Inequalities for q-gamma function ratios

Das, Sourav

doi:10.1007/s13324-017-0198-0

Cited by 4 publications

(1 citation statement)

References 13 publications

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“…We use the following lemma to establish the upper bound of binomial term α m : Lemma 2. [30] Let Γ(x) be the gamma function and let R(x, y) = Γ(x + y)/Γ(x), then…”

Section: Proof Of Theoremmentioning

confidence: 99%

Computationally Efficient Approximations for Matrix-based Renyi's Entropy

Gong¹,

Yan²,

Yu³

2021

Preprint

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The recently developed matrix-based R ényi's α-order entropy enables measurement of information in data simply using the eigenspectrum of symmetric positive semi-definite (PSD) matrices in reproducing kernel Hilbert space, without estimation of the underlying data distribution. This intriguing property makes the new information measurement widely adopted in multiple statistical inference and learning tasks. However, the computation of such quantity involves the trace operator on a PSD matrix G to power α (i.e., tr(G α )), with a normal complexity of nearly O(n 3 ), which severely hampers its practical usage when the number of samples (i.e., n) is large. In this work, we present computationally efficient approximations to this new entropy functional that can reduce its complexity to even significantly less than O(n 2 ). To this end, we first develop randomized approximations to tr(G α ) that transform the trace estimation into matrix-vector multiplications problem. We extend such strategy for arbitrary values of α (integer or non-integer). We then establish the connection between the matrix-based Renyi's entropy and PSD matrix approximation, which enables us to exploit both clustering and block low-rank structure of G to further reduce the computational cost. We theoretically provide approximation accuracy guarantees and illustrate the properties of different approximations. Large-scale experimental evaluations on both synthetic and real-world data corroborate our theoretical findings, showing promising speedup with negligible loss in accuracy.

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“…We use the following lemma to establish the upper bound of binomial term α m : Lemma 2. [30] Let Γ(x) be the gamma function and let R(x, y) = Γ(x + y)/Γ(x), then…”

Section: Proof Of Theoremmentioning

confidence: 99%

Computationally Efficient Approximations for Matrix-based Renyi's Entropy

Gong¹,

Yan²,

Yu³

2021

Preprint

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Optimal Randomized Approximations for Matrix-Based Rényi’s Entropy

Dong¹,

Gong²,

et al. 2023

IEEE Trans. Inform. Theory

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Computationally Efficient Approximations for Matrix-Based Rényi's Entropy

Gong

Dong

et al. 2022

IEEE Trans. Signal Process.

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Inequalities for q-gamma function ratios

Cited by 4 publications

References 13 publications

Computationally Efficient Approximations for Matrix-based Renyi's Entropy

Computationally Efficient Approximations for Matrix-based Renyi's Entropy

Optimal Randomized Approximations for Matrix-Based Rényi’s Entropy

Computationally Efficient Approximations for Matrix-Based Rényi's Entropy

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