| 5071. |
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| 5072. |
- Jonasson, Johan, 1966
(författare)
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Invariant random graphs with iid degrees in a general geography
- 2009
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Ingår i: Probability Theory and Related Fields. - : Springer Science and Business Media LLC. - 0178-8051 .- 1432-2064. ; 143:3-4, s. 643-656
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Tidskriftsartikel (refereegranskat)abstract
- Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester (Adv Appl Probab 38:287-298) and Deijfen and Jonasson (Electron Comm Probab 11:336-346), we seek an Aut(G)-invariant random graph model with V as vertex set, iid degrees distributed as D and finite mean connections (i.e., the sum of the edge lengths in the graph metric of G of a given vertex has finite expectation). It is shown that if G has either polynomial growth or rapid growth, then such a random graph model exists if and only if double strok E sign [D,R(D)] 0. All known transitive graphs have either polynomial or rapid growth. It is believed that no other growth rates are possible. When G has rapid growth, the result holds also when the degrees form an arbitrary invariant process. A counter-example shows that this is not the case when G grows polynomially. For this case, we provide other, quite sharp, conditions under which the stronger statement does and does not hold respectively. Our work simplifies and generalizes the results for G,=Z in [4] and proves, e.g., that with G,=Zd, there exists an invariant model with finite mean connections if and only if E D [(d+1) d]
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| 5073. |
- Jonasson, Johan, 1966
(författare)
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Mixing time bounds for overlapping cycles shuffles.
- 2011
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Ingår i: ELECTRONIC JOURNAL OF PROBABILITY. - 1083-6489. ; 16:46, s. 1281-1295
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Tidskriftsartikel (refereegranskat)abstract
- Consider a deck of n cards. Let p(1), p(2), ... , p(n) be a probability vector and consider the mixing time of the card shuffle which at each step of time picks a position according to the p(i)'s and move the card in that position to the top. This setup was introduced in [5], where a few special cases were studied. In particular the case p(n-k) = p(n) = 1/2, k = circle minus(n), turned out to be challenging and only a few lower bounds were produced. These were improved in [1] where it was shown that the relaxation time for the motion of a single card is circle minus(n(2)) when k/n approaches a rational number. In this paper we give the first upper bounds. We focus on the case m := n - k = left perpendicularn/2right perpendicular. It is shown that for the additive symmetrization as well as the lazy version of the shuffle, the mixing time is O (n(3) log n). We then consider two other modifications of the shuffle. The first one is the case p(n-k) = p(n-k+1) = 1/4 and p(n) = 1/2. Using the entropy technique developed by Morris [7], we show that mixing time is O(n(2) log(3) n) for the shuffle itself as well as for the symmetrization. The second modification is a variant of the first, where the moves are made in pairs so that if the first move involves position n, then the second move must be taken from positions m or m + 1 and vice versa. Interestingly, this shuffle is much slower; the mixing time is at least of order n(3) log n and at most of order n(3) log(3) n. It is also observed that results of [1] can be modified to improve lower bounds for some k = o(n).
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| 5074. |
- Jonasson, Johan, 1966
(författare)
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Mixing times for the interchange process
- 2012
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Ingår i: Alea-Latin American Journal of Probability and Mathematical Statistics. - 1980-0436. ; 9:2, s. 667-683
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Tidskriftsartikel (refereegranskat)abstract
- Consider the interchange process on a connected graph G = (V,E) on n vertices .I.e. shuffle a deck of cards by first placing one card at each vertex of G in a fixed order and then at each tick of the clock, picking an edge uniformly at random and switching the two cards at the end vertices of the edge with probability 1/2. Well known special cases are the random transpositions shuffle, where G is the completegraph, and the transposing neighbors shuffle, where G is the n-path. Other cases that have been studied are thed-dimensional grid, the hypercube, lollipop graphs and Erdos-Renyi random graphs above the threshold for connectedness. In this paper the problem is studied for general G. Special attention is focused on trees, random trees and the giant component of critical and supercritical G(N, p) randomgraphs. Upper and lower bounds on the mixing time are given. In many of the cases, we establish the exact order of the mixing time. We also mention the cases when G is the hypercube and when G is a bounded-degree expander, giving upper and lower bounds on the mixing time.
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| 5075. |
- Jonasson, Johan, 1966, et al.
(författare)
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Noise sensitivity and stability of deep neural networks for binary classification
- 2023
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Ingår i: Stochastic Processes and Their Applications. - 0304-4149. ; 165, s. 130-167
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Tidskriftsartikel (refereegranskat)abstract
- A first step is taken towards understanding often observed non-robustness phenomena of deep neural net (DNN) classifiers. This is done from the perspective of Boolean functions by asking if certain sequences of Boolean functions represented by common DNN models are noise sensitive or noise stable, concepts defined in the Boolean function literature. Due to the natural randomness in DNN models, these concepts are extended to annealed and quenched versions. Here we sort out the relation between these definitions and investigate the properties of two standard DNN architectures, the fully connected and convolutional models, when initiated with Gaussian weights. (c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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| 5076. |
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| 5077. |
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| 5078. |
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| 5079. |
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| 5080. |
- Jonasson, Jenny, 1976, et al.
(författare)
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Pixel-based analysis of FRAP data with a general initial bleaching profile
- 2010
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Ingår i: Journal of Microscopy. - : Wiley. - 0022-2720 .- 1365-2818. ; 239:2, s. 142-153
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Tidskriftsartikel (refereegranskat)abstract
- Jonasson et al. (2008), we presented a new pixel-based maximum likelihood framework for the estimation of diffusion coefficients from data on fluorescence recovery after photobleaching (FRAP) with confocal laser scanning microscopy (CLSM). The main method there, called the Gaussian profile method below, is based on the assumption that the initial intensity profile after photobleaching is approximately Gaussian. In the present paper, we introduce a method, called the Monotone profile method, where the maximum likelihood framework is extended to a general initial bleaching profile only assuming that the profile is a non-decreasing function of the distance to the bleaching centre. The statistical distribution of the image noise is further assumed to be Poisson instead of normal, which should be a more realistic description of the noise in the detector. The new Monotone profile method and the Gaussian profile method are applied to FRAP data on swelling of super absorbent polymers (SAP) in water with a Fluorescein probe. The initial bleaching profile is close to a step function at low degrees of swelling and close to a Gaussian profile at high degrees of swelling. The results obtained from the analysis of the FRAP data are corroborated with NMR diffusometry analysis of SAP with a polyethylene glycol probe having size similar to the Fluorescein. The comparison of the Gaussian and Monotone profile methods is also performed by use of simulated data. It is found that the new Monotone profile method is accurate for all types of initial profiles studied, but it suffers from being computationally slow. The fast Gaussian profile method is sufficiently accurate for most of the profiles studied, but underestimates the diffusion coefficient for profiles close to a step function. We also provide a diagnostic plot, which indicates whether the Gaussian profile method is acceptable or not.
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