Quantitative Stability of the Entropy Power Inequality

Courtade, Thomas A.; Fathi, Max; Pananjady, Ashwin

doi:10.1109/tit.2018.2808161

Cited by 25 publications

(31 citation statements)

References 48 publications

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“…where C σ X ,σ Y depends only on σ X and σ Y . Up to this constant, this is precisely the main result of [6]. In fact, our method can reproduce their exact result, which we present as a warm up in the next section.…”

Section: Our Contributionsupporting

confidence: 55%

“…To compare this with the main result of [6] we recall the transportation-entropy inequality due to Talagrand ( [19]) which states that…”

Section: Our Contributionmentioning

confidence: 99%

“…It seems plausible to conjecture that the dependence on the minimal eigenvalue and Poicnaré constant could be replaced by a quantity which would take into consideration all eigenvalues. Some other known stability results, both for log-concave vectors and for other classes of measures, may be found in [5,6,20]. The reader is referred to [6, Section 2.2] for a complete discussion.…”

Section: Our Contributionmentioning

confidence: 99%

“…One might expect that, in light of the equality cases, a small deficit in (3) should imply that X and Y are both close, in some sense, to a Gaussian. A recent line of works has focused on an attempt to make this intuition precise (see e.g., [6,20]), which is also our main goal in the present work.…”

Section: Introductionmentioning

confidence: 99%

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Stability of the Shannon–Stam inequality via the Föllmer process

Eldan

Mikulincer

2020

Probab. Theory Relat. Fields

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We prove stability estimates for the Shannon-Stam inequality (also known as the entropypower inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X, Y ∈ R d , the deficit in the Shannon-Stam inequality is bounded from below by the expressionwhere D (· ||G) denotes the relative entropy with respect to the standard Gaussian and the constant C depends only on the covariance structures and the spectral gaps of X and Y . In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.Our approach is based on ideas somewhat related to the ones which appear in [8]: the very highlevel plan of the proof is to embed the variables X, Y as the terminal points of some martingales and express the entropies of X, Y and X+Y as functions of the associates quadratic co-variation processes. One of the main benefits in using such an embedding is that the co-variation process of X + Y can be easily expressed in terms on the ones of X, Y , as demonstrated below. In [8]

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Section: Our Contributionsupporting

confidence: 55%

“…To compare this with the main result of [6] we recall the transportation-entropy inequality due to Talagrand ( [19]) which states that…”

Section: Our Contributionmentioning

confidence: 99%

Section: Our Contributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability of the Shannon–Stam inequality via the Föllmer process

Eldan

Mikulincer

2020

Probab. Theory Relat. Fields

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“…For any given continuous density f over R 2 , there exists a differentiable transformation T : R 2 → R 2 with symmetric positive definite Jacobian T ′ (noted T ′ > 0) such that X = T ( X * ) has density f . Courtade et al [18] noted that the Brenier map can be used in the transportation proof of [5] to prove Shannon's vector EPI. We find it also convenient to prove the complex matrix EPI: Theorem 2: The matrix EPI (3) holds for any m × n complex matrix A and any random vector X of independent complex components X i , where X * is a c-normal vector with independent components X * i satisfying (4).…”

Section: Extension To Complex Matrix and Variablesmentioning

confidence: 99%