

Suppose $X,Y$ are independent random variables with values in a compact abelian group $(G,+)$. We examine the following two entropy power-type inequalities: $h(X+Y)\geq \frac{1}{2}h(X)+\frac{1}{2}h(Y)$ and $h(X+Y)\geq \max\{h(X),h(Y)\}$, where the entropy $h(Z)$ of a $G$-valued random variable $Z$ is defined in terms of its density with respect to Haar measure on $G$. For groups that are either connected or finite with no nontrivial subgroups, we precisely characterize the cases of equality and establish explicit, quantitative stability estimates in terms of relative entropy for these two inequalities. The main tools are a generalization of an entropic inequality obtained by Green, Manners and Tao (2023) for discrete entropy, and a harmonic-analytic estimate for the chi-squared contraction coefficient in connected compact groups. As an application, we derive exponential convergence rates to the uniform distribution in relative entropy for random walks on connected compact abelian groups.

math.PR↗

arXiv2026

Entropic analogues of Grünbaum's inequality

The classical Grünbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Grünbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all Rényi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

math.PR↗

arXiv2025

On the monotonicity of discrete entropy for log-concave random vectors on $\mathbb{Z}^d$

We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ where $o(1)$ vanishes as $H(X_1) \to \infty$. Moreover, for the $o(1)$-term, we obtain a rate of convergence $ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$, where the implied constants depend on $d$ and $n$. This generalizes to $\mathbb{Z}^d$ the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy $H(X_1+\cdots+X_{n})$ is close to the differential (continuous) entropy $h(X_1+U_1+\cdots+X_{n}+U_{n})$, where $U_1,\dots, U_n$ are independent and identically distributed uniform random vectors on $[0,1]^d$ and to apply the theorem of Artstein, Ball, Barthe and Naor (2004) on the monotonicity of differential entropy. In fact, we show this result under more general assumptions than log-concavity, which are preserved up to constants under convolution. In order to show that log-concave distributions satisfy our assumptions in dimension $d\ge2$, more involved tools from convex geometry are needed because a suitable position is required. We show that, for a log-concave function on $\mathbb{R}^d$ in isotropic position, its integral, barycenter and covariance matrix are close to their discrete counterparts. Moreover, in the log-concave case, we weaken the isotropicity assumption to what we call almost isotropicity. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which extends to dimensions $d\ge1$ a result of Bobkov, Marsiglietti and Melbourne (2022).

math.PR↗

arXiv2025

Entropic additive energy and entropy inequalities for sums and products

Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums, products, and sum-product combinations of continuous random variables. Partly motivated by recent work by Goh on the discrete entropic version of the notion of "additive energy", we introduce the additive energy of pairs of continuous random variables and prove various versions of the statement that "the additive energy is large if and only if the entropy of the sum is small", along with a version of the Balog-Szemerédi-Gowers theorem for differential entropy. Then, motivated in part by recent work by Máthé and O'Regan, we establish a series of new differential entropy inequalities for products and sum-product combinations of continuous random variables. In particular, we prove a new, general, ring Plünnecke-Ruzsa entropy inequality. We briefly return to the case of discrete entropy and provide a characterization of discrete random variables with "large doubling", analogous to Tao's Freiman-type inverse sumset theory for the case of small doubling. Finally, we consider the natural entropic analog of the Erdös-Szemerédi sum-product phenomenon for integer-valued random variables. We show that, if it does hold, then the range of parameters for which it does would necessarily be significantly more restricted than its anticipated combinatorial counterpart.

cs.IT↗

arXiv2025

Pragmatic lossless compression: Fundamental limits and universality

The problem of variable-rate lossless data compression is considered, for codes with and without prefix constraints. Sharp bounds are derived for the best achievable compression rate of memoryless sources, when the excess-rate probability is required to be exponentially small in the blocklength. Accurate nonasymptotic expansions with explicit constants are obtained for the optimal rate, using tools from large deviations and Gaussian approximation. When the source distribution is unknown, a universal achievability result is obtained with an explicit ''price for universality'' term. This is based on a fine combinatorial estimate on the number of sequences with small empirical entropy, which might be of independent interest. Examples are shown indicating that, in the small excess-rate-probability regime, the approximation to the fundamental limit of the compression rate suggested by these bounds is significantly more accurate than the approximations provided by either normal approximation or error exponents. The new bounds reinforce the crucial operational conclusion that, in applications where the blocklength is relatively short and where stringent guarantees are required on the rate, the best achievable rate is no longer close to the entropy. Rather, it is an appropriate, more pragmatic rate, determined via the inverse error exponent function and the blocklength.

cs.IT↗

arXiv2025

Conditions for equality and stability in Shannon's and Tao's entropy power inequalities

We show that there is equality in Shannon's Entropy Power Inequality (EPI) if and only if the random variables involved are Gaussian, assuming nothing beyond the existence of differential entropies. This is done by justifying de Bruijn's identity without a second moment assumption. Part of the proof also relies on a re-examination of an example of Bobkov and Chistyakov (2015), which shows that there exists a random variable $X$ with finite differential entropy $h(X),$ such that $h(X+Y) = \infty$ for any independent random variable $Y$ with finite entropy. We prove that either $X$ has this property, or $h(X+Y)$ is finite for any independent $Y$ that does not have this property. Using this, we prove the continuity of $t \mapsto h(X+\sqrt{t}Z)$ at $t=0$, where $Z \sim \mathcal{N}(0,1)$ is independent of $X$, under minimal assumptions. We then establish two stability results: A qualitative stability result for Shannon's EPI in terms of weak convergence under very mild moment conditions, and a quantitative stability result in Tao's discrete analogue of the EPI under log-concavity. The proof for the first stability result is based on a compactness argument, while the proof of the second uses the Cheeger inequality and leverages concentration properties of discrete log-concave distributions.

math.PR↗

arXiv2025

Entropic versions of Bergström's and Bonnesen's inequalities

We establish analogues of the Bergström and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergström inequality and generalize a proof of Bergström's inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.

cs.IT↗

arXiv2024

The entropic doubling constant and robustness of Gaussian codebooks for additive-noise channels

Entropy comparison inequalities are obtained for the differential entropy $h(X+Y)$ of the sum of two independent random vectors $X,Y$, when one is replaced by a Gaussian. For identically distributed random vectors $X,Y$, these are closely related to bounds on the entropic doubling constant, which quantifies the entropy increase when adding an independent copy of a random vector to itself. Consequences of both large and small doubling are explored. For the former, lower bounds are deduced on the entropy increase when adding an independent Gaussian, while for the latter, a qualitative stability result for the entropy power inequality is obtained. In the more general case of non-identically distributed random vectors $X,Y$, a Gaussian comparison inequality with interesting implications for channel coding is established: For additive-noise channels with a power constraint, Gaussian codebooks come within a $\frac{\sf snr}{3{\sf snr}+2}$ factor of capacity. In the low-SNR regime this improves the half-a-bit additive bound of Zamir and Erez (2004). Analogous results are obtained for additive-noise multiple access channels, and for linear, additive-noise MIMO channels.

cs.IT↗

arXiv2024

Finite de Finetti bounds in relative entropy

We review old and recent finite de Finetti theorems in total variation distance and in relative entropy, and we highlight their connections with bounds on the difference between sampling with and without replacement. We also establish two new finite de Finetti theorems for exchangeable random vectors taking values in arbitrary spaces. These bounds are tight, and they are independent of the size and the dimension of the underlying space.

math.PR↗

arXiv2024

A Third Information-Theoretic Approach to Finite de Finetti Theorems

A new finite form of de Finetti's representation theorem is established using elementary information-theoretic tools. The distribution of the first $k$ random variables in an exchangeable vector of $n\geq k$ random variables is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided. This bound is tighter than those obtained via earlier information-theoretic proofs, and its utility extends to random variables taking values in general spaces. The core argument employed has its origins in the quantum information-theoretic literature.

cs.IT↗

arXiv2024

Relative entropy bounds for sampling with and without replacement

Sharp, nonasymptotic bounds are obtained for the relative entropy between the distributions of sampling with and without replacement from an urn with balls of $c\geq 2$ colors. Our bounds are asymptotically tight in certain regimes and, unlike previous results, they depend on the number of balls of each colour in the urn. The connection of these results with finite de Finetti-style theorems is explored, and it is observed that a sampling bound due to Stam (1978) combined with the convexity of relative entropy yield a new finite de Finetti bound in relative entropy, which achieves the optimal asymptotic convergence rate.

math.PR↗

arXiv2024

On the entropy and information of Gaussian mixtures

We establish several convexity properties for the entropy and Fisher information of mixtures of centered Gaussian distributions. First, we prove that if $X_1, X_2$ are independent scalar Gaussian mixtures, then the entropy of $\sqrt{t}X_1 + \sqrt{1-t}X_2$ is concave in $t \in [0,1]$, thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in $\mathbb{R}^d$. As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

cs.IT↗

arXiv2023

Approximate Discrete Entropy Monotonicity for Log-Concave Sums

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{1}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} - o(1) $$ as $H(X_1) \to \infty$, where $H$ denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if $U_1,\ldots,U_n$ are independent continuous uniforms on $(0,1)$, then $$ h(X_1+\cdots+X_n + U_1+\cdots+U_n) = H(X_1+\cdots+X_n) + o(1) $$ as $H(X_1) \to \infty$, where $h$ stands for the differential entropy. Explicit bounds for the $o(1)$-terms are provided.

math.PR↗

arXiv2022

Information in probability: Another information-theoretic proof of a finite de Finetti theorem

We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mixing measure is characterised as the law of the empirical measure of the original sequence, and de Finetti's result is recovered as a corollary. The proof is nicely motivated by the Gibbs conditioning principle in connection with statistical mechanics, and it follows along an appealing sequence of steps. The technical estimates required for these steps are obtained via the use of a collection of combinatorial tools known within information theory as `the method of types.'

math.PR↗

arXiv2021

An Information-Theoretic Proof of a Finite de Finetti Theorem

A finite form of de Finetti's representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $n\geq k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.

cs.IT↗

arXiv2021

Entropy and the Discrete Central Limit Theorem

A strengthened version of the central limit theorem for discrete random variables is established, relying only on information-theoretic tools and elementary arguments. It is shown that the relative entropy between the standardised sum of $n$ independent and identically distributed lattice random variables and an appropriately discretised Gaussian, vanishes as $n\to\infty$.

math.PR↗

arXiv2021

Fundamental Limits of Lossless Data Compression with Side Information

The problem of lossless data compression with side information available to both the encoder and the decoder is considered. The finite-blocklength fundamental limits of the best achievable performance are defined, in two different versions of the problem: Reference-based compression, when a single side information string is used repeatedly in compressing different source messages, and pair-based compression, where a different side information string is used for each source message. General achievability and converse theorems are established for arbitrary source-side information pairs. Nonasymptotic normal approximation expansions are proved for the optimal rate in both the reference-based and pair-based settings, for memoryless sources. These are stated in terms of explicit, finite-blocklength bounds, that are tight up to third-order terms. Extensions that go significantly beyond the class of memoryless sources are obtained. The relevant source dispersion is identified and its relationship with the conditional varentropy rate is established. Interestingly, the dispersion is different in reference-based and pair-based compression, and it is proved that the reference-based dispersion is in general smaller.

cs.IT↗