arXiv research

Paper presents a robust transfer learning method for active level set estimation.

Estimating the level set of a signal from measurements is a task that arises in a variety of fields, including medical imaging, astronomy, and digital elevation mapping. Motivated by scenarios where accurate and complete measurements of the signal may not available, we examine here a simple procedure for estimating the…

BDMBC clusters data with varying densities using a new PLLS measure.

The level set tree approach of Hartigan (1975) provides a probabilistically based and highly interpretable encoding of the clustering behavior of a dataset. By representing the hierarchy of data modes as a dendrogram of the level sets of a density estimator, this approach offers many advantages for exploratory analysis…

A new method optimizes spatial sampling for level set estimation in one dimension.

We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set o…

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumblin…

This paper introduces a more efficient method for estimating level sets with a stopping criterion.

In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.

In this paper we establish a uniform $C^{2,θ}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $n\leq 10$ (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the $C^{2,θ}$ estimate …

We study the connections between spectral clustering and the problems of maximum margin clustering, and estimation of the components of level sets of a density function. Specifically, we obtain bounds on the eigenvectors of graph Laplacian matrices in terms of the between cluster separation, and within cluster connecti…

In this paper, the problem of estimating the level set of a black-box function from noisy and expensive evaluation queries is considered. A new algorithm for this problem in the Bayesian framework with a Gaussian Process (GP) prior is proposed. The proposed algorithm employs a hierarchical sequence of partitions to exp…

New algorithms estimate function levels with near-optimal efficiency.

The clusters of a distribution are often defined by the connected components of a density level set. However, this definition depends on the user-specified level. We address this issue by proposing a simple, generic algorithm, which uses an almost arbitrary level set estimator to estimate the smallest level at which th…

Bayesian Neural Networks improve high-dimensional level set estimation.

A novel dose-finding design for cancer clinical trials using level set estimation.

Forecasts of multivariate probability distributions are required for a variety of applications. Scoring rules enable the evaluation of forecast accuracy, and comparison between forecasting methods. We propose a theoretical framework for scoring rules for multivariate distributions, which encompasses the existing quadra…

Adaptive batching improves Gaussian process surrogates for noisy level set estimation.

We show that DBSCAN can estimate the connected components of the $λ$-density level set $\{ x : f(x) \ge λ\}$ given $n$ i.i.d. samples from an unknown density $f$. We characterize the regularity of the level set boundaries using parameter $β> 0$ and analyze the estimation error under the Hausdorff metric. When the data …

We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from circular interfaces immersed in uniform grids of various resolutions. These multilayer …

We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus is on strongly stochastic samplers, in particular with heavy-tailed simulation noise and low signal-to-no…

Sharp gradient estimate for scalar curvature on 3-manifolds.

SLS optimizes minimum-volume regions for conditional quantiles, bypassing density estimation.

New acquisition functions improve Bernoulli LSE.

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

New method improves level set estimation with theoretical guarantees.

High density clusters can be characterized by the connected components of a level set $L(λ) = \{x:\ p(x)>λ\}$ of the underlying probability density function $p$ generating the data, at some appropriate level $λ\geq 0$. The complete hierarchical clustering can be characterized by a cluster tree ${\cal T}= \bigcup_λ L(λ)…

The paper tackles optimal level set estimation in crowdsourcing and tournaments.

New method improves transductive learning predictions with multiplicative oracle inequalities.

Minimal graph level sets are concave if boundary is concave.

In this paper we prove that if $γ$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $γ$ in $\mathcal{C}^0$ as $t\to 0^+$. Another perspective is that the level-set flow of $γ$ is smooth. This is a generalization of the author's previous work where t…

Sharp estimate for 2-systole on Kähler surfaces with positive scalar curvature.

In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in the plane. If $K$ is locally-connected, connected and compact, then the level set…

Study functional confounders in causal inference, enabling estimable effects.

Unified framework for active learning problems using information theory.

Study on transnormal functions and their level sets on Finsler manifolds.

Following Hartigan, a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two step…

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper a…

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …

Find conditions for starshapedness of level sets in Heisenberg group.

Proves a function's locally least gradient property if its level sets are minimal laminations.

Proves convexity of level sets of general inverse σ_k equations.

Proves smoothness of conical singularities in mean curvature flow.

Paper studies generic dynamics of MCFs with spherical singularities.

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and f…

New method for analyzing elliptic and parabolic equations.

Proves positive mass theorem for 3-manifolds with a boundary.

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they e…