The Statistics of Circular Optimal Transport

Hundrieser, Shayan; Klatt, Marcel; Munk, Axel

doi:10.1007/978-981-19-1044-9_4

Cited by 7 publications

(6 citation statements)

References 43 publications

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“…Pairwise similarity between the discrete density distributions across categories, as well as their similarity to uniform distributions and ones corresponding to idealised trot and bound, was quantified through the circular optimal transport (COT) [42]. In short, given two probability distributions µ and v , COT reflects the minimum cost of transporting the mass from one distribution to the other by taking into account the geometry of the space, as: where S 1 is a circle parametrised by [0, 1) and equipped with geodesic distance .…”

Section: Methodsmentioning

confidence: 99%

“…Brain sections were registered to the Allen Mouse Brain Common Coordinate Framework (CCFv3, [44]) using the ABBA plugin in ImageJ (BioImaging And Optics Platform, Ecole Polytechnique Federale de Lausanne). The positions of virally labelled cells and optic fibre implants were subsequently determined [42] to transform the observed limb phase distributions into the simulated distributions of idealised trot and bound from (D) in speed quintiles, with the costs averaged across limbs and mice. As in (D), data from optogenetically induced head-fixed locomotion at variable incline and head height is shown in light and dark teal respectively, whereas the grey traces represent head-free locomotor data acquired on a motorised treadmill at zero (dark grey) or variable (light grey) incline.…”

Section: Histological Quantificationmentioning

confidence: 99%

“…(C) Simulated phases of non-reference limbs, characteristic of an idealised trot (solid lines), bound (dotted), and transverse gallop (dashed) with the homologous as the leading leg. (D) Circular optimal transport costs [42] to transform the observed limb phase distributions into the simulated distributions of idealised trot and bound from (D) in speed quintiles, with the costs averaged across limbs and mice. As in (D), data from optogenetically induced head-fixed locomotion at variable incline and head height is shown in light and dark teal respectively, whereas the grey traces represent head-free locomotor data acquired on a motorised treadmill at zero (dark grey) or variable (light grey) incline.…”

Section: Supplementary Informationmentioning

confidence: 99%

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Speed-independent modulation of locomotor gait preference by sensory feedback in mice

Mitrevica

Murray

2023

Preprint

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Locomotion is one of the most ubiquitous motor actions in the animal kingdom, essential for behaviours as diverse as foraging, migration, and escape. Successful execution of all these tasks relies on continual adjustment of locomotor gait in line with the behavioural demand for speed as well as the terrain. Failure in this process would disrupt locomotor smoothness, raise its energetic cost, and increase the risk of injury due to skeletal stress [1, 2]. Animals avoid these scenarios, in part, by transitioning from left-right alternating (walk, trot) to synchronous (gallop, bound) gaits as they increase the speed [3, 4]. However, this relationship is not deterministic [5, 6] and its connection to biomechanical factors, like the loading of limbs [7, 8], is unclear. To address this, we developed a head-fixed locomotor paradigm that decouples the speed- and leg loading-related influences on gait by combining optogenetic stimulation of an established speed-control pathway [9, 10] with head height or surface incline modulation. We found a pronounced speed-independent shift in homolateral limb coordination from strict alternation to a gallop-like pattern at upward oriented body postures and upsloping terrains. Both conditions are associated with greater relative loading of the hindlimbs and have a consistent effect on gait preference during head-fixed and head-free locomotion. These results suggest that mice use proprioceptive feedback from the limbs to coordinate their gait across speeds and environments, and implicate ipsilateral control mechanisms in this process. More broadly, our work serves as a principled entry point to a behaviour-driven study of gait circuits.

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Section: Methodsmentioning

confidence: 99%

Section: Histological Quantificationmentioning

confidence: 99%

Section: Supplementary Informationmentioning

confidence: 99%

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Speed-independent modulation of locomotor gait preference by sensory feedback in mice

Mitrevica

Murray

2023

Preprint

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“…One can express the metric W 1 in terms of these cdfs (Hundrieser et al 2022). for various choices of N , illustrating the convergence of spectral measures.…”

Section: Convergence To Continuous Spectramentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

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“…In this case, arguments from functional derivation of the supremum, proves that CLT holds, which suggests that the complexity of the optimal transport problem adapts to the one of the discrete probability. This adaptability of the complexity is fulfilled in a much broader way, see Hundrieser et al (2022b). This culminates in the work of Hundrieser et al (2022a), where, using the above observations, a complete description of the possible bounds of the transport problem with convergence ratio √ n is given.…”

Section: Considerations and Further Workmentioning

confidence: 88%

Statistical analysis of the optimal transport problem

González Sanz

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Optimal transportation is a resource allocation problem present in fields such as economics, finance, physics or artificial intelligence. From a probabilistic point of view, the optimal transport cost endows the space of probability measures with a metric topology. In particular, this topology is equivalent to the weak topology of probability measures together with the convergence of moments. This makes the transport cost an appropriate tool for measuring discrepancies between distributions. On the other hand, the solution of the transport problem is known as optimal plan. That is, an unambiguous way to relate two distributions following an optimality criterion. This optimal plan, when deterministic, is called a transport map.However, in many cases the probability distribution is a theoretical, unattainable entity. It is only visible to the practitioner through its empirical version, i.e. a finite data set of size n. This work examines the asymptotic behaviour of the transport cost in its empirical version.In other words, we study the limits of the empirical cost and plans when the data grows to infinity. It is well-known that the empirical transport cost converges to the population one. Moreover, for continuous measures it does so at a rate that decreases with dimension. In this thesis we prove the consistency of the transport map using topology of set-valued maps. This leads, indirectly, to being able to state that the rate at which the fluctuations-difference between the expected empirical cost and the empirical cost itself-approximate zero is the parametric n − 1 2 , irrespective of the dimension. Moreover, these fluctuations multiplied by n 1 2 RésuméLe transport optimal est un problème d'allocation de ressources que l'on retrouve dans des domaines tels que l'économie, la finance, la physique et l'intelligence artificielle. D'un point de vue probabiliste, le coût de transport optimal dote l'espace des mesures de probabilité d'une topologie métrique. Cela fait du coût de transport un outil approprié pour mesurer les écarts entre les distributions. D'autre part, la solution du problème de transport est connue comme le plan optimal. C'est-à-dire une manière non ambiguë de mettre en relation deux distributions suivant un critère d'optimalité. Ce plan optimal, lorsqu'il est déterministe, s'appelle une application de transport. Cependant, la distribution de probabilité est souvent une entité théorique, irréalisable. Elle n'est visible pour le praticien qu'à travers sa version empirique, c'est-à-dire un ensemble de données fini de taille n. Ce document examine le comportement asymptotique du coût de transport dans sa version empirique. En d'autres termes, nous étudions les limites du coût empirique et de le plan lorsque les données croissent à l'infini. Les travaux précédents ont montré que le coût de transport empirique converge vers le coût théorique. De plus, pour les mesures continues, elle le fait à un taux qui diminue avec la dimension. Dans cette thèse, nous démontrons la cohérence de l'application de transp...

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The Statistics of Circular Optimal Transport

Cited by 7 publications

References 43 publications

Speed-independent modulation of locomotor gait preference by sensory feedback in mice

Speed-independent modulation of locomotor gait preference by sensory feedback in mice

Residual dynamic mode decomposition: robust and verified Koopmanism

Statistical analysis of the optimal transport problem

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