Celebratio Mathematica

T. Gao, S. Z. Kovalsky, and I. Daubechies: “Gaussian process landmarking on manifolds,” SIAM J. Math. Data Sci. 1 : 1 (2019), pp. 208–236. MR 3949707 ArXiv 1802.03479 article

As a means of improving analysis of biological shapes, we propose an algorithm for sampling a Riemannian manifold by sequentially selecting points with maximum uncertainty under a Gaussian process model. This greedy strategy is known to be near-optimal in the experimental design literature, and appears to outperform the use of user-placed landmarks in representing the geometry of biological objects in our application. In the noiseless regime, we establish an upper bound for the mean squared prediction error (MSPE) in terms of the number of samples and geometric quantities of the manifold, demonstrating that the MSPE for our proposed sequential design decays at a rate comparable to the oracle rate achievable by any sequential or non-sequential optimal design; to our knowledge this is the first result of this type for sequential experimental design. The key is to link the greedy algorithm to reduced basis methods in the context of model reduction for partial differential equations. We expect this approach will find additional applications in other fields of research.

Tingran Gao	Related
Shahar Z. Kovalsky	Related

T. Gao, S. Z. Kovalsky, D. M. Boyer, and I. Daubechies: “Gaussian process landmarking for three-dimensional geometric morphometrics,” Siam J. Math. Data Sci. 1 : 1 (2019), pp. 237–267. MR 3949708 ArXiv 1807.11887 article

We demonstrate applications of the Gaussian process-based landmarking algorithm proposed in [T. Gao, S. Z. Kovalsky, and I. Daubechies, SIAM J. Math. Data Sci., 1 (2019), pp.208–236] to geometric morphometrics, a branch of evolutionary biology centered at the analysis and comparisons of anatomical shapes, and compares the automatically sampled landmarks with the “ground truth” landmarks manually placed by evolutionary anthropologists; the results suggest that Gaussian process landmarks perform equally well or better, in terms of both spatial coverage and downstream statistical analysis. We provide a detailed exposition of numerical procedures and feature filtering algorithms for computing high-quality and semantically meaningful diffeomorphisms between disk-type anatomical surfaces.

Tingran Gao	Related
Shahar Z. Kovalsky	Related
Douglas Martin Boyer	Related

S. Shan, S. Z. Kovalsky, J. M. Winchester, D. M. Boyer, and I. Daubechies: “ariaDNE: A robustly implemented algorithm for Dirichlet energy of the normal,” Methods. Ecol. Evol. 10 : 4 (April 2019), pp. 541–552. ArXiv 1901.06318 article

Shape characterizers are metrics that quantify aspects of the overall geometry of a 3D digital surface. When computed for biological objects, the values of a shape characterizer are largely independent of homology interpretations and often contain a strong ecological and functional signal. Thus shape characterizers are useful for understanding evolutionary processes. Dirichlet Normal Energy (DNE) is a widely used shape characterizer in morphological studies.
Recent studies found that DNE is sensitive to various procedures for preparing 3D mesh from raw scan data, raising concerns regarding comparability and objectivity when utilizing DNE in morphological research. We provide a robustly implemented algorithm for computing the Dirichlet energy of the normal (ariaDNE) on 3D meshes.
We show through simulation that the effects of preparation-related mesh surface attributes such as triangle count, mesh representation, noise, smoothing and boundary triangles are much more limited on ariaDNE than DNE. Furthermore, ariaDNE retains the potential of DNE for biological studies, illustrated by its effectiveness in differentiating species by dietary preferences.
Use of ariaDNE can dramatically enhance assessment of ecological aspects of morphological variation by its stability under different 3D model acquisition methods and preparation procedure. Towards this goal, we provide scripts for computing ariaDNE and ariaDNE values for specimens used in previously published DNE analyses.

Julie M. Winchester	Related
Shan Shan	Related
Shahar Z. Kovalsky	Related
Douglas Martin Boyer	Related

S. Z. Kovalsky, N. Aigerman, I. Daubechies, M. Kazhdan, J. Lu, and S. Steinerberger: Non-convex planar harmonic maps. Preprint, January 2020. ArXiv 2001.01322 techreport

We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Radó–Kneser–Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte’s embedding theorem for planar graphs–RKC’s discrete counterpart–which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.

Jianfeng Lu	Related
Michael Kazhdan	Related
Shahar Z. Kovalsky	Related
Stefan Steinerberger	Related
Noam Aigerman	Related

year	title	people

			clear

Ingrid Daubechies

Complete Bibliography

Works connected to Shahar Z. Kovalsky

Filter the Bibliography List