Limit distribution theory for smooth $p$-Wasserstein distances

Goldfeld, Ziv; Kato, Koji; Nietert, Sloan; Rioux, Gabriel

doi:10.48550/arxiv.2203.00159

Cited by 4 publications

(12 citation statements)

References 43 publications

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“…The proof of this result follows that of Theorem 4 in [GG20] and [GKNR22] with only minor changes and is hence omitted. Note that when the limiting π is unique (e.g., when p > 1 and µ has a density), then extraction of a subsequence is not needed.…”

Section: Smooth Wasserstein Distance With Compactly Supported Kernelsmentioning

confidence: 86%

“…The differentiability result follows by adapting the Gaussian kernel case (cf. Lemma 3.3 of [GKNR22]). To prove weak convergence of the smoothed empirical process √ n(μ n − µ) * η σ in ℓ ∞ (B), we employ the CLT in L 2 (X σ ) and use a linear isometry from L 2 (X σ ) into ℓ ∞ (B).…”

Section: 31mentioning

confidence: 99%

“…Relations between W (γσ ) p and maximum mean discrepancies were studies in [ZCR21], and nonparametric mixture model estimation under W (γσ ) p was considered [HMS21], again demonstrating scalability of error bounds with dimension. The study of limit distributions for empirical W (γσ) p was initiated in [GGK20] for p = 1 in the one-sample case, extended to the two-sample setting in [SGK21], and generalized to arbitrary p > 1 via a non-trivial application of the functional delta method in [GKNR22]. These works also considered bootstrap consistency and applications to minimum distance estimation and homogeneity testing.…”

Section: Smooth Wasserstein Distance With Compactly Supported Kernelsmentioning

confidence: 99%

“…The functional delta method was also used in [SM18,TSM19] to derive limit distributions for OT between discrete population distributions by parametrizing them using simplex vectors. Another recent application can be found in [GKNR22], where this approach was leveraged for Gaussian-smoothed W p by embedding the domain of the Wasserstein distance into a certain dual Sobolev space.…”

Section: Introductionmentioning

confidence: 99%

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Statistical inference with regularized optimal transport

Goldfeld¹,

Kato²,

Rioux³

et al. 2022

Preprint

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Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing, and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator, and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of: (i) average-and max-sliced p-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples.

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Section: Smooth Wasserstein Distance With Compactly Supported Kernelsmentioning

confidence: 86%

Section: 31mentioning

confidence: 99%

Section: Smooth Wasserstein Distance With Compactly Supported Kernelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Statistical inference with regularized optimal transport

Goldfeld¹,

Kato²,

Rioux³

et al. 2022

Preprint

Self Cite

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show abstract

“…Let U ∼ P U and P Y |X be a given transition kernel. Throughout this proof we employ the tools of Gaussian smoothing developed in [83] (see also [84][85][86][87][88]). To this end, we denote the isotropic d xdimensional Gaussian distribution with N σ := N (0, σ 2 I dx ) with the corresponding PDF ϕ σ .…”

Section: Proof Of Theoremmentioning

confidence: 99%

Neural Estimation and Optimization of Directed Information over Continuous Spaces

Tsur¹,

Aharoni²,

Goldfeld³

et al. 2022

Preprint

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This work develops a new method for estimating and optimizing the directed information rate between two jointly stationary and ergodic stochastic processes. Building upon recent advances in machine learning, we propose a recurrent neural network (RNN)-based estimator which is optimized via gradient ascent over the RNN parameters. The estimator does not require prior knowledge of the underlying joint and marginal distributions. The estimator is also readily optimized over continuous input processes realized by a deep generative model. We prove consistency of the proposed estimation and optimization methods and combine them to obtain end-to-end performance guarantees. Applications for channel capacity estimation of continuous channels with memory are explored, and empirical results demonstrating the scalability and accuracy of our method are provided. When the channel is memoryless, we investigate the mapping learned by the optimized input generator.

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Smoothing of Binary Codes, Uniform Distributions, and Applications

Pathegama,

Barg

2023

Entropy

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The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by the Rényi divergence of order α∈(1,∞]. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner’s transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed–Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel.

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Limit distribution theory for smooth $p$-Wasserstein distances

Cited by 4 publications

References 43 publications

Statistical inference with regularized optimal transport

Statistical inference with regularized optimal transport

Neural Estimation and Optimization of Directed Information over Continuous Spaces

Smoothing of Binary Codes, Uniform Distributions, and Applications

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