We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. First, we show that the heights given by the maximum likelihood estimate are generically transcendental. For a cell in one dimension, the maximum likelihood estimator is expressed in closed form using the generalized W-Lambert function. Even more, we show that finding the log-concave maximum likelihood estimate is equivalent to solving a collection of polynomial-exponential systems of a special form. Even in the case of two equations, very little is known about solutions to these systems. As an alternative, we use Smale's alpha-theory to refine approximate numerical solutions and to certify solutions to log-concave density estimation.
We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP$_2$) distributions and log-$L^\#$-concave (LLC) distributions. In both cases we also assume log-concavity in order to ensure boundedness of the likelihood function. Given $n$ independent and identically distributed random vectors in $\mathbb R^d$ from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when $n\geq 3$. This holds independently of the ambient…
We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $\mathbb{R}^d$. Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much stronger types of convergence -- in particular, it implies convergence in Hellinger distance and even in certain exponentially weighted total variation norms. In our main result, we prove the existence…
We study the smoothed log-concave maximum likelihood estimator of a probability distribution on $\mathbb{R}^d$. This is a fully automatic nonparametric density estimator, obtained as a canonical smoothing of the log-concave maximum likelihood estimator. We demonstrate its attractive features both through an analysis of its theoretical properties and a simulation study. Moreover, we use our methodology to develop a new test of log-concavity, and show how the estimator can be used as an intermediate stage of more involved procedures, such as constructing a classifier or estimating a functional o…
The log-concave maximum likelihood estimator (MLE) problem answers: for a set of points $X_1,...X_n \in \mathbb R^d$, which log-concave density maximizes their likelihood? We present a characterization of the log-concave MLE that leads to an algorithm with runtime $poly(n,d, \frac 1 ε,r)$ to compute a log-concave distribution whose log-likelihood is at most $ε$ less than that of the MLE, and $r$ is parameter of the problem that is bounded by the $\ell_2$ norm of the vector of log-likelihoods the MLE evaluated at $X_1,...,X_n$.
We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner [2018]. Here the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-\log f$ at the mode), the likelihood ratio statistic …
We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least $(\log(n)/n)^{1/3}$ and typically $(\log(n)/n)^{2/5}$, whereas the difference between the empirical and estimated distribution function vanishes with rate $o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions.
This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that includes conditional likelihood, quasi-likelihood, and pseudo-likelihood. The theory allows all feasible combinations of the spline degree, the penalty order, and the smoothness of the unknown functions. According to this theory, the asymptotic behaviors of the penalized spline estimators depends on interplay between the spline knot number and the penalty pa…
The Ohio State University（俄亥俄州立大学）；University of Southern California（南加州大学）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\mathbb{R}^d$, for all $d \geq 4$. Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions. In more detail, we prove that for any $d \geq 1$ and $ε>0$, given $\tilde{O}_d((1/ε)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\mathbb{R}^d$, the MLE outputs a hypothesis $h$ that with high p…
In this paper, we study two problems: (1) estimation of a $d$-dimensional log-concave distribution and (2) bounded multivariate convex regression with random design with an underlying log-concave density or a compactly supported distribution with a continuous density. First, we show that for all $d \ge 4$ the maximum likelihood estimators of both problems achieve an optimal risk of $Θ_d(n^{-2/(d+1)})$ (up to a logarithmic factor) in terms of squared Hellinger distance and $L_2$ squared distance, respectively. Previously, the optimality of both these estimators was known only for $d\le 3$. We a…
We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over three subclasses of log-concave densities. The first consists of densities with polyhedral support whose logarithms are piecewise affine. The complexity of such densities~$f$ can be measured in terms of the sum $Γ(f)$ of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by $f$. Given $n$ independent observations from a $d$-dimensional log-concave density with $d \in \{2,3\}$, we prove a sharp oracle inequality, which in particular implies that the Kull…
We propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function, based on the shape restriction of concavity (alternatively, convexity). Dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis, that is, studying a concave least-squares estimator satisfying a further equality constraint. We study this null hypothesis least-squares estimator (NLSE) here, and use it to study our likelihood ratio statistic. The NLSE is the solution to a convex program, and …
We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.
We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-2/5}$ when $-1 < s < \infty$ where $s=0$ corresponds to the log-concave case. We also show that the MLE does not exist for the classes of $s$-concave densities with $s < - 1$.
Phylogenetic trees are key data objects in biology, and the method of phylogenetic reconstruction has been highly developed. The space of phylogenetic trees is a nonpositively curved metric space. Recently, statistical methods to analyze the set of trees on this space are being developed utilizing this property. Meanwhile, in Euclidean space, the log-concave maximum likelihood method has emerged as a new nonparametric method for probability density estimation. In this paper, we derive a sufficient condition for the existence and uniqueness of the log-concave maximum likelihood estimator on tre…
We study semiparametric time series models with innovations following a log-concave distribution. We propose a general maximum likelihood framework which allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well-known models, including ARMA, GARCH and ARMA-GARCH. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA-GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log-return of FTSE…
Motivated by studies in biological sciences to detect differentially expressed genes, a semiparametric two-component mixture model with one known component is being studied in this paper. Assuming the density of the unknown component to be log-concave, which contains a very broad family of densities, we develop a semiparametric maximum likelihood estimator and propose an EM algorithm to compute it. Our new estimation method finds the mixing proportions and the distribution of the unknown component simultaneously. We establish the identifiability of the proposed semiparametric mixture model and…
Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we are able to reformulate the issue of computation in terms of a non-differentiable convex optimisation problem, and thus combine techniques of computational geometry with Shor's…
We study nonparametric maximum likelihood estimation of a log-concave density function $f_0$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_0$ is known, or (b) $f_0$ is known to be symmetric about a fixed point $m$. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE's), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE's pointwise limit distribution at $m$ (either the known mode or the known center of symmetry) and at a point $x_0 \ne m$. Softw…
We propose a unified framework for likelihood-based regression modeling when the response variable has finite support. Our work is motivated by the fact that, in practice, observed data are discrete and bounded. The proposed methods assume a model which includes models previously considered for interval-censored variables with log-concave distributions as special cases. The resulting log-likelihood is concave, which we use to establish asymptotic normality of its maximizer as the number of observations $n$ tends to infinity with the number of parameters $d$ fixed, and rates of convergence of $…
The most widely used technology to identify the proteins present in a complex biological sample is tandem mass spectrometry, which quickly produces a large collection of spectra representative of the peptides (i.e., protein subsequences) present in the original sample. In this work, we greatly expand the parameter learning capabilities of a dynamic Bayesian network (DBN) peptide-scoring algorithm, Didea, by deriving emission distributions for which its conditional log-likelihood scoring function remains concave. We show that this class of emission distributions, called Convex Virtual Emissions…
In this paper, we study the approximation and estimation of $s$-concave densities via Rényi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave densities exists and is unique via the procedure of minimizing a divergence functional proposed by Koenker and Mizera (2010) if and only if $Q$ admits full-dimensional support and a first moment. We also show continuity of the divergence functional in $Q$: if $Q_n \to Q$ in the Wasserstein metric, then the projected densities converge in weighted $L_1$ metrics and uniformly on closed subsets of the continuity…
The estimation of a log-concave density on $\mathbb{R}^d$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $n$ can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order $n^{-4/5}$, when $d=1$, and order $n^{-2/(d+1)}$ when $d \geq 2$. In particular, this reveals a sense in wh…
The estimation of a log-concave density on $\mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We also present two computationally…
The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, e.g., squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is $k$-affine (i.e.\ made up of $k$ affine pieces), provided $k$ is not too large. Our results use two different techniques: the first relies on a new Marshall's inequality for log-concave dens…
In recent years, log-concave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density estimation, which require the choice of one or more bandwidths. The purpose of this article is to describe some of the properties of the class of log-concave densities on $\mathbb{R}^d$ which make it so attractive from a statistical perspective, and to outline the latest methodological, theoretical and computational advances in the area.
We present a new approach for inference about a log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf $F$. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for $F$ which we …
Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.
We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if and only if $P$ has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on $P$ with respect to Mallows distance $D_1(\cdot,\cdot)$. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove e…
In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov…
Nonparametric statistics for distribution functions F or densities f=F' under qualitative shape constraints provides an interesting alternative to classical parametric or entirely nonparametric approaches. We contribute to this area by considering a new shape constraint: F is said to be bi-log-concave, if both log(F) and log(1 - F) are concave. Many commonly considered distributions are compatible with this constraint. For instance, any c.d.f. F with log-concave density f = F' is bi-log-concave. But in contrast to the latter constraint, bi-log-concavity allows for multimodal densities. We prov…
Log-concave distributions are an attractive choice for modeling and inference, for several reasons: The class of log-concave distributions contains most of the commonly used parametric distributions and thus is a rich and flexible nonparametric class of distributions. Further, the MLE exists and can be computed with readily available algorithms. Thus, no tuning parameter, such as a bandwidth, is necessary for estimation. Due to these attractive properties, there has been considerable recent research activity concerning the theory and applications of log-concave distributions. This article give…
In this article, we revisit the problem of fitting a mixture model under the assumption that the mixture components are symmetric and log-concave. To this end, we first study the nonparametric maximum likelihood estimation (NPMLE) of a monotone log-concave probability density. By following the arguments of Rufibach (2006), we show that the NPMLE is uniformly consistent with respect to the supremum norm on compact subsets of the interior of the support. To fit the mixture model, we propose a semiparametric EM (SEM) algorithm, which can be adapted to other semiparametric mixture models. In our n…
A novel computational approach to log-concave density estimation is proposed. Previous approaches utilize the piecewise-affine parametrization of the density induced by the given sample set. The number of parameters as well as non-smooth subgradient-based convex optimization for determining the maximum likelihood density estimate cause long runtimes for dimensions $d \geq 2$ and large sample sets. The presented approach is based on mildly non-convex smooth approximations of the objective function and \textit{sparse}, adaptive piecewise-affine density parametrization. Established memory-efficie…
Penalized likelihood methods are fundamental to ultra-high dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models, such methods possess model selection consistency with oracle properties even for dimensionality of Non-Polynomial (NP) order of sample size, for a class of penalized likelihood approaches using folded-concave penalty functions, which were introduced to ameliorate the bias problems of convex penalty functions. This fills a long-standing gap in the literature whe…
The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. Prior work by D{ü}mbgen et al. (2011) establishes that, with suitable metrics on the underlying spaces, this projection is continuous, but not uniformly continuous. In this work we prove a local uniform continuity result for log-concave projection -- in particular, establishing that this map is locally H{ö}lder-(1/4) continuous. A matching lower bound verifies that this exponent cannot be improved. We also examine the implications of this continuity result for the empiric…
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measu…
We show that the density of $Z=\mathop {\operatorname {argmax}}\{W(t)-t^2\}$, sometimes known as Chernoff's density, is log-concave. We conjecture that Chernoff's density is strongly log-concave or "super-Gaussian", and provide evidence in support of the conjecture.
We establish the first tight lower bound of $Ω(\log\logκ)$ on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number $κ$ in one dimension. Whereas existing guarantees for MCMC-based algorithms scale polynomially in $κ$, we introduce a novel algorithm based on rejection sampling that closes this doubly exponential gap.
We study functional inequalities (Poincaré, Cheeger, log-Sobolev) for probability measures obtained as perturbations. Several explicit results for general measures as well as log-concave distributions are given.The initial goal of this work was to obtain explicit bounds on the constants in view of statistical applications for instance. These results are then applied to the Langevin Monte-Carlo method used in statistics in order to compute Bayesian estimators.
Log-concave distributions include some important distributions such as normal distribution, exponential distribution and so on. In this note, we show inequalities between two Lp-norms for log-concave distributions on the Euclidean space. These inequalities are the generalizations of the upper and lower bound of the differential entropy and are also interpreted as a kind of expansion of the inequality between two Lp-norms on the measurable set with finite measure.
This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the…
A canonical noise distribution (CND) is an additive mechanism designed to satisfy $f$-differential privacy ($f$-DP), without any wasted privacy budget. $f$-DP is a hypothesis testing-based formulation of privacy phrased in terms of tradeoff functions, which captures the difficulty of a hypothesis test. In this paper, we consider the existence and construction of both log-concave CNDs and multivariate CNDs. Log-concave distributions are important to ensure that higher outputs of the mechanism correspond to higher input values, whereas multivariate noise distributions are important to ensure tha…
In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices $XX^*$ with isotropic log-concave $X$-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures \cite{chen, KL22}, we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. …
In this paper we show that the family P_d of probability distributions on R^d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. Hence the nonparametric model P_d has similar properties as parametric models such as, for instance, the family of all d-variate Gaussian distributions.
University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains). Given a set of samples drawn from an unknown target distribution, we want to compute a log-concave hypothesis distribution that is as close as possible to the target, in total variation distance. In this work, we give the first computationally efficient algorithm for this learning problem. Our algorithm achieves the information-theoretically optimal sample size (up to a constant factor), runs in polynomial time, and is robust to model misspecification with nearly-optimal er…
Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry, multinomial allocation models and multivariate hypergeometric sampling. The work generalizes and improves upon previous results in a number of directions.
University of Southern California（南加州大学）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d \geq 1$. Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of $d>3$. In more detail, we give an estimator that, for any $d \ge 1$ and $ε>0$, draws $\tilde{O}_d \left( (1/ε)^{(d+5)/2} \right)$ samples from an unknown target log-concave density on $\mathbb{R}^d$, and outputs a hypothesis that (with high probability) is $ε$-close to the ta…
