Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data

Jung, Sungkyu

doi:10.1214/21-ejs1807

Cited by 3 publications

(5 citation statements)

References 39 publications

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“…The distribution of hitting points, X T , on the response sphere is given by the von Mises–Fisher distribution (Gatto, 2013; Hillen et al, 2017), which has probability density functionwhere (boldμfalse¯·trueX¯bold-italicT) is the dot product of the unit vectors boldμfalse¯ = μ/ ‖ μ ‖ and trueX¯T=XT/a and κ is again given by Equation 3. Jung (2021) showed that the marginal distribution of a von Mises–Fisher distribution projected onto a geodesic (a great circle) of the sphere is not a von Mises distribution but is sufficiently close to be indistinguishable from it numerically. So, while the marginal distributions of the spherical diffusion model are not exactly von Mises in form—unlike the circular diffusion model—they approximate it sufficiently well to be plausible alternative models for the distribution of errors in continuous-outcome tasks.…”

Section: A Spherical Generalization Of the Circular Diffusion Modelmentioning

confidence: 99%

Diffusion theory of the antipodal “shadow” mode in continuous-outcome, coherent-motion decisions.

Smith¹,

Corbett²,

Lilburn³

2023

Psychological Review

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Continuous-outcome decisions, in which responses are made on continuous scales, are increasingly used to study perception and memory for stimulus attributes like color, orientation, and motion. This interest has led to the development of models of continuous-outcome decision processes like the circular diffusion model that predict joint distributions of decision outcomes and response times (RTs). We use the circular diffusion model and a new spherical generalization of it to model performance in a continuous-outcome version of the random-dot motion task. The task is a benchmark test of decision models because it yields bimodal distributions of decision outcomes: In addition to a peak or mode in the true direction of motion, there is a secondary, antipodal, mode at 180° to the true direction. Models like the circular diffusion model, in which evidence is accumulated by a single process, are thought to be unable to predict bimodality. We compared diffusion models for the continuous motion task in which evidence is accumulated in either a two-dimensional (2D) or a three-dimensional (3D) representational space. We found that performance was well described by a spherical (3D) diffusion model in which the drift rate encodes perceived motion direction and strength and the points on the bounding sphere representing the decision criterion are projected onto a 2D circle to make a response. A model with an antipodal component of drift rate and drift-rate variability successfully predicted bimodal distributions of decision outcomes and the joint distributions of decision outcomes and RT for individual participants.

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Section: A Spherical Generalization Of the Circular Diffusion Modelmentioning

confidence: 99%

Diffusion theory of the antipodal “shadow” mode in continuous-outcome, coherent-motion decisions.

Smith¹,

Corbett²,

Lilburn³

2023

Psychological Review

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“…In the particular case in which we want to compute SSW 2 between a measure µ and the uniform measure on the sphere ν = Unif(S d−1 ), we can use the appealing fact that the projection of ν on the circle is uniform, i.e. P U # ν = Unif(S 1 ) (particular case of Theorem 3.1 in [55], see Appendix B.3). Hence, we can use the Proposition 1 to compute W 2 , which allows a very efficient implementation either by the closed-form (13) or approximation by rectangle method of ( 12).…”

Section: Methodsmentioning

confidence: 99%

“…Hence, to define a sliced-Wasserstein discrepancy on this manifold, we propose, analogously to the classical SW distance, to project measures on great circles. The most natural way to project points from S d−1 to a great circle C is to use the geodesic projection [40,55] defined as…”

Section: Definition Of Sw On the Spherementioning

confidence: 99%

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Spherical Sliced-Wasserstein

Bonet¹,

Berg²,

Courty³

et al. 2022

Preprint

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Many variants of the Wasserstein distance have been introduced to reduce its original computational burden. In particular the Sliced-Wasserstein distance (SW), which leverages one-dimensional projections for which a closed-form solution of the Wasserstein distance is available, has received a lot of interest. Yet, it is restricted to data living in Euclidean spaces, while the Wasserstein distance has been studied and used recently on manifolds. We focus more specifically on the sphere, for which we define a novel SW discrepancy, which we call spherical Sliced-Wasserstein, making a first step towards defining SW discrepancies on manifolds. Our construction is notably based on closed-form solutions of the Wasserstein distance on the circle, together with a new spherical Radon transform. Along with efficient algorithms and the corresponding implementations, we illustrate its properties in several machine learning use cases where spherical representations of data are at stake: density estimation on the sphere, variational inference or hyperspherical auto-encoders.Preprint. Under review.

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“…where μ • XT is the dot product of the unit vectors μ = µ/ µ and XT = X T /a and κ is again given by Equation 3. Jung (2021) showed that the marginal distribution of a von…”

Section: A Spherical Generalization Of the Circular Diffusion Modelmentioning

confidence: 99%

Diffusion Theory of the Antipodal "Shadow'" Mode in Continuous-Outcome, Coherent-Motion Decisions

Smith¹,

Corbett²,

Lilburn³

2022

Preprint

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Continuous-outcome decisions, in which responses are made on continuous scales, are increasingly used to study perception and memory for stimulus attributes like color, orientation, and motion. This interest has led to the development of models of continuous-outcome decision processes like the circular diﬀusion model that predict joint distributions of decision outcomes and response times (RT). We use the circular diﬀusion model and a new spherical generalization of it to model performance in a continuous-outcome versionof the random dot motion task. The task is a benchmark test of decision models because it yields bimodal distributions of decision outcomes: In addition to a peak or mode in the true direction of motion, there is a secondary, antipodal, mode at 180◦ to the true direction. Models like the circular diﬀusion model, in which evidence is accumulated by a single process, are thought to be unable to predict bimodality. We compared diﬀusion models for the continuous motion task in which evidence is accumulated in either a 2D or a 3D representational space. We found that performance was well-described by a spherical (3D) diﬀusion model in which the drift rate encodes perceived motion direction and strength and the points on the bounding sphere representing the decision criterion are projectedonto a 2D circle to make a response. A model with an antipodal component of drift rate and drift-rate variability successfully predicted bimodal distributions of decision outcomes and the joint distributions of decision outcomes and RT for individual participants.

show abstract

Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data

Cited by 3 publications

References 39 publications

Diffusion theory of the antipodal “shadow” mode in continuous-outcome, coherent-motion decisions.

Diffusion theory of the antipodal “shadow” mode in continuous-outcome, coherent-motion decisions.

Spherical Sliced-Wasserstein

Diffusion Theory of the Antipodal "Shadow'" Mode in Continuous-Outcome, Coherent-Motion Decisions

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