arXiv research

Log-density gradient estimation is a fundamental statistical problem and possesses various practical applications such as clustering and measuring non-Gaussianity. A naive two-step approach of first estimating the density and then taking its log-gradient is unreliable because an accurate density estimate does not neces…

Study online monotone density estimation with expert aggregation and log-optimal calibration.

Proposes log density gradient to improve reinforcement learning sample complexity.

Estimates log-concave densities in graphical models using tent functions.

Geodesic balls are isoperimetric in hyperbolic spaces with certain densities.

We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such…

We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.

Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attain low classification errors. The resulting classifier is linear in the log-transformed univariate and b…

Study minimax risk of score estimation for log-concave distributions.

Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.

Improves sequence modeling with a flow-based recurrent mixture density network.

A new sampling method using log-concave Markov chains.

The Riemannian Langevin Algorithm samples from manifolds efficiently.

New method for estimating diffusion model densities without solving flows.

New study shows low-degree polynomial algorithms struggle at clause densities close to Fix's.

Partition Tree estimates conditional densities for mixed continuous and categorical variables.

We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where $l$ is the relator length, going to infinity. (a) $1 + 1/C < \Cdim(…

We consider the problem of sampling from a strongly log-concave density in $\mathbb{R}^d$, and prove an information theoretic lower bound on the number of stochastic gradient queries of the log density needed. Several popular sampling algorithms (including many Markov chain Monte Carlo methods) operate by using stochas…

MALA mixes efficiently under smoothness and isoperimetry assumptions.

DPS uses PINNs to estimate drift in diffusion models for sampling.

A new sampling method, RC-LMC, reduces computational cost for high-dimensional log-concave distributions.

New algorithm samples neural network posteriors efficiently.

The COS method for European options pricing is improved with a new bound for the number of terms.

Mean shift clustering finds the modes of the data probability density by identifying the zero points of the density gradient. Since it does not require to fix the number of clusters in advance, the mean shift has been a popular clustering algorithm in various application fields. A typical implementation of the mean shi…

Smart Bayes integrates generative and discriminative features for improved classification.

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $φ(x,y,z)=αx+βy+γy$, where $α,β,γ$ are real numbers not all zero. We prove that if a surface is $φ$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(α,β,γ)$ and the surface must…

The main results are two characterisations of log-concave densities in terms of the collection of lift zonoids corresponding to a peacock. These notions are recalled and connected to arbitrage-free asset pricing in financial mathematics.

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves conv…

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density…

GBHT uses gradient boosting for density estimation with theoretical guarantees.

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 complex…

Improved VI with Price's gradient estimator for target log-density.

Algorithm samples from composite log-concave distributions using gradient evaluations and restricted Gaussian oracles.

The paper connects complex normalizing flows to Kähler-Ricci flows using geometric and statistical perspectives.

BBVI converges nearly dimensionally independent for log-concave targets.

Modal regression is aimed at estimating the global mode (i.e., global maximum) of the conditional density function of the output variable given input variables, and has led to regression methods robust against heavy-tailed or skewed noises. The conditional mode is often estimated through maximization of the modal regre…

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a seri…

We consider the problem of sampling from a strongly log-concave density in $\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin dif…

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforci…

Survival MDN uses invertible functions to speed up survival analysis models.

The $\textit{volume density}$ of a hyperbolic link $K$ is defined to be the ratio of the hyperbolic volume of $K$ to the crossing number of $K$. We show that there are sequences of non-alternating links with volume density approaching $v_8$, where $v_8$ is the volume of the ideal hyperbolic octahedron. We show that the…

Many sequence-to-sequence generation tasks, including machine translation and text-to-speech, can be posed as estimating the density of the output y given the input x: p(y|x). Given this interpretation, it is natural to evaluate sequence-to-sequence models using conditional log-likelihood on a test set. However, the go…

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we assume to be strictly log-convex. For $N=1$ we show that the solution is sometime…

We analyze the data on personal income distribution from the Australian Bureau of Statistics. We compare fits of the data to the exponential, log-normal, and gamma distributions. The exponential function gives a good (albeit not perfect) description of 98% of the population in the lower part of the distribution. The lo…

Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, there…

Non-Gaussian component analysis (NGCA) is aimed at identifying a linear subspace such that the projected data follows a non-Gaussian distribution. In this paper, we propose a novel NGCA algorithm based on log-density gradient estimation. Unlike existing methods, the proposed NGCA algorithm identifies the linear subspac…

Pathfinder uses quasi-Newton optimization for variational inference.

Improves GANs by sampling from an energy-based model induced by discriminator scores.