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Thomas A. Courtade

Publications and source records attributed to Thomas A. Courtade.

At least 37 records · Page 2Linked to original sources
arXiv2019

A Family of Bayesian Cramér-Rao Bounds, and Consequences for Log-Concave Priors

Under minimal regularity assumptions, we establish a family of information-theoretic Bayesian Cramér-Rao bounds, indexed by probability measures that satisfy a logarithmic Sobolev inequality. This family includes as a special case the known Bayesian Cramér-Rao bound (or van Trees inequality), and its less widely known entropic improvement due to Efroimovich. For the setting of a log-concave prior, we obtain a Bayesian Cramér-Rao bound which holds for any (possibly biased) estimator and, unlike the van Trees inequality, does not depend on the Fisher information of the prior.

cs.IT↗
arXiv2019

Transportation Proof of an inequality by Anantharam, Jog and Nair

Anantharam, Jog and Nair recently put forth an entropic inequality which simultaneously generalizes the Shannon-Stam entropy power inequality and the Brascamp-Lieb inequality in entropic form. We give a brief proof of their result based on optimal transport.

cs.IT↗
arXiv2018

Stability of the Bakry-Emery theorem on $\mathbb{R}^n$

We prove stability estimates for the Bakry-Emery bound on Poincaré and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting that if a $1$-uniformly log-concave measure has almost the same Poincaré constant as the standard Gaussian measure, then it almost splits off a Gaussian factor, and establish similar new results for logarithmic Sobolev inequalities. As a consequence, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are based on Stein's method, optimal transport, and an approximate integration by parts identity relating measures and approximate optimizers in the associated functional inequality.

math.FA↗
arXiv2018

Bounds on the Poincaré constant for convolution measures

We establish a Shearer-type inequality for the Poincaré constant, showing that the Poincaré constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincaré constants corresponding to convolutions of subsets of measures. This implies, for example, that the Poincaré constant is non-increasing along the central limit theorem. We also establish a dimension-free stability estimate for subadditivity of the Poincaré constant on convolutions which uniformly improves an earlier one-dimensional estimate of a similar nature by Johnson (2004). As a byproduct of our arguments, we find that the monotone properties of entropy, Fisher information and the Poincaré constant along the CLT find a common root in Shearer's inequality.

math.PR↗
arXiv2018

Existence of Stein Kernels under a Spectral Gap, and Discrepancy Bound

We establish existence of Stein kernels for probability measures on $\mathbb{R}^d$ satisfying a Poincaré inequality, and obtain bounds on the Stein discrepancy of such measures. Applications to quantitative central limit theorems are discussed, including a new CLT in Wasserstein distance $W_2$ with optimal rate and dependence on the dimension. As a byproduct, we obtain a stability version of an estimate of the Poincaré constant of probability measures under a second moment constraint. The results extend more generally to the setting of converse weighted Poincaré inequalities. The proof is based on simple arguments of calculus of variations. Further, we establish two general properties enjoyed by the Stein discrepancy, holding whenever a Stein kernel exists: Stein discrepancy is strictly decreasing along the CLT, and it controls the skewness of a random vector.

math.PR↗
arXiv2017

Information-Theoretic Perspectives on Brascamp-Lieb Inequality and Its Reverse

We introduce an inequality which may be viewed as a generalization of both the Brascamp-Lieb inequality and its reverse (Barthe's inequality), and prove its information-theoretic (i.e.\ entropic) formulation. This result leads to a unified approach to functional inequalities such as the variational formula of Rényi entropy, hypercontractivity and its reverse, strong data processing inequalities, and transportation-cost inequalities, whose utility in the proofs of various coding theorems has gained growing popularity recently. We show that our information-theoretic setting is convenient for proving properties such as data processing, tensorization, convexity (Riesz-Thorin interpolation) and Gaussian optimality. In particular, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove several results on Gaussian optimality. Several applications are discussed, including a generalization of the Brascamp-Lieb inequality involving Gaussian random transformations, the determination of Wyner's common information of vector Gaussian sources, and the achievable rate region of certain key generation problems in the case of vector Gaussian sources.

cs.IT↗
arXiv2017

Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons

Pairwise comparison data arises in many domains, including tournament rankings, web search, and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutation-based models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. The rates achieved by these estimators are shown to be optimal for a large class of graphs. Our results are corroborated by simulations on multiple comparison topologies.

cs.LG↗
arXiv2017

Denoising Linear Models with Permuted Data

The multivariate linear regression model with shuffled data and additive Gaussian noise arises in various correspondence estimation and matching problems. Focusing on the denoising aspect of this problem, we provide a characterization the minimax error rate that is sharp up to logarithmic factors. We also analyze the performance of two versions of a computationally efficient estimator, and establish their consistency for a large range of input parameters. Finally, we provide an exact algorithm for the noiseless problem and demonstrate its performance on an image point-cloud matching task. Our analysis also extends to datasets with outliers.

stat.ML↗
arXiv2017

A Counterexample to the Vector Generalization of Costa's EPI, and Partial Resolution

We give a counterexample to the vector generalization of Costa's entropy power inequality (EPI) due to Liu, Liu, Poor and Shamai. In particular, the claimed inequality can fail if the matix-valued parameter in the convex combination does not commute with the covariance of the additive Gaussian noise. Conversely, the inequality holds if these two matrices commute.

cs.IT↗
arXiv2016

Entropy Jumps for Radially Symmetric Random Vectors

We establish a quantitative bound on the entropy jump associated to the sum of independent, identically distributed (IID) radially symmetric random vectors having dimension greater than one. Following the usual approach, we first consider the analogous problem of Fisher information dissipation, and then integrate along the Ornstein-Uhlenbeck semigroup to obtain an entropic inequality. In a departure from previous work, we appeal to a result by Desvillettes and Villani on entropy production associated to the Landau equation. This obviates strong regularity assumptions, such as presence of a spectral gap and log-concavity of densities, but comes at the expense of radial symmetry. As an application, we give a quantitative estimate of the deficit in the Gaussian logarithmic Sobolev inequality for radially symmetric functions.

cs.IT↗
arXiv2016

Novel probabilistic models of spatial genetic ancestry with applications to stratification correction in genome-wide association studies

Genetic variation in human populations is influenced by geographic ancestry due to spatial locality in historical mating and migration patterns. Spatial population structure in genetic datasets has been traditionally analyzed using either model-free algorithms, such as principal components analysis (PCA) and multidimensional scaling, or using explicit spatial probabilistic models of allele frequency evolution. We develop a general probabilistic model and an associated inference algorithm that unify the model-based and data-driven approaches to visualizing and inferring population structure. Our algorithm, Geographic Ancestry Positioning (GAP), relates local genetic distances between samples to their spatial distances, and can be used for visually discerning population structure as well as accurately inferring the spatial origin of individuals on a two-dimensional continuum. On both simulated and several real datasets from diverse human populations, GAP exhibits substantially lower error in reconstructing spatial ancestry coordinates compared to PCA. Our spatial inference algorithm can also be effectively applied to the problem of population stratification in genome-wide association studies (GWAS), where hidden population structure can create fictitious associations when population ancestry is correlated with both the genotype and the trait. We develop an association test that uses the ancestry coordinates inferred by GAP to accurately account for ancestry-induced correlations in GWAS. Based on simulations and analysis of a dataset of 10 metabolic traits measured in a Northern Finland cohort, which is known to exhibit significant population structure, we find that our method has superior power to current approaches.

q-bio.PE↗
arXiv2016

Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities

We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasserstein distance. Further, if one of the densities is log-concave and the other is Gaussian, then the deficit in the EPI can be controlled in terms of the $L^1$-Wasserstein distance. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Wasserstein distance when densities are uniformly log-concave on sets of measure arbitrarily close to one. Our stability results can be extended to non-log-concave densities, provided certain regularity conditions are met. The proofs are based on optimal transportation.

cs.IT↗
arXiv2016

Links between the Logarithmic Sobolev Inequality and the convolution inequalities for Entropy and Fisher Information

Relative to the Gaussian measure on $\mathbb{R}^d$, entropy and Fisher information are famously related via Gross' logarithmic Sobolev inequality (LSI). These same functionals also separately satisfy convolution inequalities, as proved by Stam. We establish a dimension-free inequality that interpolates among these relations. Several interesting corollaries follow: (i) the deficit in the LSI satisfies a convolution inequality itself; (ii) the deficit in the LSI controls convergence in the entropic and Fisher information central limit theorems; and (iii) the LSI is stable with respect to HWI jumps (i.e., a jump in any of the convolution inequalities associated to the HWI functionals). Another consequence is that the convolution inequalities for Fisher information and entropy powers are reversible in general, up to a factor depending on the Stam defect. An improved form of Nelson's hypercontractivity estimate also follows. Finally, we speculate on the possibility of an analogous reverse Brunn-Minkowski inequality and a related upper bound on surface area associated to Minkowski sums.

cs.IT↗
arXiv2016

Linear Regression with an Unknown Permutation: Statistical and Computational Limits

Consider a noisy linear observation model with an unknown permutation, based on observing $y = Π^* A x^* + w$, where $x^* \in \mathbb{R}^d$ is an unknown vector, $Π^*$ is an unknown $n \times n$ permutation matrix, and $w \in \mathbb{R}^n$ is additive Gaussian noise. We analyze the problem of permutation recovery in a random design setting in which the entries of the matrix $A$ are drawn i.i.d. from a standard Gaussian distribution, and establish sharp conditions on the SNR, sample size $n$, and dimension $d$ under which $Π^*$ is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of $Π^*$ is NP-hard to compute, while also providing a polynomial time algorithm when $d =1$.

math.ST↗
arXiv2016

Brascamp-Lieb Inequality and Its Reverse: An Information Theoretic View

We generalize a result by Carlen and Cordero-Erausquin on the equivalence between the Brascamp-Lieb inequality and the subadditivity of relative entropy by allowing for random transformations (a broadcast channel). This leads to a unified perspective on several functional inequalities that have been gaining popularity in the context of proving impossibility results. We demonstrate that the information theoretic dual of the Brascamp-Lieb inequality is a convenient setting for proving properties such as data processing, tensorization, convexity and Gaussian optimality. Consequences of the latter include an extension of the Brascamp-Lieb inequality allowing for Gaussian random transformations, the determination of the multivariate Wyner common information for Gaussian sources, and a multivariate version of Nelson's hypercontractivity theorem. Finally we present an information theoretic characterization of a reverse Brascamp-Lieb inequality involving a random transformation (a multiple access channel).

cs.IT↗
arXiv2016

Partial DNA Assembly: A Rate-Distortion Perspective

Earlier formulations of the DNA assembly problem were all in the context of perfect assembly; i.e., given a set of reads from a long genome sequence, is it possible to perfectly reconstruct the original sequence? In practice, however, it is very often the case that the read data is not sufficiently rich to permit unambiguous reconstruction of the original sequence. While a natural generalization of the perfect assembly formulation to these cases would be to consider a rate-distortion framework, partial assemblies are usually represented in terms of an assembly graph, making the definition of a distortion measure challenging. In this work, we introduce a distortion function for assembly graphs that can be understood as the logarithm of the number of Eulerian cycles in the assembly graph, each of which correspond to a candidate assembly that could have generated the observed reads. We also introduce an algorithm for the construction of an assembly graph and analyze its performance on real genomes.

cs.IT↗
arXiv2016

Strengthening the Entropy Power Inequality

We tighten the Entropy Power Inequality (EPI) when one of the random summands is Gaussian. Our strengthening is closely connected to the concept of strong data processing for Gaussian channels and generalizes the (vector extension of) Costa's EPI. This leads to a new reverse entropy power inequality and, as a corollary, sharpens Stam's inequality relating entropy power and Fisher information. Applications to network information theory are given, including a short self-contained proof of the rate region for the two-encoder quadratic Gaussian source coding problem. Our argument is based on weak convergence and a technique employed by Geng and Nair for establishing Gaussian optimality via rotational-invariance, which traces its roots to a `doubling trick' that has been successfully used in the study of functional inequalities.

cs.IT↗