On the geometry of geodesics in discrete optimal transport

We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest

Ny kollektiv trafikplan i Aalborg

Den kollektive trafik i Aalborg afvikles efter principper, der har været gældende siden 1989, hvor den seneste kollektive trafikplan blev vedtaget af byrådet. Denne plan var i høj grad baseret på, at den kollektive trafik skal konkurrere med andre trafikformer overalt og på alle tidspunkter. Den generelle trafikudvikling i de senere år har medført, at der nu må sættes spørgsmålstegn ved, om det fortsat er rimeligt. Velstandsudviklingen har medført, at folk vil have bil, hvis de har mu- lighed for det. Biltrafikken vokser år for år - og det ser ud til at fortsætte. Mange busafgange af- vikles med ingen eller ganske få passagerer. Derfor foreslås i den nye plan en omlægning af den overordnede strategi, der indebærer omlæg- ning af ressourcer fra de trafiksvage tidspunkter og områder til tidspunkter og dele af byen, hvor der er mange potentielle passagerer. Den kollektive trafik bliver mere markedsorienteret. Der skal naturligvis fortsat være mulighed for transport af dem, der ikke har adgang til bil, men det skal ske efter andre principper. Den direkte anledning til, at en ny kollektiv trafikplan udarbejdes netop nu er, at en ny fælles ter- minal for bybusser og regionale busser tages i brug ved Aalborg Banegård, og at byrådet i den forbindelse har besluttet, at bybussernes fælles omstigningssted - der hvor der er korrespondance mellem bybuslinierne - skal flyttes hertil fra Nytorv /Østerågade i byens centrum. Samtidig åbner den nye nærbane mellem Skørping og Lindholm, hvor det er væsentligt, at der etableres gode bus- forbindelser til nærbanestationerne, så der kan skabes optimale omstigningsmuligheder. Planen kan gennemføres uden udvidelse af de driftsmæssige ressourcer, der er i den kollektive trafiksektor i Aalborg Kommune og hos Nordjyllands Trafikselskab. Den indeholder samtidig en robusthed, der indebærer, at den - inden for visse grænser - kan tilpasse sig ændrede vilkår med hensyn til efterspørgsel efter kollektiv trafik samt ændrede økonomiske vilkår

How to Center Deep Boltzmann Machines

Abstract This work analyzes centered Restricted Boltzmann Machines (RBMs) and centered Deep Boltzmann Machines (DBMs), where centering is done by subtracting offset values from visible and hidden variables. We show analytically that (i) centered and normal Boltzmann Machines (BMs) and thus RBMs and DBMs are different parameterizations of the same model class, such that any normal BM/RBM/DBM can be transformed to an equivalent centered BM/RBM/DBM and vice versa, and that this equivalence generalizes to artificial neural networks in general, (ii) the expected performance of centered binary BMs/RBMs/DBMs is invariant under simultaneous flip of data and offsets, for any offset value in the range of zero to one, (iii) centering can be reformulated as a different update rule for normal BMs/RBMs/DBMs, and (iv) using the enhanced gradient is equivalent to setting the offset values to the average over model and data mean. Furthermore, we present numerical simulations suggesting that (i) optimal generative performance is achieved by subtracting mean values from visible as well as hidden variables, (ii) centered binary RBMs/DBMs reach significantly higher log-likelihood values than normal binary RBMs/DBMs, (iii) centering variants whose offsets depend on the model mean, like the enhanced gradient, suffer from severe divergence problems, (iv) learning is stabilized if an exponentially moving average over the batch means is used for the offset values instead of the current batch mean, which also prevents the enhanced gradient from severe divergence, (v) on a similar level of log-likelihood values centered binary RBMs/DBMs have smaller weights and bigger bias parameters than normal binary RBMs/DBMs, (vi) centering leads to an update direction that is closer to the natural gradient, which is extremely efficient for training as we show for small binary RBMs, (vii) centering eliminates the need for greedy layer-wise pre-training of DBMs, which often even deteriorates the results independently of whether centering is used or not, and (ix) centering is also beneficial for auto encoders

DETC2005-84864 FRACTIONAL MODEL OF A GASTROCNEMIUS MUSCLE FOR TETANUS PATTERN

ABSTRACT This study talks about gastrocnemius muscle identification. During biological activation, every contractile structure is unsynchronized. Likewise, contraction and relaxation phases depend on all contractile elements, the activation type and the state of health. Moreover, gastrocnemius muscle is composed of three fibre types: Fast Fatigable (FF), Fast Resistant (FR) and Slow (S) fibres. Some recent works highlight a fractal structure of the muscle, which consolidate the approach based on the use of a non integer (or fractal) model to characterize its dynamic behavior. A fractional structure model, due to its infinite dimension nature, is particularly adapted to model complex systems with few parameters and to obtain a real time exploitable model. According to its complexity, muscle structure and activation mechanisms, and to these previous considerations, an identification based on fractional model is presented. A model is proposed for the tetanus pattern response in a high tiredness state. It is based on a multi-model structure, which corresponds to the decomposition in contraction and relaxation phases. This multi-model structure is expected to be included subsequently in agonist-antagonist structure