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Modality-Constrained Density Estimation via Deformable Templates

Dasgupta, Sutanoy; Pati, Debdeep; Jermyn, Ian H.; Srivastava, Anuj

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Sutanoy Dasgupta

Debdeep Pati

Anuj Srivastava


Estimation of a probability density function (pdf) from its samples, while satisfying certain shape constraints, is an important problem that lacks coverage in the literature. This article introduces a novel geometric, deformable template constrained density estimator (dtcode) for estimating pdfs constrained to have a given number of modes. Our approach explores the space of thus-constrained pdfs using the set of shape-preserving transformations: an arbitrary template from the given shape class is transformed via a shape-preserving transformation to obtain the final optimal estimate. The search for this optimal transformation, under the maximum-likelihood criterion, is performed by mapping transformations to the tangent space of a Hilbert sphere, where they are effectively linearized, and can be expressed using an orthogonal basis. This framework is first applied to (univariate) unconditional densities and then extended to conditional densities. We provide asymptotic convergence rates for dtcode, and an application of the framework to the speed distributions for different traffic flows on Californian highways.


Dasgupta, S., Pati, D., Jermyn, I. H., & Srivastava, A. (2021). Modality-Constrained Density Estimation via Deformable Templates. Technometrics, 63(4), 536-547.

Journal Article Type Article
Acceptance Date Nov 23, 2020
Online Publication Date Feb 1, 2021
Publication Date 2021
Deposit Date Feb 9, 2021
Publicly Available Date Feb 1, 2022
Journal Technometrics
Print ISSN 0040-1706
Electronic ISSN 1537-2723
Publisher American Statistical Association
Peer Reviewed Peer Reviewed
Volume 63
Issue 4
Pages 536-547


Accepted Journal Article (560 Kb)

Publisher Licence URL

Copyright Statement
This is an Accepted Manuscript version of the following article, accepted for publication in Technometrics. Dasgupta, Sutanoy, Pati, Debdeep, Jermyn, Ian H. & Srivastava, Anuj (2021). Modality-Constrained Density Estimation via Deformable Templates. Technometrics 63(4): 536-547. It is deposited under the terms of the Creative Commons Attribution-NonCommercial License (, which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

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